COSMIC NUMBER and GEOMETRY; an introduction.

1. The Square and the Circle.

2. The Vesica Piscis and the hexagon (p.23)

3. The geometric harmonics of the Sun, Earth and Moon. (p.40)

In the study of cosmic number, and (sacred) geometry, there are many symbols of geometric character, which enable the examination of various relationships and concepts…one of the simplest, oldest,yet most important and subtly complicated is that of the Squared Circle; a symbol dating back to the ancient Egyptian and Greek cultures at the least.

At heart to symbolize the ‘marriage’ of the heavens, and the earth; though other reasons for this complex symbol lay within the (often overlooked) relationships which exist between Pi, and Phi, in particular.

Pi, as we all know, is the constant which gives us the circumference of a circle. Phi gives us the ‘Golden Ratio’ between sections of a straight line. The Greeks,and probably much earlier cultures,knew that the circumference of a circle is 22 units to a diameter of 7,which they proved visually by showing 7 circles drawn across the diameter of a circle will enable exactly 22 circles around the circumference; and mathematically by variousprocedures for finding ever more detailed enumerations of PI.

Such as –     3 + 1/7 + 1/9 + 1/11 + 1/13 and so on,to infinity.

 The relationship of Pi to the circle,and spherical geometry, makes it one of the most important values in geometry, possibly even in mathematics. And it’s relationships to many other values found in the study of many fields of mathematics, and physics, are of central importance to the questions raised in cosmic# of how energy and matter inter-relate and evolve; Pi, as well as Phi(1.618,etc), the square roots of 2,3,5 etc; the geometries of the cube, or hexagon; alpha – the fine structure constant – as well as quite a few other significant values such as 123.456789, 864/432,  1.111, 1.272 (√φ), 369; and others.

So, we have the two central values in the circle and the line (the 0 and the 1),Pi and Phi. The 0 and 1 can be connected to many concepts; of ‘female’ and ‘male’; the two components of binary code,(from which came the possibility of the computer age); to the binary symbols of the I-Ching which are ‘yin’ and ‘yang’ (which inspired Wilhelm Leibniz in the 18th century in his creation of binary code); to quantum physics where 1 is the polarity,and 0 is the spin; or in atomic terms, 1 is the central proton, and 0 is the orbiting electron… it is even to be found united in the Greek symbol for Phi–φ – which comprises a 1 passing through a 0 so the 0 bisects the line at one or two points.

Further,from the historic perspective, there is the excellent symbolism at neolithic sites across Europe, indeed the world over, of stone circles and lines, or edifices/menhirs; examples such as at Stonehenge, or Avebury, where central upstanding structures are located within circles of standing stoneswhich together have been shown to be sophisticated astronomic calculators – and often near or in conjunction with stone lines. Such rows and circles of standing stones (or pairs of lines) are to be found throughout Britain, and France; Carnac in north-western France, for example, is one of the most extensive sites in Europe. In south-west England one of the most numerous sites for such antiquities is across the wide area of Dartmoor,in Devon. This c.600square miles area (now a National Park), as well as being home to hundreds of neolithic and historic homesteading, farming and mining sites, also has nearly as many more ‘symbolic’ constructions, includingcairns, standing stones, circles (concentric in some cases), and (double) rows of stones, placed in the ground at various periods of the neolithic era -by who, and for what reason these ‘intellectually significant’ edifices were strung across Dartmoor, and Britain, is completely unknown – and yet in all probability considering the siting of individual megalithic structures and features to celestial orientations, as well as to each other in geometric alignments, it is more than likely the overall network of sites was laid out and constructed by the druids of Britain. These were the religious ‘leaders’ of ancient Britain circa 3600 Bce, who were a nature-worshipping elite, who formed a nationwide brotherhood which was so sophisticated in terms of wisdom attained that they formed a northern European‘college’

which even the Chaldeans sent groups of students to learn at. (These people of the marsh areas of ancient SE Iraq, became the rulers of Babylon in the 7th century Bce, and were famed for/ gave their name to; their stargazing activities in Babylon especially, where the first Zodiac was noted from decades of studies of the night skies. The Chaldeansare mentioned in the (early books of) the Bible- see Genesis 22.22 mentioning the son of Abraham’s brother Nahor, confirming in this way the Sumerian origins of the Abrahamic legacy, a fact already noted in the Bible, and Egypt sections, indeed likely a part of the biblical message, Sumer being the most important, and unique founding civilization of early history); likewise the Greeks, and the Pythagoreans were thought by some to have learnt from the British druids, an amazing possibility. Many sites in Scotland in particular appear to provide some support to this, namely, where from antiquity have survived many ‘geometrically’ shaped stone balls, or polygons, which resemble in all details the five Platonic Solids, believed in ‘sacred geometry’ to be the five basic building-blocks of reality; detailed studies of them show the incredibly complex and sophisticated geometric nature of these symbols. There are over 400 such stones from various Scottish neolithic sites, which are believed to have had several purposes, for matters of learning primarily. The Towie Stone (right) is one of the most accomplished of these strange mathematical and artistically beautiful artefacts, and dates to 3000Bce !

In druidic schools of development, the aspiring neophyte was required to undergo up to twenty years of training – in astronomy, herbs and healing, in self-negation and discipline, etc – before they could attempt to undergo the final initiation into the mysteries of their religion; a ‘rite-of-passage’ bearing some similarities to the Norse myth in the Eddaof Odin (right) hanging on the World-Tree, Yggdrasil for nine days and nights. And a good part of this lengthy preparation was in the arts of astronomy, geometry, and the understanding of sacred number, ie the qualities possessed by the different numbers present within maths and the world. Michell writes; “In their colleges of science, magic and philosophy they taught astronomy, astrology, mathematics and geometry. To those who had mastered these subjects they were able to demonstrate the physical dimensions of the earth and of the heavenly bodies” (p174, The view over Atlantis”).

He also notes that most of what we know of their culture comes from second hand reports by the Greek and Roman historians who wrote of them, even as the druidic era faded into the past. He writes that in the Christian era there are two phenomenon which have their roots in druidic wisdom; namely, the Arthurian grail cycles of the 12th and 13th centuries, as well as the early works of British mythology, such as the Welsh Barddas – and the medieval cathedrals and abbeys, especially of the Gothic style, often built on sites given sanctity from druidic times. Chartres for example, one of the best examples of sacred architecture in the world, was built on the site of the druidic temple of the Carnutians.

(below) Two fine examples of stone circles, most probablyextant on Dartmoor since the time of the building of the pyramids (!);

Yellowmeads circle, Sheep Tor/ Burrator; Attribution Wikimedia, R.Knight/ Scorhill stone circle, near Gidleigh, north-east Dartmoor.

There are several examples on Dartmoor of stone rows (usually double rows) leading up to a stone circle; thus indicating the symbolic ‘marriage’ of the 0 and the 1, or ‘male’ and ‘female’ energies. To speculate, perhaps such examples are sites of particular significance in terms of energies; as we examine in the Ley-lines of England section, the Michael (and Mary) line stretching from Lands End Cornwall, to Great Yarmouth Norfolk, measuring 366.2 miles long, passes through the centre of Dartmoor, its route at the western edge of the area marked by the hill-top St Michael’s church at Bren Tor.

In noting the centrality of 0 and 1, and in support of the importance of this easily-overlooked symbolism, in the Bible Jesus says of himself;

I am the alphaand the  omega, the beginning and the end, the first and the last

(Rev. 22.13), symbolizing thereby his spiritual ‘completeness’ and significance where all dualities disappear…

So these concepts, the 0 and the 1 are a much overlooked thread in the historyof civilizations’ ways of using  metaphors,(numbers) and symbols to express cosmic phenomena… and in the relationship between Pi and Phi the two forms,of circle and line, or heavens and earth, may be studied.

And this central thememay be why of the Egyptians based so much of their design of Giza complex and the Great Pyramid upon the form and geometry of the Squared Circle.

One reason that Pi,like other numerical conceptssuch as Phi and the square roots of 2,3,and 5 are seen to be so ‘significant’ compared to simple fractions such as 3/16, or 8/11 etc, is that they are all quantities which can never be finally and completely numerically defined…there is no ‘end numeral’ which completely defines them; so effectively,they are all continually ‘evolving’ – providing a link(in the eyesof many therefore) to the aspects of reality which are linked to the primordial roots of ‘eternity’…

In this way these square roots, like Pi and Phi, are known as the ‘generative’ values of sacred geometry and cosmic number, connecting them closely to the energies of life which vitalize the material world.

 These fully unknowable numbers are called by mathematicians the irrational numbers,in that they cannot be formed by using twointegers to give a definite ratio, or by a discrete algebraic formula…   

From the times of the builders of Egypt,the irrational roots,and theconstants Pi, and Phi have been incorporated into the geometry of the architecture of ‘sacred’ buildings. As we shall see in this section and that of Egypt and the Great Pyramid, the ‘canon’ of thought which continued on through the Greek and Roman eras was largely already defined by the geometric architectural understanding of the early Egyptian temples and pyramids; buildings such as the Temple of the Sphinx at Giza, the Temple at Thebes, and at Karnak and many others, which were built upon the subtlest of geometric and architectural canonswhile being gigantic and awe-inspiring in their actual constructions.

Sacred architecture based upon the geometries of the square roots found much support in Greece, home of the great mathematical and geometric works by  found expression in such iconic buildings as the Acropolis at Athens, in terms of the ratios of significant aspects of the buildings ; in the belief that the main proportions of a building determine how energy ‘resonates through’, or is ‘channelled’ through that edifice.  ie. the use of (cosmically) harmonic proportions – as in music – creates a positive energetic environment. . . so the adage that (sacred) architecture is ‘music in stone’ comes from this understanding.

Roman works and writings devised by Vitruvius(left, Vitruvian Man by Leonardo da Vinci, 1492, showing the innate geometric harmony of the divine/ human figure) created a ‘school’ of ‘sacred architecture’ which was used in temples and buildings around the world by the Romans – before the decline of the empire consigned the theories contained in his works to virtual oblivion, for much of the first millennium… it was not until the 14th century and the early Renaissance that the works of Vitruvius which had been found guardianship in the Byzantine and Islamic civilizations of the Near East were reintroduced to western civilization. One source of this ‘cultural osmosis’ was from the Spanish regions ruled by Islamic authorities, cities such as Toledo and Cordova, which were for centuries significant centres of learning and understanding. There was another point of contact between west and east however which saw a similar rebirth of sacred architecture in Europe, and that was through the events of the Crusades, some three centuries before the beginnings of the Renaissance.

One of the key developers of the Gothic cathedrals and architectural style was Bernard of Clairvaux, the Benedictine monk who was the founder or leader of the Cistercian order of monks in France; the network of nobility and clergy from the Burgundy region of France had close links toone of the key routes of East-West transmission of ‘cosmic wisdom’; the Knights of the Temple of Solomon – the Templars.

SIDE-BAR; Founded in 1118 AD the Poor Soldiers of Christ and the Temple of Solomon – the  Knights Templar – existed for 11 years as a small tightly knit group of 9 founder members, knights and leaders of religious orders such as Bernard of Clairvaux, in their home region of Champagnein France; it was only in 1130 they were given official status of an order,and their full nameby Rome in a papal decree.

  Part of the curious natureof the Templar story is the fact that they werepresent in the centre of Jerusalem, among the rulers and palaces and temples,even during this pre-official period of 1119-1131, as the small and compactunit of nine ‘founder-members’…

  Whatsoever, when domiciled in Jerusalem they were given space within the headquarters of the city’s palace connected to the al-Aqsa mosque,directly upon the Temple Mount (see above left for the Temple Mount shown on the Templars insignia) at the centre of Jerusalem…. whether they displaced the Islamic authorities there, or co-habited the area containing several buildings is not clear.

It is from this circumstance that their name comes – for the al-Aqsa mosqueon the Temple Mount was rumoured to be built upon the foundations of theoriginal Temple of Solomon, central to the Hebrew faith from biblical times… in fact it was revered in Hebrew texts as the place where the Earth first arose out of the waters, where Adam and Eve were created, where the floodwaters of the Deluge subsided, and was the ‘centre of the world’ therefore from time immemorial. In keeping with the sanctity of the place, Mohammed the Prophet of Islam is believed to have been at this place at the start of ‘The Night Journey’ (as detailed in the Holy Qur’an) when an angel lifted him up through the heavens to see God in his heavens…

  So again we find a temple built upon a site accorded the greatest significancefrom the earliest of times; which over the course of millennia was re-built, or built over,or assigned to a new religion’s stewardship, while always retaining it’s (energetic) purpose, as a ‘navel-point’ of the land, a place connecting heaven and earth.

  And the Knights Templars put forward the narrative from around 1135-40Ad, that they had excavated beneath the mosque’s foundations,whilst purposely looking forevidence of Solomon’s Temple. And furthermore,that they had uncovered 5 trunks worth of documents and so forth, from which they claimedtheir authorityas guardians of such ‘ancient secrets’.

   Many theories exist as to what – if anything – the Templars foundon orbeneath the Temple Mount…but whatcannot be doubted is the effect that their ‘spiritual overseer, ‘or ‘patron’, St Bernard of Clairvaux – had on westernarchitecture over the following three centuries or so. For Bernard, and the Cistercian Order played a significant role in the development of the ‘gothic’ style of architecture, from which some of the most sublime, and highly regarded cathedrals in Europe ever built came to fruition… (left; the complex geometries within the ceiling of Peterborough Cathedral, Cambridgeshire, England, built 1118-1237Ad).

A clear example of the transmission of ‘cosmic wisdom’ contained within sacred geometry from this example is the incorporation into Gothic churches and cathedrals of the pointed arch, used in windows and over entranceways etcetera; for this geometrically derived development came directly from Islam, indeed is called the ‘Islamic arch’. What the inner meaning of the arch achieves is to route the energies formed by the pressures of the building and the earthupwards into the atmosphere. In contrast the Roman arch, which is a rounded semi-circle routes the rising energies around and back down to the earth…

   And at the heart of many of these churches and cathedrals, was the siting of theedifice upon a navel point – and from the place on the floor directly over this pointall following aspects of the building were derived, in geometrical proportions of thesacred ratios of the irrational square roots, of √2,√3,and √5 – in keeping with the ancient canon of virtually all the major civilization from Sumer onwards, particularly the those of Egypt… wisdom which passed to the Greeks and Romans by and large, and was preserved by the Byzantine Church in the East, as well as by Islam when it arose in the 7th century.

So speculation existstherefore that the source of the Templar’s knowledge was not fiveboxes or containers of ‘ancient documents’ from the foundations of the original Temple – but a (secret) brotherhood, or order, of the nation of Islam present in Jerusalem at this time…

 So, like many different examples covered in this book, wisdom was actively transmitted across the boundaries between cultures, be it consciously, orby cultural osmosis; as Gurdjieff characterized/represented the historic presence or absenceof cosmic wisdom at different points in history to a river flowing above ground for a period, then below ground for a while,andso on.

 The point is noted by Clive Prince and Lynn Picknett that for the first eleven years of the Templars, there were only nine ordermembers, indicates this was a symbolic feature of some sort; particularly considering the importance of the Enneagram in Eastern and Islamic philosophy, and likewise in features of the architectural plan of edifices such as Chartres – so from (even before) their incorporation by the Papacy in 1129Ad, the order seems to possibly have had a close relationship to cosmic #, and questions of sacred geometry and architecture.

   A study of the esoteric/ mathematic/ architectural sources of the pre-Jerusalemfounding members of the Templars, would be certainly be of great value. This propensity for ‘philosophical number’ in it’s many guises may have come from the associated brotherhoods of Hugue de Payens,a French nobleman with connections to the order ofCistercian monks,led by Bernard of Clairvaux, instrumental in developing and spreading the ‘gothic’ style of architectureas mentioned.

There is very little information on the  Muslim source of the canon involved,apparently the continuance of a very long line of ‘conscious mathematics and architecture ‘(re-)introduced into the West’ at that time in the environs of Jerusalem and therearound.

  Conjecture may  tend towards it being some form of early Sufi brotherhood, although as such the sufic orders did not begin until the 13th or 14th centuries. Nevertheless, the inner circles of the Islamic faith, as found in Jerusalem in 1118 Ad, were skilled in geometry, architecture, and geodetic study of the surface of the earth. This subtletymay be seen in the design and placing of the Dome on the Rock (with a cubic/ octagonal base and circular dome representing the Earth and the heavens); that of the cubic geometric design of the Kaaba in Mecca, Saudi Arabia;the predominance of (octagonal) geometric artof the highestmathematical standards; the pre-eminence in the 10-12th centuries of Islamic science, such as mathematics, chemistry, architecture; and so on.

Mecca, the central holy site of Islam is believed by some to be sited at the ‘golden point’ on the earth’s surface between North and South Poles, ie the phi (φ) point.

This is worked out as follows; between the two poles lies 180°; and 180/ 1.618033 (φ) = 111.246. Subtracting the 90 degrees from North Pole to equator leaves 21.246°for the phi point. Mecca is placed at 21.422°S, ie is approximately 0.176 of a degree from the phi point, a variance of less than 1/10 of a per cent, or 10 miles or so… although the southern city limit of Mecca is at 21.278°S, making it just a small distance of 0.03 of a degree.

Strikingly – and in confirmation of this website’s theory that various cosmic# values of significance are encoded within chapter and verse numbers of sacred texts such as the Bible etcetera – the coordinates of the latitude of the Kaaba within Mecca in the minutes and seconds form are 21°25”21’ – this is noteworthy because the first reference to the Kaaba in the Holy Quran is at Sura 2.125.

There are some incredible examples of encoding within the numbers of chapters and verses within the Bible, as the section on it explores.

  The Egyptian civilization, from it’s earliest expressions, circa 3000Bcethroughout many centuries,and ruling dynasties, placed the design of it’scentral spaces, religious and political centres, on this mathematical,cosmic and harmonic basis, using related irrational roots, plus of course, Pi and Phi.

  To take an example of how important throughout time these 3 or moresquare roots have been – and are – to the studies of templearchitecture and number,it may be hypothesized that the Egyptian geometers portrayed thesignificance of these cosmic # roots in artworks in the following way;

The first diagram comes from ancient Iran, being an Iranian winged sun-disc called the Farhavar, symbol of the Zoroastrian religion (c.600Bce), stemming from identical Assyrian and Sumerian symbols of 800-1500 Bce.Confirming this the figure holds a ‘rod-and-ring’ symbol within his hand, extant from Sumerian artworks of the 2nd millennium Bce, and Egyptian ‘Shen ring’ depictionsalso. (In fact the image disappeared from Iran some time after the end of the Achaemenid Empire founded by Cyrus the Great which lasted from c.550-330Bce, and was only rediscovered in the 20thcentury by dint of the researches of the Parsi scholar J.M.Unvala, in 1925!)The picture below is of an Egyptian version of the same winged sun disc, and to the right, an Egyptian stela of Akhenaten and Nefertiti, created circa 1330 Bce, with the rays of the sun tipped with serpents heads; all such examples pointing to the nature of the sun’s rays as being connected to intelligence, as well as to the energetic serpent lines through the Earth’s surface through which these cosmic forces run, vitalizing the world as they do so.And lastly, the geometric diagram of the derivation of the square roots of 2, 3 and 5, the ‘generative’ roots linked to Pi and Phi, as can be seen by the 50 (or 0.5) units (ie half-square) diagonal extending to 111.81 units horizontally; by adding 50 units Phi of 161.81 is obtained. Thus √5 + 1 / 2 = 1.618; ie 2.236 + 1/ 2 = 3.236/ 2 (or 1.118 + 0.5) = Phi. So, A = 111.81, B = 141.41 (√2), and C= 223.6 (√5)

Birds nearly always signified the skies, the heavens, and divine intelligence, so that hieroglyphs of divine beings often saw them represented as bird/ human hybrids. This is a natural association of higher matters with the skies within the division of material reality into air/ earth / underworld / waters… an allegory first used by the Sumerian deities the Anunnaki, when assigning the three most important leaders –  Enlil, Anu and Enki – to the three bands of the world, North, Equator and South, with associated metaphors of sky; earth; and waters. And as rulers they represented in human terms the mind, heart and body/instinct accordingly.

 And it is important to remember that geometrywas at the heart of much that the Egyptian civilization did,in terms of building design,sitedesign and location ,their artworks,and even city location – for example the ceremonial and political centre Heliopolis was located across the Nile from Giza -about 17 miles- to create a mirror of the heavenly plan whereby the (significant) stars of the heavens are located either side of the great ‘river’ of the Milky Way…

The history of different cultures derivation of increasingly accurateestimates of Pi (π) is quite well documented, including the Babylonian,theGreek(Pythagoras school) at Cronos, c.530Bce, the Indian mathematicians efforts; and fromthe complex constructions of Neolithic circles and monuments, such as Stonehenge,Avebury,the Orkneysand also the mentioned ‘teaching tools’ such as the Scottish Platonicsolids carvings, there are certainly arguments that some (working) knowledge of Pi and its applications existed across Europe and the Middle East well beforePythagoras’ school outlined it’s features.

 So, from the period of the Sumerian civilization onwards, considering the myths (histories passed down) from then of the Anunnaki and their creation of the aspects of society, including the earliest forms of writing, literature, wisdom and learning established by them, it is fair to say that many of thepathways of civilization’s evolution laid down then were rooted in a mathematic framework; hence the Anunnaki each had numbers between 1 and 60 with which they were represented. The significance of number to these deities, said to have created mankind from mixing their celestial genetics with lower ‘earthly’ genes, as well as then giving mankind all the blessings of civilization. There is good support for this in the establishment in Sumer of thesexagesimal system of  measurement/division (base-60 system); something which since it’s inception has been the most perfect mathematical system for representing and measuring time, space and degrees of the circle.As such, the 360° of the circlehas been used toplace every location on the earth, and outwards, in thevast reaches of space for virtually the entire history of mankind.

The sexagesimal system relates space numerically with time; a circle of 360 degrees, with each degree having 60 minutes,and each minute 60 seconds; and of time, as a day of 24 hours has60 minutes in each hour, and 60 seconds in each minute… placing the measurement of the circles of the planets as they circle the sun which circles the galaxy centre, within an equable framework of time, is an incredibly sophisticated mental step forwardfor early civilization- as well as providing  a trigonometric system for measuring angles of both conceptual studies as well as of the planets and stars.Sohow the firstreal system of mathematical reckoning in human civilization discovered the optimal way to link the dimensions of earth and the heavens, and time and space and geometry–(at a stage of history priorto writing itself, as far as we know)* – is nearlyunfathomable. And a very good indication that some sort of higher influence was giving this wisdom to mankind in it’s formative stages, as all the main myths and religions of the time stated. . . ie, the Sumerian /ABA cultures, Egyptian, Indian and Chinese, in addition to many other less widely influential cultures with similar celestial mythologies.

To return to the orbits of the planets circling the Sun, their ‘year’ is defined as the period of the time taken for any planet to reach the same position in relation to the Sun and the stars in a circle; thus a complete revolution is thus taken as being that planet’s ‘year’; and the number of 360° circles it turns on it’s axis in that period constitute the number of days of the year. So a year on Mercury is 58 earth days; but from one sunrise to the next is 176 days! Making the Mercury day longer than the Mercury year! Jupiter takes 29.5 ‘earth’ years to complete one circle around the Sun, and one Saturn day lasts just 10 ½ hours, as it rotates at 22,000 mph…

So, both the length of the day, and of the month, and the year, are derived from measuring how much time passes during the period of the turning of one complete circle around the axis of (elliptical) orbit…

The sexagesimal system (‘6-based system)’is called this because though in Base 10, it is centred upon multiples of 6, and 10 of the decimal system, thus  6 x 10 = 60 x 6 = 360 x 10 = 3600, and so on.

  This system provides us with 60 seconds in a minute, 60 minutes in an hour, 2 x 12 hours in a day (and night), and 365 days within a year, etc. (The early designers of mankind’s astronomy and maths could conceivably have created a circle of 366° to reflect the ‘reality’ of the sidereal year; but a circle of 360° is far more effective in it’s purposes due to the great number of numbers which sub-divide it;

The number 360 is perhaps the optimal number to divide the circle into; for its connections to the measurement of time, (60 x 60 x 24) as well as because of the quantity of whole numbers with which it can be divided, (let alone significant fractions);


Some of the first records mankind has of sexagesimal mathematics are on clay tablets found during archeological digs in what was Babylon. These used the process of notation whereby a whole number was then added to with a number divided by 60; then the next number was divided by 60², then 60³ and so on.

So for example the square root of two – 1.4141 as we have seen, the diagonal of a square of unit 1 width which at the corner equals the length of the radius of an equal circle – has been found to have been worked out in Babylon using this system to an accuracy of 2 parts in a million.

The notation is thus; 1; 24; 51; 10 = 1 + 24/60 + 51/60² + 10/60³ = 30547/21600 = 1.41421296. This system of first fractions ‘minuta’, then seconds, thirds, fourths and so on is how the ‘minute’ and the ‘second’ got their name! This clay tablet called has been dated to around 1800Bce!  Greek mathematicians of the first millennia Bce such as Ptolemy in the 2nd century Ad used the same system while using Greek alphabetic numerals and geometry, and only for the fractional part of the number – but was able to create in this way a table of chords, which was a trigonometric table, the only example of such for over a millennium.

 Plato in his ‘Republic’ used a value of 60⁴ – 12,960,000 – to illustrate an allegory of a marriage, denoting the value easily therefore as 1,0,0,0,0. The number 1296 crops up repeatedly in cosmic number related subjects, as does its fractions 648, 324, 162 and 81. An example is the fact that 360 / 32.4 = 11.111 ; something of significance in analysing world geometries and relationships between sites, as we see in the Ley-Lines of England section.

Medieval astronomers, led by the Arabic mathematicians also refined or standardised the use of sexagesimal fractions to denote minutes and seconds and thirds(!) of time, around 1000Ad. Developments in this notation enabled works such as the Alphonsine Tables (c.1320Ad) in Paris to create a standard unit for time, using base-60 values. This provided data for calculating the positions of the main planets in the solar system, especially Sun, Moon and Earth relative to the fixed stars.

Another usage of the method was by Ptolemy for describing the value of Pi as 

3 ; 8 ; 30, meaning 3 + 8/60 + 30/60² = 377/120 = 3.1416 66.

A Persian mathematician in the 15th century, Jamshid Al-Kashi calculated 2π all the way to 9 sub-digits ie to 1/60⁹…

Another circle (or spiral) ‘of the heavens’ the Earth exists within is that of the solar system within the galaxy, the Milky Way. This is around 100,000 light years across, (ie 600,000 trillion miles), and we are about 26,000 light years from the centre of the galaxy. The galaxy is disc-shaped in general,like the solar system,and has four or so spiral arms emanating from out of the centre. We are situated on the Orion spiral arm, which contains most of the stars of the Milky Waywe can see unaided in the night-sky.

If it takes the earth 365 days or so to complete one year to circuit the Sun (to return the same place in relation to the backdrop of the stars), it takes the solar system around 225 million years to complete one

turn around the galaxy. This is what is called a ‘Great Year’ (of the Galaxy – not to be confused with the Great Year of the Sun; this is the period of the Sun’s Precession of the Equinoxes, estimated to be around 25,920 years in total for the Sun to circle the Earth against it’s back-drop of the stars). Interestingly, the period of the Sun’s Great Year is equal to the period of time it takes for light to travel from the galactic centre, to our position on the Orion arm, ie 26,000 light-years away.

Further, 225,000,000/ 25,920 = 8680.5555

close again to the 864-866 valuesconsidered to be connected to many of the Sun’s manifestations; indeed,with a difference of 868 to 866 of 1.002:1 – a surprisingly neat ratio, especially considering the galactic ‘year’ length is an estimate. In fact 25,920 x 8660 = 224,467,200 . . . well within the range of what the Galactic Year may measure. And if this estimate is divided by 224,466,880, the result is 1.00000 142. Connecting the size of the Sun, the period it ‘turns’ around the Earth’s axial wobble for 26,000 years, the solar system’s distance from the centre of the Milky Way, and cos# related to Pi and phi etc,in a way that is indicative such values result from inner harmonics of energy rather than by random chance(!) As we see shortly, the square roots of 2 (1.414), phi (1.272), and 1.23456789 (1.1111) all relate intrinsically to the Squared Circle. Of course it might be said that gravity may have naturally created such relationships between the galaxy and solar system, still according to mathematical principles, but without inner meaning…

TheMilky Way in sacred architecture has often been associated with the Vesica Piscis, which resembles in it’s inner area the disc of the Milky Way, as viewed from Earth in the night-sky, as we see in the Vesica section.This is due to the axial nature of both. Each have been compared to the female yoni, as symbolic of creative fertility, of birth within the matrix of reality.For example, at Chartres cathedral we have the statue of Jesus at the main steps, where he sits within the Vesica Piscis. And likewise in the depiction of Jesus and his disciples at the church of St-Sernin (Saturnin) in Toulouse, France, (another UNESCO World Heritage Site), dating from around 1125 Ad (below right).

Notice too(left) the 4 rays within a circle behind his head at Chartres; this is a symbol which is representative of the joining of the square and the circle, the heavens and the earth, as we shall examine shortly, akin to the four rivers of the Garden of Eden flowing to each quarter of the compass.

Interestingly as well, some New Age as well as Indian or Eastern thought maintainsor speculates, that as we are circuiting the galaxy and coming into contact with other systems and regions of the galaxy not experienced‘before’, these regions may be causing  gravitational and higher-dimensional interactions within the solar system whichlinkto the quickening of events the Earth is currently going through. This may sound ridiculous – but it isnevertheless possible. No two circumstances are ever identical…

The distance from the galactic centre we occupy on the Orion Arm spiralsituates us in a stably-oriented orbit around the galaxy’s centre; it has been noted before thatif we were much nearer to the denselypacked centre, or on an orbit which took us nearer to the other arms of the galaxy, this would potentially have destabilized the millions of comets and objects in the Oort Cloud to then bombard the solar system; indeed this scenario is conjectured to be a possible cause of some of the periodic mass-extinctions on our planet, such as the dinosaurs 65 million years ago.

So from the Sumerian civilization onwards,mankind has fortuitously used in it’s various forms and types the most efficient,sophisticatedand harmonious system of countingand understanding time and space.  Which is linked to the widespreadlegends and myths of divine transmission of (‘cosmic’) understanding, present in all the major civilizations; Sumerian (and related), Egyptian, Indian and Chinese, as well as in early civilizations such as Native American, South American, Tibetan, European druid-based societies, and so on.

The tidiness of the sexagesimal system in part lies in the link it allows between the base-10 mathematics of measuring distances on earth and the base 12-friendly mathematics of the heavens – be it the division of periods of time; 12 hours in a day and a night, 12 months in a year – 12 houses in the zodiacthrough which the Sun travels over the course of the Great Year of Precession(which isc.25,920 years each cycleor 2160 years per sign of the zodiac…the newest era of which – Aquarius – the world is entering within the first century or so of thismillennium).

 So the relationship between 12 and 10 starts to become central to our study; andthe ratio of 10:12, (or 6/5, or 1.2) becomes one worth looking forin all cultures’ artworks,religious imageries, textsand so on.

 .                  .                 .

 Before examining the squared circle though, it is worth quickly looking at how Phi may be derived using a pen and compass drawing just two circles and a square around them;

In this diagram the square is 200 units across, ie the diameter of the outer circle.

BE = 100 units, and BF = 150.

AE = 111.8 units as one half of the diagonal √5, (as 1² + 2² = hypotenuse²) = √5 /2.

EF = 50 units, therefore AF = 111.8 + 50 = 161.8 = Phi.

Additionally, if AB = 161.8-150 = 11.8, therefore AM = 11.8, or 50 – 38.19.

If AS = 150, then MS = 138.19, ie 200 – 61.8, or 38.19 + 100.

So, to create a squared circle, the first step is to create the optimal ‘key’; in other words, by drawing the circle to a size of 1, or for ease of study, 100 units.

Thus the circle has a radius, from centre to edge, of 50 units, and a diameter of 100 units.

Now to find a square which is equal in perimeter to the circle, we must divide 314.159 by 4, as the circumference of a circle is 2πr, (ie 2 x 3.14159 x 50; or 314.159 in this instance). So a square of equal perimeter = 314.159/ 4, which = 78.54 units.

Now a square equal to the circle’s circumference can be added, of radius 39.27, and diameter of 78.54 units.

And the eight cross-over points can be joined through the circle’s centre-point, to form a ‘four-rayed’ star as such, or

a basic four armed ‘Maltese’ cross. The measurements of this diagram of the squared circle are as follows;

If we take the triangle which reaches from centre to the middle of the side of the square, (ie the radius of the square) along the central line, thislinePL = 39.27. Line LM(from square side midpoint to where it intersects the circle) = ?, and (the hypotenuse) MP = 50 units. By Pythagoras’ law we know that PL²+ LM² = MP².

Thus 39.27² + LM² = 50², or

1542.054 + LM² = 2500; thus LM = 30.950

With the same measures shown beneath the diagram for points A – D, from the intersection to the square’s corner AC = 19.05, as from square side to circle edge AB = 10.73, from square corner to height of circle BC = 8.32.So as the diagram shows, the measurements are;

AB = 10.73

BC = 8.32

AC = 19.05

CD = 30.95

AD = 50

BD = 39.27

If we double these for the diameter measures we get some idea of how close to significant cos# values nearly every measure is; (2x) AB = 21.46 (~21.6), BC = 16.64,(=1/6) AC = 38.10 (~38.19), CD = 61.90 (~61.8), AD = 100,and BD = 78.54, ie Pi/ 4.

The three angles of this right-angled triangle are; a = 38.24°, b = 51.76°, and c = 90°. Thus a is very close to phi of 38.19, and b to phi of 61.803 – 10 = 51.803 degrees, at a difference of 0.05, and 0.04 degrees each. Indeed for the lengths to give angles of 38.19 and 51.803 degrees, CD and BD would measure 30.916 and 39.295 respectively. (then AB = 10.705, and AC = 19.025). So this marginal difference at many points to the values of Phi is a central feature of the squared circle.

Additionally 39.297 x 2 x Pi = 246.91, ie. twice 123.456…

So what else do the values of the squared circle tell us?

Well intriguingly, virtually every value has cos# related values, again in close proximity; so 10.73;10.8 or 108, 21.46; 21.6/216, 38.10/38.19 (phi), 61.9/61.8033 (phi), and so on. And BC of 16.64 is very close to the hexagon value of 1/6 of 0.16666, or 100/6 = 16.666, a mere 0.02 difference… additionally, AB + CD = 41.68. One quarter of 16.666 = 4.1666.

            So the values of 8.32, 16.64, and 41.68 are all close to perfect fractions of the hexagon, for some reason hard to discern… thus incidentally the entire 100 unit length – 2 x BC of 8.32 = 100 – 16.64 = 83.3 6, ie 5 x 16.68. This makes BC exactly 1/6 of the 50 units length, and 1/12 of the 100 units length. Or BC of 8.32 = 1/5 of the 41.68 remainder of the 50 units length;so 50/ 41.68 = 1.2 (or 6/5), ie BC is 1/6 of 50, while AB + CD = 41.68 = 5/6 of 50.

Also, while 100/ 41.66 = 2.4, and 360/41.66 = 8.64, the quarter circle of 90 degrees / 41.666 = 2.16 003456, indicating it to be in relation to thecos# harmonics shown.

So every measurement derived from the squared circle is so close to values related to Phi, or to fractions of the hexagon, or cosmic # staples such as 108, 216 and so on as to be identical; and of course, the measures all stem from the original Pi/4 value of 78.54 diameter/ 39.27 radius. Thus Pi and Phi are intimately linked in this diagram, as the angles of each right angled triangle show, likewise.

 And this is by no means the only point of interest within the symbol. Considering the centrality of the Great Pyramid to cosmic#, and the ability to derive it’s height and length/ angle of slope from both Pi and Phi, it appears likely the designers of the pyramid had both values in mind when it’s measures were decided; and very possibly those of the Squared Circle too.

As we have seen, the solar numbers of the sun, 864 and 432 are at the heart of cosmic#. So too the first five prime numbers, of 2, 3, 5, 7 and 11. And within the Great Pyramid the ratios of 7:11 are believed to have been intentionally encoded into it’s internal proportions. Why might this be? One possibly relevant idea is that they are both capable of increasing or decreasing the base values of 22/7 of Pi. Moreover 14/11 = 1.272, 11/14 = 0.78 57, 7/11 = 1.272/2, and 11/7 = (22/7) /2, obviously…

And the perimeter and height of the Great Pyramid reflect these ratios; each side = 110 cubits, and the perimeter = 440 cubits;while the height equals 280 cubits, so 440/ 280 = 11/7.

Some researchers have also noted the ability to measure the area of the three main and six minor pyramids at Giza within a squared circle; in particular the two sets of minor pyramids are linked by this, as well as the Sphinx, theoretically.

So in the squared circle, if Pi /4 = square length, this means 22/7 / 4 = 0.7854; or alternatively, 314.159/ 4 = 78.54.

If we take 22/7 / 4 and multiply it by the value of 22, this is equal to 7/4,

So 7/4 = 1.728.

If we take theharmonic of 11 (or 1.1), the following figures are obtained;

78.54 x 1.1 = 86.4 (Sun/ seconds in a day)

                                      x 2.2 = 172.8

                                      x 3.3 = 259.20 (Precession)

                                      x 4.4 = 345.6

                                      x 5.5 = 432

                                      x 6.6 = 518.4 (angle of Great Pyramid = 51.84°; angle squared circle 51.8°; 72²).

                                      x 7    = 550

                                      x 8.8 = 691.1 (1°Ea circumference = 69.11 miles)

                                      x 9.9 = 777.6

These are all cosmic # values related to the measures of the Earth, the Moon, and the Sun, as well as many other instances…

And there is more curious harmony between the values obtained by dividing 7 / 4 (ie 22/7 / 4 x 7) = 1.728, and then dividing this by multiples of 1 or 1.1, (or 11/10);

                  172.8 / 1.1 = 157.09 ie Pi /2

                            / 2    = 86.4

                            / 2.2 = 78.54

/ 3    = 57.6 (see Vesica units of hexagon, 14.4 x 4)

                            / 3.3 = 52.36 (= Egyptian Royal Cubit in cm, plus Pi/6)

                            / 4    = 43.2 (ie sun’s radius etc)

                            / 4.4 = 39.27 (squared circle square radius where circle = 100 diameter)

                            / 5    = 34.56 ( Pi x 11, ½ of one degree of earth equatorial circumference)

                            / 5.5  = 31.4159 (ie Pi x 10)

                            / 6     = 28.8 (Vesica 2 x 14.4; linked in Bible to Temple; and to universe’s frequency)

                            /6.6   = 26.181 (ie 2.6181 = Phi squared – φ²)

                            / 7     = 24.68 (ie 2 x 12.34 etc)

                            / 7.7  = 22.441556 (ie 2 x 11.22 0 77)

                            /  8    = 21.6 (see Moon diameter 2160 miles)

And so on; 8,8 = half of 39.27 as the harmonics reduce in size as new inner squares and circles are formed.

More permutations involving 7 and 11 reveal other relationships within the symbols main values and proportions;

        Pi/4 ie. 78.54 x 7 / 3.3 = 166.6 (ie the 1/6 ratios linked to the hexagon in the Vesica and squared circle)

                                  / 5.5  = 100 (showing the 5.5/11/22 ratios derivable from Pi) 

                                  / 6.6  = 83.3 (as noted, 5/6 of 100, or ½ of 1/6th – ie. 1/12th of 1).

                                  /7.77 = 70.7 (ie half √2, and the diagonal of the square measuring 50 units in diameter)

                                  / 9.9  = 55.5 (which appears in so many examples, as 1/3 of 1/6 – or 1/18th  – of 1 etc).

Meanwhile 78.54 (ie 314.59 / 4)  x 11 = 864   

                                                              “  / 3.3 = 261.8

                                                                 / 7.77 = 111.18 (and by 7.77556 = 111.1111)

                                                                 / 2 = 432, /4 = 216, /8 = 108

                                                                 /7 = 123.456789

To show some of the quite amazing relationships existing between all these central factors in the Squared Circle, the Vesica, and as both appear when applied to the proportions of the Great Pyramid, which seems to have been built to some inner canon related to these values…

One reason for this – and the circumstance in the Great Pyramid’s inner proportions – is that 7 / 11 = 0.636, which is very close to being half the square root of phi (being 127.27 /2), while 11 / 7 = 1.57 4 which ~π/2, ie, 22/7 /2.

Meanwhile 11 divided by the square root of phi (√1.61803), namely 1.272 03 gives the result of 8.64 7, midway between 864 of the solar harmonic, and 86.6% of the Vesica geometries. Additionally, 11/ 1.61803 ie phi itself, = 6.79.

If the corner of the square is examined, each RA triangle at the corner has an angle of 51.76°; thus being 6.76 degrees more than the 45°of the corner line’s angle.  Thus the difference between the two values is exactly 0.03 degrees; if 1 / 0.03, this gives the result of being 1/ 33.333. Also, 0.3819 x 3.33333 = 1.272 9, a close value to √φ, and also 4/π of 1.273; as we shall see, the ratio of an inner: equal circle within the squared circle.

So amongst other things the designers of the Great Pyramid certainly may be seen to have chosen the most appropriate inner proportions possible in balancing 7 and 11.

  Graham Hancock prints this diagram showing the 7 and 11 proportions in the design of the GP.


  In fact,           14 / 11 x 2.468  =   3.1416

                      or π / 1.23456789 = 1.272 345 036   (x 11 = 14)

If a square is drawn of 11 units diameter (5.5 radius) the CIRCLE with EQUAL circumference has a diam of 14 units (radius of 7 units) and circumference 44 units. If 2 more circles are drawn, inside and outside the square, they are diam.11/radius 5.5/circumference 34.56,and diameter 14 (radius 7.77) circumference 48.85.

So what deductions can be drawn from this size of square and circle?

 – firstly, a pyramid drawn of base 11, and height 7 units, is an exact representation of the Great Pyramid.The actual measurements are 756 feet base, and 481.3 feet height. This gives a width:height ratio of 1.57074,or 3.1415/2, a very close approximation of π/2.

 It is stated that in Egyptian Royal Cubits, the base = 440 RC, and the height = 280 RC. This gives a proportion of 3.14 28/2 (ie. slightly different ratio to the measurements in feet…). This value in RC is very close to 3.14 2696 7806 (which when squared = 9.87654321; a number we have seen is related in various ways to many of the other values that repeatedly occur in cos#…)

 There is equally interesting data contained within the 5.5/7 Squared Circle diagram; if a triangle is drawn by extending a line from the centre along the horizontal axis the length of 5.5, and 90º up to the 7 units circle line, a RA triangle is formed of the ratios; 

 if the Hypotenuse (c) = 7, and base line (a) = 5.5,

then 7² = 5.5² + b² = 49 = 30.25 + b²

                              =>b² = 18.75

                              =>b  = 4.33  (!)

The solar radius of 432,000 miles, the Vesica value of 43.3 and 86.6% on the vertical axis, and so on. So the triangle on each side of the square towards the centre point therefore = 8.66 units, with each side’s angle equalling 51.76°, ie effectively the angle of the Great Pyramid. While the angle at the apex between each side and the central axis = 38.2 units, ie close to matching exactly the 38.19/ 61.81 phi values. So this proportion on a squared circle of 5.5 and 7 radius, (and 11 and 14 diameter) opens up some fascinating data. Likewise that the circle inside square of radius 5.5/diameter 11 has a circumference of 34.56; and the square equal to the circle of 5.5 radius has a radius of 4.32/ diameter of 8.64/ perimeter of 34.56…!


The Squared Circle, combining the circular geometries of the Heavens, and thefour-squareness of the material world has long been foundin some of the most important edifices of all culturesand civilizations;

 For example, the architecture of the already mentioned Dome on the Rock, one of the religion ofIslam’s pre-eminent sites of worship, in Jerusalem, and one of the most important religious sites in the world, for

Hebrew, Christian and Islamic religions, has been argued to be basedupon this ‘inner meaning’ of uniting the Heavens and Earth.

(left) the Dome on the Rock, since antiquity the site has been said to be the ‘navel of the world’. The geometric sophistication of the temple’s design allows the energies of the higher dimensions to permeate the site, and on into the world, changing human consciousness as it does so..

The dome of St Paul’s cathedral is likewise designed exactly for that reasonof uniting the spherical geometries of the heavens with those of the material world. The dome ‘of the heavens’ is joined to the ‘square’ of the body of the church by an octagonal structure within the walls of the building; considering the ethos of esoteric and mathematical philosophy present within England during the period Sir Christopher Wren designed the church (c.1690) it would seem likely this key 8-sided structure was chosen for both conceptual and practical reasons. Another example from the building in this regard is that it is exactly 365 feet high, or 111metres.

  A study of symbols throughout history shows how widespread this combining ofthe square and the circle has been from the earliest times; so for example,in representations of the year as a circle, the four seasons is a natural wayto sub-divide the circle; this has always been the easiest and clearest way to represent theseasons – as an example, a photograph of a stone stelae from the British Museum (below); while the Assyrian stela is stated to be a military campaign record, the significant thing is that the four seasons are clearly delineated, and if the outer markings areexamined,it shows each quarter as being sub-divided by 13 weeks, making 52 in total and into 3 months per quarter, giving the weeks and months in the year; and of course each quarter of the circle represents a season, with it’s allotted activities.This stela is from around 1200 Bce, in Akkadian Sumer. So the circle divided by a cross of two lines at right angles is one of the oldest, and most ubiquitous symbols of civilization, with examples not just from earliest Sumer, but from the Indian subcontinent, and Egypt too.

showing just how entrenched in mankind’s consciousness the squared or quartered circle is. Soboth timeand space in their simplest representations are 4-fold divisions of the circle; thehours of the clock 3 6 9 and 12, and points of the compass N S E and W…

 And from the Bible…the division of Eden by the 4 rivers symbolizes the earthly expression of the divine higher dimensions, the forms the ‘creative energies of the heavens’ take as they energize matter. (right) the Labyrinth on the floor of Chartres Cathedral, symbol of the spirit’s journey (situated centrally beneath the 12-fold Rose Window).

But as with most things, this started with the Sumerians, whorepresented certain starsas being 4-rayed, again almost certainly including some symbolic inner meaningto such symbols…sometimes differentiating the different planets of the solar system and Niburu by the number of rays. So the image shown right is from the Tablet of Shamash (the sun deity) made in the city of Sippar in the 9th century Bce, when it was under Assyrian rule. Looking at this Tablet in more detail, there are several other highly meaningful symbols; the two ‘marker-stones’ the star rests upon are omphalos stones as depicted in Egyptian and Greek mythology and artworks, pointing to locations and sacred sites where higher dimensional energies are more abundant; the date-palm tree is a symbol of the unknowable reaches of the divine, (ie higher dimensions, something perhaps indicated by the mathematical regularity of the pattern of the tree bark, similar to the phi spirals of a pine-cone – something encoded into the Tree of Life stela, as we see elsewhere!); two angels holding cords to protect and measure the star/ Earth (debating what is represented – it certainly doesn’t look like a planet beyond Earth, as it’s resting on the stool indicates it is part of the material reality of this planet); Shamash holding a (measuring) rod-and-ring as symbol of divine authority and stewardship of the earth and circle of the solar system; the human figure to the left of the picture is the sovereign of Assyria introduced by a junior deity (with horned headwear) receiving the blessings of the Anunnaki deity.Shamash too wears the symbol of the gods, what is interpreted widely as horns, but may be serpents in the depiction, as the Egyptian gods wore what is called the ‘Uraeus’, the small serpent at the front of their head. In keeping with the symbolism mentioned in this section, serpents were associated with the energies of the sun, and with light. Thus ‘Ur’, in both the Egyptian usage and original Sumerian sense meant ‘the light of the heavens’, or similar.It is highly possible that Orion takes it’s name from this meaning, being one of the most central constellations (among many) in antiquity… Shamash wears clothes made up or decorated with wave patterns; what this means is debateable, as he was not often depicted as a marine or piscine (orserpent-related) god, as Enki, his relative and friend was. The two lions within the box or square that Shamash sits on may again represent solar energies – but within the human being, ie as the energies of life. Repeated depictions of kings conquering lions were based upon this symbolism, meaning that the powers of the sun invigorate the human being with great force, but without control may lead to unbalanced characteristics within the person. This is basis of the story of Samson in the Old Testament, who is a figure of immense (solar) vitality, but little self-control or restraint…So as such the ‘box’ symbol represents the instinctive centre of the stomach, which absorbs and directs the physical life-force of the Earth’s energies.

And the star (or symbol of ‘Earth’) may well be interpreted to represent the four-square material dimension of this planet as the divine energies received at key points are distributed to the four corners of the globe. Thus the verse in Genesis 2.10 of the river (singular) of the Garden of Eden splitting into four rivers and flowing to the points of the compass (as it implies);

And a river went out of Eden to water the garden; and from thence it was parted, and became into four heads. The name of the first is Pison; that is it which compasseth the whole land of Havilah, where there is gold…”

To the left is an early example of the usage of this four rayed symbol, used in adepiction of Inanna (Ishtar) in an Akkadian clay cylinder seal from circa 2300 Bce. (That she has a lion controlled by a cord again points in allegory to the Sun; and thus to the duality ofthe energies of the sun, as being vitalizing and essential, while also equally dangerous/ destructive if unbalanced or unrefined).

The natural division of the circle by four points has led to it beingused time and again, often for it’s natural geometric use as the orderly division of the circles of the globe, and the heavens, as well as for time; material reality, in other words,(perhaps as connected to the formless higher energetic spheres by the centre of the circle)…Examples throughout history include, the Maltese Cross, Knights of St John Hospitallersorder,the United Nations symbol,the NATO sign, etcetera;

Interestingly, the St John’s/ Maltese Cross is formed bylinking the four arms of the cross where circle and square meet, and thenturning each outward-pointing corner of the square inwards.


    To look at this extension of the squared circle in more detail it is necessary to begin with drawing a square, and then 3 circles as follows, namely;g

 an INNER circle (A) = 100 units

  an EQUAL circle (B) = 127.3 units

&   an OUTER circle (C) = 141.41 units

If square of equal diameter to circle A has a perimeter of 400, then divided by Pi gives us the diameter of circle B which is equal in circumference to the square. This gives B the diameter of 127.32 units -so basically, the square, and circle A = 100 units, B = 127.3 (√φ), C= 1.414 (√2).

and I:E:O = 1:127.3:1.414, while E:O = 1.1111 – ie√1.23456…

Well, if we ascribe the Inner circle (A) = 100units diameter/ circumference = 314.159.

andas before, the square outside circle A = 100 diameter/ 400 perimeter, and a square equal to circle A(= 314.159/4) = 78.54 units length diameter.

The Outer circle (C) of 141.41diameter (√2) has 70.70 radius/ 444.22 circumference.(So a circle drawn within the square within circle A, of diameter 70.7 units, will have a circumference of 222.1 units).

Thus the squared circle diameters of A:B:C as I:E:O are in proportions of 1: 1. 273: 1.414. (1:√φ : √2), plus the ratio of 1; 1.111 (1:√1.23456…)

The√123.456789is alternatively 10/9  = 1.111 (√1.23456789)

                                           and 10²/9² = 1.23456789

So in short, this shows that√2 / √1.234 = √φ

   1.414 / 1.111 = 1.272-3

– although the square root of phi equals 1.272, in the squared circle the difference between the Inner to Equal is closer to 1.273; so a squareof r.100 has an inner circle of r.100,and an Equal circle of 127.32, plus outer circle of 141.41. But considering all the other phi measures within the squared circle, plus the other cos# values associated with the square roots of 2 3 and 5, and Pi and phi, it is fair to consider 1.273 in this case to be effectively the square root of phi, even as Pi / 1.272 4 = 2.469 ie 1.2345 x 2.

Whether the inner circle is designated as 50 units radius or the square, leads obviously to a different set of values, even though still in harmonic progression. If circle B = 100 units, the square equal to it = 78.54 units, circle A = 78.54, and C = 111.055. If C = 100, B = 78.54 and A = 70.7) If every observed circle can be said to have an equal square,plus aninner circle within the square,and outer circle around the square(or vice-versa,squares I:E:O to a circle!), an infinite progression may be assumed from each resulting circle and square – in what can be a very confusing process, ie. a progression of I:O circles goes up or down in increments of √2 (1.4141).  

This process occurs with both circles:squares and vice-versa; so starting with a circle of 50 units diameter, we can draw 3 squares Inside : Equal to : Outside of the circle. And so on indefinitely. . .

It is noticeable that whereas   –   the Circles progress  Inner:Equal:Outer in 1 : 1.273 : 1.414 (so Equal : Outer = 1 : 1.111)

the Squares progress  Inner:Equal:Outer in 1 : 1.111 : 1.414 (so Equal : Outer = 1 : 1.273)

Of central note is the Pi/φ square/circle relationship;   π / φx 4 = 7.77    

ie.3.14159/ 1.618033 x 4 = 7.77  

In this way showing why the values of 5.5, 7, 7.77, 11, 14 etcetera, have such significance.         

So a Square of radius 5.5 will be equal to a Circle of radius 7, will have an inner circle of radius 5.5, and an outer circle of 7.77.

All these figures are straightforward divisions of22 and 7, thus relating the proportions to Pi and φ, within the framework of the Squared Circle…

One of the most notable aspects of this network of values within the harmonics of the squared circle when extended infinitely, is the curious symmetry about these numbers and proportions, both between the various cos# values and 100 – and likewise 90 ie the number of degrees in each quadran of the circle; if we trace the ratios between 1 and 2, giving the increase in proportions before a value is doubled – as in the increase of a frequency or note by one exact octave –  we find the following relationships;

Ratios in progressionbetween 1 and 2

Inner octave >< 1 and 2, of 100 units.Inner octave >< 1 and 2, of 90 units.  Divided by therefore –
1/ 1.111 = .900.9/ 1.111 = .81             (√1.23456)
 / 1.234 = .81/ 1.234 = .729  1.23456    
/ 1.273 = .7854 (=π/4)/ 1.273 = .707     (√2/2)~√φ
  / 1.414 = .707    (=√2/2)/ 1.414 = .6365   (~√φ/2)√2
       / 1.5708 = .6365 (~√φ/2)/ 1.5708= .573     (114.6/2)π/2
       / 1.62    = .6172  (123.45/2)/ 1.62    = .555 (111.1/2)       (φ)
       / 1.80     = .555  (90/1.62)/ 1.80    = .50        ( 100/2)         360/2
       / 2.0= .50  (ie new octave)/ 2.0      = .45      (90/2)         2             (100/2.222 = 45)

Clearly the two tables show the close relationship in some essential way between 100 and 90, (on a proportion of 1:1.1111). So whether in straightforward decimal calculations of base-100, or the 90 degrees of the circle quarter the same proportions between 1 and 2 repeatedly occur in various ways.This may stem from the close relationship between 10 and 9 ,

ie  10  x 10 : 10 x 9 (100:90) = 10:9    = 1.1111

and10 x 10 :  9 x 9 (100:81) = 1.23456789

And therefore√1.23456789 =1.111111

Further to this, if the value π 3.14159 2653 is divided by these key harmonics we find;

3.14159 / 1.111   =  2.828       (2√2)
                / 1.2345 =  2.544       (2√φ) 
                / 1.273   =  2.468     2(1.234)
                / 1.414   =  2.222     2(√1.234)
                / 1.5708 =  2
                 /1.818   =  1.728              etc.

(ie.π /(10/5.5) =   1.728 =  2x .864, 4 x .432, or 3.456/2, etc).

Also 360/ 86.4 = 4.166, which is ¼ of 16.666 (ie 100/6 or the basis of the geometries of the hexagon).

And these relate essentially to each other as;    22.222 : 16.666: 11.111 : 5.5555 = 4:3:2:1 (ie. of 100 divided

by; 4.5/ 6/ 9/ 18, etcetera).

It can be seen these are all related fractions of 1; 100/ 3   =  33.333;100/ (3²/2) ie 4.5 = 22.222; 100/ (3 x 2) ie 6 = 16.666; 100/3² = 11.1111; 100/ (3 x 4) = 8.333; 100/(3 x 5) = 6.666; 100/ (3 x 6) = 5.555; 100 / (3 x 9) ie 27 = 3.70370370.

How is this last value of significance?

The number 369 is another cos# value which is central to the Enneagram (the 9 sided figure introduced to the West by Gurdjieff), wherein the 3 6 and 9 are the primary points of the 3 ‘sections’. Nikolai Tesla said of the value; “If you only knew the magnificence of the 3, 6, and 9, then you would have a key to the universe”. And we see in the Bible section how the significance of this number may beincluded metaphorically in the narrative of the crucifixion of Christ, when the narrative notes the time at the 3rd, 6th and 9th hours… thus pointing to the divine nature of Jesus, (then and when ascended), confirming as such his statement that he is “the Alpha and the Omega”. Alpha (α) is coincidentally the modern letter used to denote the Fine Structure Constant, an intrinsic aspect of reality related to the speed of light, and the essential structure of atoms… the value for alpha is 1 / 1.37035999206.

See diagrams; left, Enneagram of Gurdjieff / Ouspensky, showing the symbol used to understand the workings of the notes within the octave, with key steps between mi-fa, and si-do… Gurdjieff said that any phenomena occurring under conformable conditions within a temporal framework is subject to the laws of the enneagram..! The subject of inner notes, and inner octaves within inner notes, and octaves, can easily become incredibly difficult to understand (!) The symbol shows the progress of 1/7 to 2/7 etc,

from 0.14285714 to 0.28571428 to 0.428571 etc. To right is a diagram of one aspect of the meanings found by Tesla in his understanding of the Enneagram or nine-pointed division of the circle. What is frankly incredible is that two of the most profound and influential thinkers of the early 20th century should independently have confirmed the centrality of a much overlooked symbol…

How does 369 relate to the values we are looking at in relation to the squared circle, and cos#?

1111.111 / 3 = 370.370
                              “      – 369 = 1.370 3 33
                               “     – 36.9 = 333.47
                               “     – 3.69 = 366.6
                               “      -.369 = 370.0013666  (² = 1 369 01.01 12)
Further to this, 123.456789 x 3 = 370.370 367
                                                                  “ – 1.370359992 (∝) = 369.0000070 !
ie. 3 x 123.456789 = 370.370 367
369 = 1.37 0367 (~∝)

So to find the exact value to give Alpha is as follows;          3 x 123 = 369

                                                                              +     3 x .45678 666 = 1.37 035999 206 (α)

– note the first change to the upward sequence = 866 (!) –  and so on. Also, as a theory, the closeness of the 35999 figure to 360 is in some way relates to the geometry of the 360-degree circle and to the circle and the heavens.

That all these cos# values are embedded within the Square and the Circle shows that they are not only closely related to each other, and the maths of the (squared) circle; 360/90 but also directly to Pi (π) – and of course, the (somehow incredible) value of 114.6 (see below) which appears to be linked not only intrinsically to Pi, Phi and the irrational roots but alsoto Alpha(α)- the Fine Structure Constant…

The value of 114.59, which can be seen to be equivalent to the ratio of 360/ 3.14159, ie. the degrees of a circle divided by Pi (the proportion of the circle’s circumference divided by it’s diameter).

But this is by no means the limit of it’s significance; there are several other essential relationships expressed by this value of 114.6;

360 / π=   114.59
3 / 2.618 (φ²)  = 1.14 59
365 x π (/10)   =   114.6 6
90 x 127.3 3  =   114.59
90/ 78.54       =   1.1459
141.4/1.234   =   114.59
162 / 1.414    =   114.59
369 /1.609/2  =   114.6
120 / π            =  38.19  (φ)
π / 2.618 (φ²)   =  1.20
114.6 / 3         =  38.19  (φ)
38.19 x 3.33    = 127.3  √ φ                (3.33 = √11.111)
38.19 x 1.414  = 54 = 864/16
127.33 /1.111 = 114.6       etc.

The 127.2-3 value represents the square root of Phi mid-point as in the squared circle the value is 127.3 whereas the actual square root of 1.618033 = 1.272 02. And yet the upper value is undoubtedly significant, being the product of 400/π = 127.32 4 making it the diameter of the circle equal to a square of side 100. As mentioned also, Pi / 1.272 4 = 2.469 ie 1.2345 x 2.

And re Egyptian Royal Cubit:  114.59  x  0.5236 =  60

π/ 6 = 0.5236,  x 5  = 2.6181 (φ²)

  – furthermore (!)  π / 2 / α(1.370 35999) = 114.6 26

Thuspointing to the mysterious nature of this value –  certainly a key number in the study of the square and the circle.

Relevantly here, a circle of diameter 78.54 has a circumference of 246.76 units, ie  close to 123.4 (x 2); so for a circle of half the length radius (39.27) = to a square of length 50 units, which is part of the series of inner/outer derived circles, the circumference will be ~123.4 (at 12.33703435units)

So the relationship between the 3 can be summarized as;


————-= √φ


  or 1.4141

   ———— = 1.2728                  


To express the relationship between Pi, φ, and the square it can be shown that –   


                            —- x 4 = 7.77   (or π / φ x 400 units = 776.6)


This is a highly likely reason for the importance of the 5.5/7/7.77 square/circle series, linking the various cos# values perfectly as it does. Having noted the perimeter of a square of side 11/radius 5.5 = 44 ,it follows that the equal circle of 7 radius/ 14 diameteralso has circumference of 44; 

7x 2 x π (or 22/7) = 44.  And for the inner circle, with radius equal to the Sq. of 5.5,                             

              11 x π = 34.56 circumference  (and 11.0033 x π = 34.5678 864!)

While as the diagram shows, a triangle of 5.5 height and 7 hypotenuse has 4.33 base.

We can also see that a circle of 34.56 circumferenceis equal to a square of equal Perimeter; with the length/diameter of each side therefore 8.64 units, and the centre-corner side’s ‘radius’ = 4.32 units.

 So this apparently ‘conceptual value, of 34.56 is related to 864 and 432, and 123.45 and 987.654, as well as being theresult of 11 x π, two more cosmic-# staples. Yet it turns out to be closely related to the dimensions of the Earth,as well as the Sun. The latter by 34.56 being the product 8.64 x 4, 4.32 x 8; to the Earth by the following;(in fact the radius of the sun, taken at 432,500 miles according to most recent estimates by NASA, if multiplied by 8 = 3,460,000; or 3,456,789 / 8 = 432, 098).

If the Earth’s circumference is divided by the # of degrees in the circle (360), the distance of 1° at the Equatoris obtained –

                                24,883.2 / 360 = 69.12 m

Which turns out to be exactly   34.56   x 2    = 69.12

So 1/2 of an equatorial degree of longitude  = 34.56 m – in other words,

24,883.2 / 360/ 2 = 34.56

 (  And incidentally, 1° of Equatorial/Cmf = 111.21 km..! A familiar figure).  

Or alternatively ; 3960/360 = 11

                                    “  /720  = 5.5

  • And 1 mile = 5280 feet => GtPyramid height (480” ) is 1/11 of a mile)

Plus – Earth Cmf (2π x radius) / 22 x 4 (88) = 282.8 = 2√2 x 10² or; π / 1.111 x 10² ! These are all cos# values.

  • Earth Cmf in feet = 131,382,240.
  • Divided by 114.6 = 1,146,441 feet.
  • (or Earth Cmf in ft/11,462 = 11,462)
  • π/2/? = 114.626
  • 360/314.159 = 114.6 etc.

Notice from the Table of Squared Circles (in the Appendix) the occurrence of 720 in the progression of squares wherethe OUTER square of circle radius 90 = 720 Perimeter/180 Diameter/90 Radius, and the Perimeter of EQUAL square of circle of 114.57 Radius/229.15 Diameter/720 Circumference.

As half of 144 (12 x 12) 72 has significance, but it is in its being 1/5 of 360 that it comes tothe fore, as a phi pentagon has five equal angles of 72 degrees; this isrelated closely to how phyllotaxis and the growth of plants according to phi proportions.

And in the cosmic cycle of the Precession of the Equinoxes – the Great Year (the Sun’s ‘year’ by which it completes one revolutionthrough the heavens before returning to it’s starting point) – it taking 25,920 years, means that for the Sunto precess 1° (25920/360 = 1°of the great circle), takes 72 years. A figure which was difficult for the astronomers ofantiquity to observe/study, due to the almost imperceptible changes taking place in the night-skies over whatis nearly each century in time… it is considered that the Greek mathematician/astronomer Hipparchus discovered/examined the process of Precession, though it was possibly known before then. Certainly it was known to the philosophers of India who incorporated it into their myths of the ‘Churning of the Heavens’ by the gods and demi-gods playing tug-of-war with the cosmic serpent, around the axis of the sacred mountain, Mount Mandara).

This myth indicates that cosmic energy – the milk of Ambrosia – is created by the (gravitational) forces emanatingfrom the centre of the Galaxy we inhabit which turn the Sun on its path of Precession…again,another highly advanced ‘mythology’ created by a sophisticated culture claiming its advances were the result of cosmic influences (including the beings known as the Nagas)…

Incidentally,the vast galaxies of the universe, such as our own, show in the (fractal) spirals across 600 trillionmiles (600,000 billion) of the Milky Way, for example, the ubiquity of gravity, and it’s effect on all matter; as water may be shaped by gravitational forces. This sensitivity to gravity may highlight the biblical and other references to the ‘waters of  the heavens/ solar system’; in fact this practice of conjoining the waters and their depths with the concept of space dates from Sumerian times where the concept of the Ab-Zu or Ap-su meaning ‘the depths’ may be applied to subterranean water aquifers or reservoirs, the underworld, the subconscious, or the depths of space. In fact the word ‘zodiac’ which depicts the ‘celestial zoo’ of the constellations comes from this Sumerian stem.

Further to this, who is to say that the many Phi proportions in the human body are precisely also incorporated within our growing body through the effect of gravity acting upon the 70% of us which is water..? Especially as much of the formation of the body occurs within the ‘waters’ of the womb.

 Another interesting aspect of 72 is that;       72 x 72 = 5184       

 one of the significant values which keeps reoccurringin this subject – ie the 51.84° ~angle of the Great Pyramid, etc.

In fact 1° of Precession takes apparently just under 72 years, the figure being more like 71.7 or so. Curiously, the Speed of Light is related here;

 186 282 / 5184  (72²)  x  2    = 71.86

 or   186 282 / 71.86               = 2592.0   [ie. Precession/ 10]

All of which strongly support the idea that these incredible coincidences, such as the equatorial circumference of Earth 24,883.2 / 720 = 34.56 miles,or24,883.2 /288 = 86.4. Theseare by no means an accidental or random results;indeed are all  evidence of the conjoining of the energetic and material planes, according to the cosmic principles contained within maths and physics; in other words, for life to take root on a planet such as ours the Sun, theEarth and Moon must all be harmonically related in size and distance between them, and moreover all be individually in proportion with the key mathematical constants of Pi, Phi, the Speed of Light, Alpha and others. See the Sun:Earth:Moon Harmonics section for more on this).


It has been stated often in this section that these relationships and geometries are why the builders of the Great Pyramid used the basicproportions of 5.5:7:(7.77):11:14, and so on, up to 22,28,44,and so. Likewise as the ratio of 440 RC to 280 RC fits perfectlywith these Pi harmonics related to 22 and 7, and extends to the ratios and lengths connected to Cos# which describe how energy/light interact within the material plane of the world. In this way the designers ensured that (whatever the purposeof the Great Pyramid/Giza site), it would be in perfect resonance with the energies it transforms within the world grid of which itoccupies a central site.

One example of the way there are curious correspondences or synchronicities between related subjects in this field, is the tuning/frequency of the modern piano.

This instrument holds an enormous amount of complexity within the subject of the tuning of its strings to make 8 octaves, of 7 notes each, with each octave (ie from C to the next C) being a doubling of the strings’ frequency. This doubling or halvingof frequencies within octaves is at the heart of cos# in it’s measurement of vibrational rates and changes, as we haveseen in this chapter already. The list of people who have studied the maths and physics of the vibration rate ofstrings across the octave includes the Chinese (using stringed instruments such as the gu-qin). It was a Chinese scholar who devised Equal Temperament Tuning, Cu Tsai-Yu publishing his ideas c.1580; these were picked up by Jesuit missionaries in Shanghai, and transmitted back to Europe; it was 25 years later that they were published by  the Flemish mathematician S.Stevin; and when they were championed by J.S.Bach in the 18th century, they were widely adopted.

 The gu-qin is much like the single stringed instrument of Pythagoras’ school, in that it was used to study the harmonic undertones and overtones of notes produced by changing the length of the instrument’s string (by use of frets in the West, and tuning pegs on the gu-qin). This led to many early advances in Chinese music theory, and the widespread adoption of the gu-qin in all classes of Chinese society, but especially among that of the philosophers.

The music made by the instrument is so tonally ‘pure’ and characterized by the production of harmoniously-related tones, that a gu-qin was included in the collection of ‘mankind’s artistic and intellectual achievements’ which were put on the Golden Voyager Record, as part of the NASA Voyager missions into space, in 1975 – and still moving through inter-stellar space today, as the furthest man-made object has ever travelled from earth(at a distance now of estimated 14.483 billion miles* (155.8AU) as of Jan 21,2022, according to NASA/JPL) –  and  Pythagoras,who saw reality – the energetic and material levels – as consisting of ‘strings’ which vibrated according to the influences/forcesbrought to bear on them, thus creating the material differences seen in the universe; a conception not too far removed from modern ‘string theory’, incredibly. (With apologies to any Pythagoras experts for any ‘misquoting’ of his basic tenets and beliefs).

This portrayal of Pythagoras working with his stringed instrument (with pegs visible) to examine (cosmic) frequencies and octaves is found at Chartres cathedral, where he is depicted among many figures within the right bay of the Royal Portal entranceway and steps. The image below Pythagoras is a wider view of the right side of the west portal of the cathedral, showing some of the amazing work of the edifice. The stone carvings here date to the original period of construction, in the 12th century…

Pythagoras’ school – and studies of geometry were in large part inspired by the same subject of the vibrational frequencieswithin the stringed instrument mentioned, the mono-chord,in termsof the harmonicovertones, and the various musical progressions and scales obtainable. Indeed pre-Renaissance piano tuning inEurope was (and still is) called ‘Pythagorean tuning’, the progression of the 12 notes of the octave in ‘perfect fifths’.12 perfect fifths = 7 octaves, except that Equal Temperament (ET) tuning flattens each fifth by a twelfth of a Pythagorean Comma (23.46 cents ie ¼ of a semi-tone, /12 = 2cents)-so instead of 3:2 (1.5), the 7th semitone, based on 12 equal logarithmic divisions of an an octave equals(ie., is) the 7th power of the 12th root of 2 or 1.4983: 1

Originally, the Pythagorean tuning using the Circle of Fifths was used as a way of organizing the 12 chromatic pitches (all the notes present in the octave) in a sequence that ascends in a cycle of ‘perfect fifths , in a ratio of 3:2 (1.5) with each fifth-  through 7 octaves (this generates the 12 notes of the octave, in different octaves, before C is reached again 7 octaves up. An Equal Temperament tuning will produce a note that is exactly 7 octaves higher. And a ET fifth increases the frequency at a rate not of 3:2, ie. 1.5 :1, but at 1.4983:1, approx.2 cents smaller than Pythagorean tuning. So, across the 7 octaves this leads to a slight excess for Pythagorean tuning of c.23.45 cents* – which is called the Pythagorean Comma. (ie the difference in a fifth times 12) – the end resulting note being not exactly 7 octaves higher from the starting note, but rather,a tiny amount higher than the 7 octaves.An ET fifth has a ratio of 2⁷/12 :1 (= 1.4983 :1). A perfect fifth is thus equivalent to 7 equal-tempered semi-tones)!

*E/T tuning divides the octave into 12 semi-tones of 100 cents each.

A perfect fifth changes the ratio of frequency by exactly 3:2,say from  E F G A B. In octave 6 on the piano E⁶ = 329.6 2Hz- using Pythagorean tuning, the B⁶ 5 notes up is tuned to 3:2 of this, giving a frequency of 494.43Hz. This process continues until each of the12 notes of the octave is present in the ‘circle of fifths’, in what is called a chromatic scale, or chromatic circle. Now using ET tuning, B⁶ = 493.9Hz, the note obtained by increasing the rate ofchange each time by 1/12th of the root of 2(1.059) (meaning that after 12 increases the notes have doubled in frequency). This tuning avoids the discrepancies caused by Pythagorean tuning, thus making some of the inherent dissonances less obvious/noticeable,at the cost of ‘flattening’ each of the notes by a tiny amount.

In fact, if we divide the two given values for the note B using the two systems we get a difference of;

                            494.43 / 493.9 = 1.001 073

The small discrepancy of the Circle of Fifths led to European composers such as Beethoven devising/ adopting the Equal Temperament tuning, so that the difference between notes was made equal. The rate in Equal Temperament tuning is defined as thus by wikipedia; “the most common tuning system (in Western music and classical) since the 18th century has been ‘twelve-tone equal temperament, 12-ET, or ‘twelve equal’, which divides the octave into twelve parts, all of which are equal on a logarithmic scale with a ratio equal to the 12th root of 2 , 1.05946. The resulting smallest interval is 1/12th the width of an octave, and is called a semi-tone or half-step”. . .

(there being 12 different notes in an octave, 7 major/5 minor) ie 1.05946 increase per all semi-tones, the interval between Si-Do and Mi-Fa is also this ratio.

 In contrast traditional pre 12-ET tuning using natural whole-number ratios of the ‘circle of fifths’ to obtain each interval were slightly different, with Si-Do and Mi-Fa being around

The octave consists of tones from a single harmonic series of an implied fundamental from which the other notes derive as ratios as follows (with the notes marked by semi-tone gaps marked in capitals);

 C (do)     D (re)       E (MI)       F (FA)      G (sol)      A (la)      B (SI)       C (DO)

1:1           9:8            5:4            4:3         3:2            5:3         15:8         2:1

  1            1.125        1.25         1.33        1.5           1.66        1.875        2

           9:8        10:9         16:15         9:8         10:9         9:8         16:15  

 Cents 204        182          112           204         182         204           112

 The ratios in the third line are the intervals between major notes, showing mi-fa and si-do to be the smallest intervals, there being no actual semi-tone. This may point to why Gurdjieff, in the lineage of cosmic consciousness stated these two intervals to be where changes are most likely to take place in the conceptual use of the Enneagram and Octave for understanding any courses of events which take place within time – or indeed, are the only points where changes may occur.

As can be seen, Do-Si and Mi-Fa are the smallest intervals, being between E and F, and B and C, as the 16:15 ratio, or 112 cents reveal.

In Pythagorean tuning, an early form of a just perfect fifth tuning the notes of the octave, Si-Do and Mi-Fa are 1.053498 (256/243) higher than preceding notes. This method preserves the octave as a factor of 2, despite being slightly less than Equal Tuning which has 1.0594 difference between all 12 intervals. Just intonation has most intervals of the circle of fifths as being higher than ET tuning.

This shortfall across the circle of fifths called the Pythagorean Comma is present not only in music. For example Gordon Strachan (the author of  ‘Hidden Cosmology in the Bible’, and ‘Chartres Cathedral’) says that a small PC was built intentionally and consciously into both Stonehenge by it’s (Celtic druid) builders, andalso into Chartres’ architectural plan/building design. Why? Strachan says that both were built clearly to ‘sacred temple’ geometric principles –so at Chartres, from the omphalos point (at the crossover point on the two axes, using√3 as its ‘key’, in the double-triangles  latticework of the two triangles), the energies expressed through the √3 are constantly expanding, and as such ‘representative of infinity’; in this flow of energy the building then had a slight imbalance built in to create a small ‘oscillating’ effect in the routing of the energies of the earth through the structures of the edifice. This to ensure that it is more than a sterile architectural plan ‘set in stone’ built without reference to the energy flows within.

*(The Pyramid of the Sun at Teotihuacan in Mexico is argued to have small discrepancies of size and angle built into its bases/sides,for similar reasons, said to relate to vorticular energies as they arise from the earth up through the structure of the pyramid).

There is a similar dissonance in the angled sides of the Pyramid of Menkaure at Giza, 2 sides of which have an angle to the ground of 51.85° and 51.19°. (Furlong citing Lehner’s measures of the pyramid, of sides343 feet, and 335 feet) As  Furlong notes, these differences are far too large to not have been noticed by thedesigners/builders of the structure at an early stage, considering the sides of the GP, for instance, of sides 756 feet have differences measured in inches. The stark difference in Menkaure’s sides may well be another example of small design’imbalances’ built in to encourage ‘oscillating’ energy flows, especially if such ‘energetic dynamics’ were at the heart of the buildings’ design.

 – this is to say that a small and marginal ‘imperfection’ enables energies to flow a little more quickly – in principle this is similar to the Typhoon fighter jet which had constant micro-imbalances built into it’s 70 or so computer flight-stabilization systems, to enable faster reaction speeds for evasive action – it is already moving one way or another off straight-line direction every micro-second…

Strachan maintains that there is a similar design of a Pythagorean comma built into the layout of Stonehenge, for similar purposes, as a small additional inlet for cosmic energies.

 The principle is akin not only to the irrational roots, but also to the subterranean water-courses which underlie many sacred sites, introducing a continual small flow of (electro-magnetic) energy to permeate the structure.


Now, this all said (!), there is modern debate around  the tuning of the A above Middle C; as can be seen, Middle C⁴ vibrates at 261.68Hz,(~ φ²), while the note A⁴, located five notes higher, vibrates at the rate of 440 Hz/sec. And it is from this note and it’s frequency that every other note on the piano is derived, it being what is called the Standard Tuning Reference note.

One feature of this is that the note oneoctave down at A³ vibrates at 220 Hz, the A⁵ (the A above A⁴ of 440Hz) at 880 Hz…and so on.(While C⁸, at 4186.006Hz, the first and only note of the 8th octave, numbered as key #88,is the highest note on the piano,and A⁰ is the lowest – at 27.5 Hz. and therefore A¹ = 55Hz, A² = 110 Hz., etc. All on a harmonic of 11 therefore, as the values of Pi, 22/ 7 are related to 2.75, 5.5, 7, 11, 14, 22, 28, 44 and so on). And as noted, the edifice at the ‘centre’ of the world’s continent’s, the Great Pyramid is designedwith ratios of 7:11 throughout, and has a base length of 440 cubits, and height of 280 cubits, thus aligning the dimensions of the pyramid intrinsically with the defining values of Pi.

The debate is that some students of cos#/SG subjects, having discovered the centrality of the A⁴ (440Hz) above mid-C (261.7Hz), which is the Standard Tuning reference note of the piano) should likewise be tuned to 432 Hz/sec, in order to align it, and thesounds/wavelengths/frequencies of the entire piano, with the 432 harmonic and all theways that it resonates through reality… some claim they can tell the difference between pianos tuned to 432 to 440, saying it is a more ‘resonant’ or ‘natural ‘sound.

  It is clearly a good idea to consider; but the ratios considered in this chapter give pause, in showing how 440 may resonate with the 864/432 harmonic, as well as with Pi, phi and so on. For the Great Pyramid,Giza,the Sphinx, and the foremost measure of all these monuments, the Royal Cubit etc, were all the product of clearly the highest levels of consciousness.

(As well as this, the note next to A² of 110Hz is B² which has a frequency of 123.4Hz; so three octaves higher, A⁵ has a frequency of 880Hz while B⁵ = 987.7Hz. . .showing that the cosmic# value of 1.23456789 x 8 = 9.87654321; and additionally we can note that 1.234…9  x 7 = 864.

The note C⁵, double the frequency of C⁴(261.63) is equal to 523.25, remarkably close to the value for the Egyptian Royal Cubit of .5236 metres; so linking the piano frequencies to the π/φ²/ERC relationship of, 3.14159(π)/6 = .5236, which x 5 = 2.618  (φ²). Or π = φ² x 6/5.

One more coincidence is that D# in octaves 2 and 3 has a frequency of 77.7 and 155.5Hz, both of which have many cosmic# relationships;

in the Ray of Creation octave D# is the semi-tone between Re (the moon) and Mi (the Earth), and in the geometries of the squared circle of diameter 7 units, as we have seen in this section already.

 As can be seen, the frequencies of the ‘key’ note A are in harmonic relation to values present in the Great Pyramidratios of 5.5 etc. There is more to this subject. If it is considered how we receive information from external reality through the senses,particularly on those of sight and hearing, it is known that the ears have three times more nerves, andtherefore sensitivity/power, than the eyes.

Thus leading to the saying  ‘the eye estimates; but the ear calculates’…

The salient point here is that sound waves are received through the structures of the outer ear, and then channelleddown into the middle/inner earto the cochlea, where they are converted via receptor groups of hairs within the ear-drum’s liquid into ‘electrical’ signals. These are sent via neurotransmitters along the nervous system, via the brainstem to the brain – where they receive further processing and ‘conversion’ into the ‘sounds’ we hear/perceive… another example of the incredible nature of the body’s structures, formed of living matter.

Now, the first point is that the cochlea is designed/ has evolved in a spiral structure along which sound-waves travel. The number of spirals affects directly the frequency-range of sounds we can hear… so a dog, or cat, or bat have ears with approximately

4 or more spirals, or something like…the human ear has 2.75 spirals,whichinterestingly is exactly 5.5/2 –  providing some measure of support potentially to the equivalence of these values in this chapter,especially in the octave frequencies from A⁰ (27.5Hz) on through to A⁷ on the piano-forte keyboard.

The tuning of the modern piano may seem irrelevant to these subjects concerning Giza and the earth, and yet it is a part of the planet’s reality, and an example of the ubiquity of the inner-frequencies and resonanceswithin the cycles, and harmonic relationships of life. Perhaps modern piano tuningsassumed the 440Hz frequency for A⁴ because it sounded ‘right’ , in repeating cosmic frequencies/harmonics; ie as quoted already, ‘life/the (material)universe follows the Tao’… or energetic principles.

Incidentally, spirals play a central part in many aspects of reality, and in the processes of life for organic life. As the etymology chapter highlights, the stem of ‘spiral’ has the following words or concepts attached to it;

 Hope –  aspire – work (perspire) – breathe –  worship  – the  heavens – spirit. 

‘Hope’ is from the French word ‘espere’;  while ‘heavens’ comes from the heavenly spheres. Plus of course the shape of galaxies as wide as 100 light-years across(ie 100 x 6,000 billion miles!) form spirals sometimes*, such as our own galaxy the Milky Way,formed in straightforward spirals – we are situated in the Orion arm of the 4 spiral arms of the Milky Way galaxy.Of the numerous examples of how (φ) spirals are a central aspect of reality, in both material and energetic terms, are; the patterns of weather systems; how water moves under the action of gravity; how hair grows outwards from thecrown of the head; the spiral whorls present upon each finger’s print; the digestive systems of animals with high-volume food intakes; the ears of mammals,birdsand reptiles,as mentioned; the shells of sea-creatures such as Nautilus shells; the spirals on organic flowers and plants such as sunflower heads, pine-cones, pineapples, cacti, acorns etcetera; and as stated, the spiral galaxies, of various types.


Of further interest are some of the many significant values found in Cos# etc, coincidentally (?) in such tables as the Tree of Life. For example the Circle of radius 39.27/78.54 radius highlighted in blue ink in the Table below if considered as an EQUAL circle to a square of 30.84/61.7 has an outer circle of radius 43.2/diam. 86.4 units. Indicating how these cosmic# values can be found within the inner geometries of the circle starting from 50 radius/100 diameter – although if every derived circle, or E:I:O square etcetera is used as a starting point, the number of new squared circle sets is effectively infinite. . .

That a circle of diameter 39.29 7 (instead of 39.27) has a Circumference of 123.45…9 points again to the close correlations between the squared circle and the values indicated;  means that when multiplied by 2³  (increased in size  x 8) this produces (diameter 314.38) giving a circle of circumference 987.654321. The two #s  of 123.456…, and 987.654 etc, raise the central question of the octave, in that the 7 notes of the octave begin a new octave, at double the frequency, with the 8th note in the series;

the A⁴ above middle C, for example has a frequency of 440hz, while the one below middle C at A³ = 220hz…  this subject is at the heart of cosmic #, and has been so from the earliest times, with diverse schools/ teachers studying the maths and physics of the doubling (etc) of vibrations of a musical instrument’s string; Pythagoras used/ developed a stringed instrument for his studies, (as depicted in the stonework of Chartres cathedral), much in the same manner as the Chinese philosophers of the same era studied the gu-qin instrument and its harmonics… more on this subject in following sections…

The figure 123.456789 has some significant features in cos#; likewise it’s ‘corollary’, 987.654321…

indeed, there are some noteworthy relationships involved around these two numbers.

 – as we have seen with the square/circle/ φ – a circle of diameter 61.78 units doubles up to 123.456789; while in the original squared circle the value of CD = 61.9 units in diameter, and 30.9 in radius. So a radius of 30.86 x4 = 123.456789; another value therefore within the first squared circle which is extremely close, but not identical to the significant cos# values… while this is clearly not an identical match, the fact so many cos# values are almost indicated is actually probably significant in some way itself…

So 123.456789/ π  = 39.29 75164 ( x2 = 78.58… significantly close in value to the square/circle measures of 39.27 and 78.54)

 In fact, considering the necessity of numbers,values and proportions in Cos# relating to the geometries and physics of reality, it is of great interest that 123.45…9  has a significant relationship to α the Fine Structure Constant, a value at the centre of modern physics studies of light, (and it’s relationship to matter);

As can be seen,the #  we derive is;

   Diam. circle of  39.27 x π  = 123.370 3435.

Not only is this close to 123.456 etc – but equally importantly –

√1.23 373 849 (ie 1.1107378)³ = 1.37 0359992 = α

Alternatively, 123.3734889 / π= 39.27 100124…

Table of some of the features of 123.456789, etc;

400 / 1.62 = 123.45679012 x 2
10² / 9²      = 1.23456789
10  /  9        = 1.1111
              x 3.60   =  444.444   ie. 4(√123.45 )
              x 3.5      = 432.09
               x 3³      =  3333.333    
               x 3⁴      =  10,000 (10⁴)
               x 18⁴    =  12,960,000
               x 18³    =  720,000
               x 18²    =  40,000
               X 18     = 2,222.222
               x 8.1     = 1000 (10³)
               x 7         = 864
               x 8         = 987.654321

 See table next page of squared circle progressions.

Appendix to Squared Circle section;
Table 1. C       I          R     CL  E    S    S   Q        U  A  R    E   S
35.35/1.414=2550157.08E. 19.6339.27157.08
    I.  17.6835.35141.4
    O. 2550200
(50/1.414) =35.3570.70222.11E.  27.7755.54222.11
    I.   2550200
    O.  35.3570.7282.8
35.35x 1.111=39.2778.54246.74E.  30.8461.68246.74
    I. 27.7755.5222.18
    O. 39.2778.54314.159
[39.27 x 1.11=43.286.4271.4E.  33.967.87271.4]
39.27×1.273=50100314.159E.  39.2778.54314.159
    I.   35.3570.70282.8
    O.  50100400
50 x 1.273 =63.65127.3400E.   50100400
    I.   45     90360
    O.  63.65127.3509.2   
63.65×1.11 =70.70141.41444.22E.  55.5111.05444.22
    I.   50           100       400
    O. 70.7141.4565.6
70.7x 1.111 =78.54157.08493.48E.  61.7123.370493.48
    I.   55.5111.05444.22
    O. 78.54157.08628.32
70.7 x 1.273 =90180565.6     E.  70.7141.4565.6
    I.   63.65127.3509.2
    O. 90180360
90 x 1.111 =100200628.32E.  78.54157.08628.32
    I.   70.7141.41565.6
    O.  100200800
90 x 1.273 =     114.57229.14719.8          E. 90180720*
    I.  81162648.2
    O. 114.57229.14916.56


Having introduced some of the most germane points concerning our representative Circle, and Square, and PI and PHI relationships within these geometric symbols, we can now progress to adding another circle to the symbol.

So the original circle may be termed the circle of ‘the heavens’, representing the flow of the highest energies into the circle of material reality, the Universe and outwards in the four directions of the compass – much like the Garden of Eden is defined by the four rivers, and the four points of the compass. Genesis 2.10-11;

And a river went out of Eden to water the garden; and from thence it was parted, and became into four heads.

The name of the first is Pison; That is it which compasseth the whole land of Havilah, where there is gold. . .“ – and so on.

The second circle represents the transmission of the energies to the material plane of the Earth;  the circle is made of equal radius to the first, and drawn from a point on the circumference of the first circle; so that the circle drawn in this way passes through the centre of the original, and both centres are on the circumference of the other, in the way shown.

This figure of two equal intersecting circles has been known since antiquity, and was named as the Vesica Piscis – the fish’s bladder (or ‘vessel of the fish’, opinions vary) – due to it’s shape, and was examined in the works of two of the most well-known academics of antiquity; the Greek (founding father of geometry) Euclid, and the Roman architect and geometer Vitruvius, (70Bce-15Ad) who wrote ‘De Architectura’; a volume which was rediscovered by Italian scholars in the Middle Ages and helped shape the architecture of the Renaissance… see Leonardo da Vinci’s Vitruvian Man, for example. His views on architecture were based on three standards – strength, utility, and beauty. But the unifying theme of his work was proportionality, the relationship of all the individual parts of a building to the whole; or of the building to it’s environment. This belief extended to the human body, of which many examples of phi proportions can be found, hence da Vinci’s drawing and related studies, which searched for universal ratios and patterns expressed in the human body. In this way affirming the dictum that man was made in God’s image, embodying cosmic principles and laws.

Archimedes too examined the Vesica, concerning the sectoring of a circle into 6 equal proportions, thus making the drawing of a hexagon possible with compass and pencil, a feature we will develop in a while.

To begin then, we draw our second circle.                                                                                                    

Thus each circle has equal radius of 50 units here, a diameter of 100 units, and circumference of 314.159 units.

The central area is the key point of interest here, as it contains some essential geometric principles – and has symbolic meanings encoded into many epics and religious texts of antiquity. Most well known is the Christian adoption of the symbol as representative of Jesus Christ (as intermediary between God and mankind), as well as Christianity and the Piscean Age.

At Chartres Cathedral, one of the most highly regarded Gothic cathedrals of the Middle Ages, the space over the central (Islamic archway) door has Jesus portrayed within a Vesica (left). To right; Plate from the Book of John, Syriac New Testament,1555, showing correlations between Jesus, the Vesica Piscis, and the Qabalah Tree of Life !

 The foremost symbolic meaning of the Vesica is that suggested by its uniting of duality  – in that the 2 circles are at this central point, conjoined as such. And so symbolically the two circles represent the two spheres of heaven and earth – or energy and matter.

 Examining the geometry of the vesica brings some fascinating points to the fore; so, to begin with, we draw a line along the vertical axis, from the top to the bottom of the central area.                                                       

Firstly,it is of relevance that this line is 86.6% of the length of the circles’ diameter.This is half the √3.                

This is worked out as follows;

By drawing a line from each circles centre to the crossover point, we create an equilateral triangle above the ‘equator’, and beneath. The ratio of the central vertical axis through the VP is this; for a circle of height 200 units (radius 100 units, diameter 200 units) the length of the line is the √3. 1.732 ie 86.6% of 2. And each triangle measures 86.6 units ie half of the vertical axis.

If the circle is diameter of 100 units (& radius of 50 units) each half of the Vesica axis (DB & BF) = 86.6 /2 ie 43.3 units high.

AB = 25 units

AC = 50

AD & DC = 50

EG = 100 (EC = 50 + CG = 50)

DF = 86.6% of EG, ie 86.6 units.

DB = 43.3 / BF also.

The ratios of BC : CD : DB = 1 : 2 : √3

As we shall see soon, AD and AF form 2 / 6 of an equal sided hexagon within the entire RH circle, by creating a mirror of the points by drawing a Vesica from the other side of the circumference.

As the radius of the circle (AC) = 50 units, the diameter therefore = 100 units, and the circumference = 314.159 units, ie Pi.

This raises the point that the segment of the circumference AHD = 314.159 / 6 = 52.36. Curiously, this is the length of the Egyptian Royal Cubit, as used and defined in the Great Pyramid (and ascertained through a process of deduction by the Victorian surveyor of the site of Giza,  Sir Flinders Petrie). Ie a Royal Cubit measure 0.5236 metres, or 52.36 cm ; or 20.6 inches. This characteristic of the ERC being 1/6 of Pi, and of the circumference of the circle within the Vesica Piscis is a good indication that the designers of the Great Pyramid knew of this symbol and it’s geometries when they designed the Great Pyramid. Several other features of the double-triangle within the Vesica are of significance too.

Incidentally A to E via H and D = Pi/4 = 78.54 units, as noted in the Squared Circle section.

So from apex to base of the Vesica we have 2 triangles of height of 86.6% of the radius of each circle, which combined are accordingly 86.6% of the circle’s diameter.

Taken together these triangles form a rhombus composed of two equilateral triangles. A rectangle placed around the rhombus is known as a root3 rectangle, and as can be seen from the diagram (left) of Chartres Cathedral, (from Strachan, op.cit), is at the heart of the geometric design of the edifice working outwards from the ‘omphalos point’, at the crossover points of the horizontal and vertical axes.

This system of ‘double triangles’, known as the Ad Triangulum system was widely used by the early Gothic-era monks and architect builders of the great cathedrals of Europe, people such as Bernard of Clairvaux, and his Benedictine and then Cistercian order. There is a clear line of transmission of this Sacred Geometry at this point from antiquity to the ‘modern European’ era; for Bernard was one of the originators of the order of the Knights Templar, founded in 1118 Ad, though only given Papal recognition at the Council of Troyes in 1128 under their full name; The Poor Fellow-Soldiers of the Order of Christ and the Temple of Solomon. Research and/or legends posit that this ‘cosmic’-based wisdom was given to the Templars when they established themselves in Jerusalem, by orders extant within Jerusalem holding the highest philosophic and scientific levels of understanding and wisdom. In other words orders within Islam, a religion immersed in matters of geometry and cosmic wisdom.

(This has the potential to be viewed as an example of positive higher-dimensional influences ‘transmitting’ (cosmic) wisdom through ‘lower’ Cainite (?) streams as represented by the (worldly) Templars; due to the position of influence the Templars held within western civilization, as seen in their connection to the development of the Gothic architecture movement , arguably the highest expression of ‘sacred’ geometry in church building seen in the West).

The architecture of the civilization of Islam from the 7th century onwards likewise was based upon very similar geometry-based tenets dating back presumably to the Egyptian schools which designed the earliest pyramids, and was transmitted to the western nations of Europe by the highest Islamic schools of architecture and religious thought within the cities of the Holy Land when the European Crusades won temporary stewardship of Jerusalem and other cities, from 1095 on and off for the next two centuries until 1267, although numerous smaller campaigns were waged after this date in not just the Holy Lands, but in the Iberian Peninsula against Muslim rule, and northern Europe around the Baltic against ‘pagan’ peoples of Slavic, Baltic and Finnish origins.

But the significant point is that when the western orders of the church, and the elites of the various European kingdom’s political/ military orders came into close cultural contact with the Islamic rulers of the Holy Lands, there was a significant amount of interchange of higher knowledge and practices of the Islamic world at that time.

As Jerusalem was one of Islam’s most revered centres of religion and culture (as a world omphalos point, or centre), where the centre(s) of ‘esoteric’ wisdom within Islam lay at this point in time lay is certainly not clear; the Sufi orders, for instance, were yet to begin from around the 13th century onwards. And as well as the Islamic religion’s own sources of cosmic wisdom, strong traces of previous transmissions of higher consciousness existed within Near East traditions of the Sumerian/ Assyrian/ Babylonian civilizations; the Hebrew religion, (with many shared concepts); Egyptian texts, architecture and religious practices from the 3rd millennia Bce onwards; influences from Iran and Persia, which themselves bore traces of the pathways trade and travellers from India and central Asia took through the Persian cities; the writings of the Greeks, preserved in communities such as Alexandria in Egypt; and so on. In such an environment with thousands of year of rising and falling religions and civilizations, with the inevitable interactions between them all in the various cosmopolitan centres of empires, delineating the origins of many of the most valuable beliefs and sciences by the year 900Ad in the Near East is by no means easy. 

While this may look complicated, in fact the complexity of the Near East was one of it’s foremost strengths at this time, indicating a rich mix of different cultures and centres of learning; so much so that as G.Strachan writes in his book ‘Chartres Sacred Geometry, Sacred Space’ (p17);

 “Today it is difficult for us in the West to realize that in those times, Islam had achieved a far higher level of civilization than north-western Europe, in almost every field of the arts and sciences. However, the historical evidence clearly indicates that it was only through contact with Islam that the full works of Plato, Aristotle and Euclid, which had been largely lost in the West, were rediscovered.

It was only after these, and other classical works had been fully translated from Arabic into Latin, and added to the curriculum in the schools of Paris, Laon, Chartres and the other centres of learning, that a renaissance of culture was triggered off in northern France. The rise of Gothic architecture was the most spectacular result of this.”

So the Islamic architectural knowledge in the lineage of the ‘wisdom of antiquity’ was transferred to the west – it is conjectured – primarily through the Gothic cathedrals of the 12th century across Europe, particularly in France. And the innovations of Gothic architecture have been shown to have come from several main figures;  such as the ‘master mason’ Villard de Honnecourt, the architect who introduced the (Islamic) ‘pointed arch’ intrinsic to the Gothic style. These arches unlike rounded Roman arches pointed skywards,, routing energies of the edifice not back into the earth but skywards. Their design was based upon the geometric creation of two curves of a circle drawn from two points within the base-line – thus angled arches could be made to a greater or lesser degree;

(left) the renowned doorway of Peterborough Cathedral, Cambs. (right) the arched doorway at Toledo cathedral, Spain.    

 Strachan also points to the numerous English knights and land-owners who brought (possibly captive) Islamic architects and masons with them when they returned from the Crusades; Richard de Granville brought one such mason who designed Neath Abbey in Glamorganshire, and then later became architect to the Norman king of England Henry I (cited from Nigel Pennick,1994). Similarly Strachan states that the ‘dating of early examples of cross-ribbed vaulting at Durham Cathedral coincides with the return of the Prince-Bishop of Durham, William of St-Calais, in the mid-1190’s. His successful campaign against the Muslims in Sicily resulted in him bringing back captured Islamic masons’ (Strachan, Chartres, p..18-19). Masons were not the only experts from the Islamic world who were brought back to England, as we shall see shortly, the influence of Islamic practices and sciences were strongly influential in modernising the western courts of it’s nation’s rulers.

So research has shown there was a considerable amount of  wisdom, science or technology transfer from the occupied Near Eastern peoples in the occupied Crusades lands to the western forces of the Europeans, at all levels, from the courts of the kings and nobility to the religious orders which were the effective disseminators of knowledge throughout society at that time. It is argued that the higher standards of learning prevalent in the Islamic world at that time led to European societies gaining knowledge of;

Chemistry and distillation, via studies of ‘alchemy’, including oil products’ derivations and usage, the creation of alcohols,  perfumes, distilled products and so on;

Banking and accounting practices such as double-entry book keeping; the Knights Templars instituted a system of credit and identity transferral of funds for Crusader noblemen travelling across Europe to the Holy Lands – thus becoming gradually ‘bankers’ to the vastly wealthy ‘nobility of Europe’, and becoming the world’s first ‘multinational corporation’  over the next century or two. But the possibility exists they received an amount of advice from their connections with the elites of Islam in the Holy Lands, as well as with Norman/Italian financial circles based in the Savoy region …one legend has it that king Richard the Lionheart brought an Islamic book-keeper back from Sicily on his return from the Crusades; the book-keeper introduced double-entry accounting to the Chancellor of the Exchequer, and was occupied in working there for several years. In order to maintain national dignity and prevent animosity the man was given the name John Brown…

whether any researchers can find evidence of this story is uncertain.

Architectural improvements such as arched windows, derived from Islamic architecture and mathematics (see right); and Gothic architecture developed the use of the Ad Triangulum system of siting (sacred) buildings from a central point, and geometrically developing the rest of the building from the navel-point. This would indicate the practice was indeed linked to the canons of (sacred) architecture in antiquity, be they Egyptian, Hebrew, Indian, Greek, and Roman, all of which placed their temples at navel-points in the land, and built from the heart of the site outwards using geometric ‘keys’. . .

Also techniques of stained glass production, involving the colouring of glass using again methods resulting from (al)chemical researches;

Courtly practices including; the more balanced and respectful treatment of ladies, which led to the European development soon after of the popularity of ‘wandering minstrels’, or ‘minnesingers’ who sang of platonic knightly love for the Queens and noble women of the courts.* Possibly Grail Quest themes and standards were also passed culturally across this

Two centuries long meeting point between the most advanced expressions of (Near) East and West.

Higher standards of cleanliness were passed to the westerners from the Islamic rulers, such as the use of linen bed-sheets, a practice little known in Europe before the Crusades.  Often pre-Crusades the sovereign courts of Europe had matted rush-floors, something which changed during this period of contact between hemispheres.

*see for example  ‘Eleanor of Aquitane, the grand-daughter of William IX, the ‘troubadour Duke’ who was influenced by Arabic love poetry’ (as stated by Jane Carroll in Beshara magazine, winter 2019)  – who mentions also that Chartres was at the forefront of a 13th century ‘renaissance’ whereby new works from Greece and Rome, with Arabic commentaries were filtering up from Islamic Spain. Another example of the advanced nature of the Arabic sciences and culture at this time.

Chartres cathedral, exterior.

In regard to the use of Ad Triangulum, it may be argued that the Islamic mathematician/geometers and architects gained knowledge of this Sacred Geometry from the works of Vitruvius and the Roman schools of architecture in the early first millennium, or via contacts with Greek culture and it’s connections with Egypt; knowledge which was then lost to Europe until the Crusades, and then the Renaissance… with each civilization refining or adding it’s own understandings of the subjects contained within.

But there is fair argument to say that this system of geometry and number-based cosmology and architecture was originally devised by both the Sumerian civilizations and the Egyptians, from where it was transmitted to both Hebrew and Greek religious philosophers, and thence to Rome…

(Though Egyptian examples of omphalos stones are to be found at the oasis at Siwa, and at Thebes, it may be argued they were later introductions during the Ptolemaic era when Greeks ruled the Egyptian ‘state’; but the precision of Egyptian geometric applications at Giza, and the Great Pyramid, pyramid complex and the Sphinx all indicate the founders of Egyptian civilization were – like the founders of Sumer – aware of the geometric significance of sites in relation to each other in terms of the proportions of the Earth. The image in Egyptian and Greek artworks of ‘omphalos’ stone in pairs is actually also present within an Assyrian stela of the Tablet of Shamash; with the presence of pairs of marker stones indicating the geodetic nature of the phenomenon as linking sites of special nature to others across the land, and the globe.

(pictures i – iii); Egyptian scene from the Book of the Dead / the Papyrus of Queen Nejmet showing two omphalos stones linked to the Sun, with ‘celestial beings’ holding lines or wands of connection or measurement.Note the two birds (crows?) sat upon the marker stones in the second image; also that the stone is effectively the ‘rod and ring’ symbol of Sumer/ Shen-ring of Egypt, used to symbolize divine and earthly authority, as a cord of protection depicting the boundaries of both the Creation and kingdom. The third image is a depiction of the source of the Nile, in Upper Egypt (from John Ward, 1902). The presence of the pair of birds may well symbolize that it is a significant geodetic point on the Earth’s surface, as with the Greek omphalos myth of the two eagles released by Zeus in different directions around the world which met at Delphos).

(pictures iv and v); Sumerian Tablet of Shamash, Assyria, c.885Bce, two deities/ higher beings hold lines to four rayed symbol (of Earth with divine energies diffusing from the central point?), identical to the Egyptian pair of ‘angels’ with measuring lines. Also note the presence of two omphalos/ marker stones as foundation points beneath the main symbol; a theme supported by the square base upon which the symbols rest ie how energy distributes into material reality.

These types of omphalos stones are similar in concept to the benben stone as formulated in Heliopolis in the Pyramid Texts, as the mound or apex that arose from primeval waters and upon which the creator, Aten then settled. Foundation points of the world in other words… similar to the Hebrew concept of the Foundation Stone in Jerusalem in the basement of the Temple, said to be placed over the waters of the abyss at the ‘centre’ or ‘navel’ of the world, to keep them in abeyance. Thus locating sacred sites at geodetically significant and related points within the individual nation, and the world.

(left) the Foundation Stone, within the Dome on the Rock, Temple Mount, Jerusalem, 1915 photograph.

Greek omphalos marker stones have also had in the past the alternative name in Greek and Roman civilizations of baetyl stones, often linked to meteorites which were then made into objects of veneration. The name derives from the word ‘Bethel’ as found in the biblical narrative of Jacob and the House of God he marked with a sacred stone.

Other examples of geomancy within the formative civilizations of the Near East in antiquity include the siting of the entire city of Heliopolis*, a major religious centre named after, and dedicated to, the Sun , across the Nile from Giza, set in a plan aimed at reflecting the presence of relevant star constellations on either side of the Milky Way; in this way creating a mirror on Earth of the heavenly dispensation above. This perspective of viewing the world as a ‘tessellated’ chequer-board upon which to mirror (and thus resonate with) celestial patterns has been present throughout the history of humanity, even in the modern era.

*Meaning ‘city of the Sun’, or it’s native name of ‘On’ from ‘iunu’; ‘the Pillars’, (or possibly from the Sumerian An-u, ie ‘heavenly’) this meaning may be linked to ‘obelisks’  (or ben-ben stones also) representing the ‘male’ energies of the Sun. So the king of Israel Solomon is said to have been so called in homage to the Sun, his name including three words for the sun, or celestial light; sol, aum, and on.

His relationship to the Queen of Sheba (the ‘Queen of Seven’) represents the (alchemical) marriage of male and female energies within the self, or within reality. The first mention of the number 666 in the Bible is in connection with the amount of gold sent to Solomon for building the Temple. The number signifies the energies of the sun according to writer John Michell in ‘The Heavenly City’, while gold itself is the metal taken as representative of the sun throughout history. The deity worshipped at On was the sun god Atun, or Ra.

To return to our study of the symbol the Vesica Piscis, one feature of this which presents itself here, is the close proximity of the figure 43.3 (BD) to the Cosmic number 432…and likewise of 86.6% (BF) to the significant number of 864. It is fair to say that this proximity of values is of some importance/ relevance, and is one indication for the prevalence of the importance of 432 and 864 in geodetic and astronomic, even energetic terms.

 One fascinating link here is brought to light when examining one of the most important measurements of 864,as found in our solar system; as we have seen, the diameter of the Sun has been measured at 864,000 miles for over a century. But recent measurements by the SOHO satellite have put the distance at a more exact distance of 864,900 miles, or 865,000 thereabouts.

 Thus linking both 86.6%, (√3/2) and 864 harmonics to the size of the Sun, and its related geometries and harmonies such as the speed of light…

  * So for example, we can see from the following some indication of the complex but harmonious relationships between the material reality of our Sun-Earth-Moon system, and the ‘cosmic’ geometries of Pi and Phi;

(Indeed the possibility that our solar system is representative of a highly unusual harmony, rather than randomness, in material terms because of the higher energetic forces at work within the system…)

 –>    864,900 / 7920 / 3.14159/ 1.272019260 = 27.32 732711**

 Or in words;

    Diam. of the Sun / Diam. of Earth / π    /   √φ  = orbital period of the Moon around Earth.

This value, 27.32 is also the size of Moon to Earth’s diameters, percentage-wise) an amazing coincidence.

( 864 -and 432 –  are ‘coincidentally’ numbers central to the Sun, and the speed of light; and as the diameter/radius difference is 864,000 and 432,000, so the 86,400 seconds in a 24-hour period, divide into 43,200 in the seconds of ‘daylight’, and 43,200 seconds in ‘night-time’ of the diurnal cycle…

Indeed, the number of years it takes for the circle of the polar axis of the Earth make the Sun appear to travel (precess) through the houses of the Zodiac (in a backward motion), arriving at its starting point in the stars relative to Earth – the Precession Cycle – is estimated to stand at around 25,920 years; and if you multiply this number by 33.33 (ie.100/3), then 864,000 is obtained; another example of just how many levels and cycles and energies are linked by this  864-866 value.

  Yet another intriguing relationship is that of ‘e’ – the constant called ‘Euler’s constant’ – a value of great mathematical importance, and another irrational value. The simplest equation links it to Pi and the 864(5/6) harmonic;

         ->             e /Pi  = 0.865 2

    (2.7182818 / 3.14159 26535 = 0.865 2 55979)  

Considering all the ways in which 864 and 432 are found to be significant in the measures and proportions of the Sun:Earth:Moon system, plus the speed of light etc, it is beyond coincidence that 864 is the number the counting system introduced by the earliest civilization (known of the post-deluge era) gave the length of the day, one daily revolution of the earth in its orbit around the sun – through using the base-60 system …ie 60 x 60 x 24 = 86,400 seconds in a day.

  In fact with this first numerical system of the circles of time and space (ie spherical  geometry)  being shown to be for most of civilization’s history as ideal for it’s purposes – of linking time and space, and the heavens and earth in their overlapping cycles –  the (historical)  assertion that the wisdom of Sumer came directly from what may be called ‘cosmic  sources’ /’visitors from the ‘heavens’ looks to be more and more likely with time.

Incidentally this perspective concerning the advanced nature of the Sumerian civilization lends some substance to the narrative to Enoch, in the Bible and in The Book of Enoch; for the assertion that he was visited by ‘angels’ who took him on a journey through the levels of the cosmos to receive God’s blessing, and then be given the task of writing down ‘the wisdom of the heavens’ for the usage of mankind once the foreseen Deluge has passed, fits the appearance of the first post-Deluge full civilization, Sumer  c.4000Bce with its ‘celestial’ leaders providing mankind with systems of agriculture, technology, craftsmanship, metallurgy, (ziggurat/ step-pyramid) temples, city creation/ architecture and building technologies, the institute of kingship, writing and mathematics (from pictographic to cuneiform around 3400 Bce) poems and (‘grail quest’) epics such as Gilgamesh, music and stringed instruments, a legal system, religious system and so on… what is also highly interesting is that Enoch describes the 200 ‘fallen angels’ who begat the Nephilim as ‘each teaching mankind’ various forms of divination, technology or wisdom, thus making the Deluge necessary; and yet this is essentially what he passed on to mankind in his tablets for rebuilding after the Flood. Similar to the serpent in Eden telling Eve if she and Adam eat of the Tree of Knowledge ‘they shall be as gods’…  (and Enki/Ea/Oannes/ FuHsi; are all versions of the hybrid-being/serpent  fulfilling the role of  ‘givers of wisdom to mankind’) – and then being cast out of Eden for the Fall. A confusingly entwined set of positive and negative meanings… and indicative of a complex understanding within the narratives, of mankind’s guidance by early cosmic beings.

  So with increasing understanding, the argument that all the immense developments of Sumer were simple evolutions/ intellectual developments attained by the first ‘societal’ grouping of mankind, c.4000 BC, begins to look increasingly unlikely.

(More on the Sun/Earth/Moon harmonics, and the importance of 864 and 432 later.)

 ** Another point of interest, from the cosmic # perspective is the relation between pi and the square root of phi – (which in squared circle = 1.272 >< 1.273;

                                 3.14159 /2     /  1.272033          =     1.234 8.

π/2     /       √φ                 =   ~1.2345

          in fact  –   π / 1.23456789 = 1.272 345 O362 !

Again reinforcing the significance of 1.234…9 etc, as a harmonic value of 10²/9²  – and relating to its square root (1.1111) resulting from 10/9…

 – both numbers 123.456 , and 111.1 (and  111.81; √3/2, and 1.6181-0.5) appear with great regularity in equations and geometry where Pi and Phi are present, for reasons unexplored really yet by mainstream mathematics.


Chartres floor-plan (left), and photograph of the open central spaces of the cathedral (right);

It is the view of many architectural and Sacred Geometry writers and researchers that the derivation of this equilateral triangle, and likewise the two equilateral triangles that form a rhombus, was the key to the design of many of the cathedrals, monasteries and churches of the middle ages, from this central point outwards. For example at Chartres (diagram from Strachan, Chartres Cathedral), where the centre of the cathedral is sited over the vertical and horizontal axis meeting point. And in keeping with the inner geometries of the Vesica the height of the vertical axis is √3 to the width; or 1: 0.866 for the height of the circle to that of the vertical axis. So a circle of radius 100 units, and diameter 200, will have a vertical Vesica axis of 173.2, which is 86% in proportion. The centre point of the site is built directly over the crypt where a subterranean stream runs through, and where a sacred chamber was dedicated from the megalithic era of the druids and the Carnutian tribe, according to various histories of the site.

Likewise later Roman and early French versions were built over this omphalos point, each being replaced through the centuries, until the 12th century when the Gothic version was built by clergy and architects associated with St Bernard of Clairvaux, the Benedictines and the Cistercians. These in their turn had connections via Bernard, Hugue de Payens and others to the Knights Templars, who gained their understanding of sacred architecture from Islamic scholars when they were based in Jerusalem at the site of the Temple Mount, and the Dome on the Rock; and to three religions of Judaism, Christianity and Islam a world navel (omphalos) point connected to the higher dimensions. 

SIDE-BAR – In what is most likely coincidence, the Ad Triangulum basis of the rhombus formed by two equilateral triangles finds a match in Lagrange gravity points found by the mathematicians Leonard Euler, and ten years later by Joseph Lagrange in 1772, as they studied the three-body problem of gravity, particularly in which two gravitational bodies are far larger than the third. These points in space are also used as points of stability with regard to equilibrium of gravitational forces between two orbiting bodies, such as the Earth and the Sun to enable enhanced observations.

So three of the 5 (‘libration’) points lie along the axis between the two bodies’ centres-of-gravity. From the diagram, L¹ is between the Earth and the Sun around 1 million miles from Earth, is currently occupied by SOHO, the Solar and Heliospheric Observatory (and Deep Space Climate Observatory) satellite.

At the L² position, also a million miles along the line away from the Sun, the James Webb telescope has assumed station there, around a million miles from earth. (This replaces NASA’s WMAP satellite measuring cosmic background radiation of the universe). Points L1-3 are still by and large precariously balanced so that continual minor adjustments have to be made to their positions; L4 and 5 are stable ‘like a ball in a bowl’ according to Neil deGrasse Tyson (quoted at

As can be seen, these symbols all have similar geometries; from the Lagrange points in space at which the gravitational fields between the Sun, Earth and Moon are in some form of stasis, meaning such zones can be used to ‘anchor’ satellites, such as the James Webb telescope; the architectural plan of Chartres cathedral; the inner are of the Vesica Piscis, one of the most important symbols of cosmic number, and ‘sacred architecture’; and from Sumerian cylinder seals and stelae, the symbol for the palm tree, and fruit, denoting that which is sacred or ‘of the gods’, and of the essence of life. The Great Pyramid, as one of the finest examples of such in the world, is based closely upon the same ‘double-triangle’ rhombus of the Vesica, as the diagrams below shows.

That the measure of the Pyramid make it 1/11 of a mile in height, and in ratio of 1: 43,200 to the polar axis of the world is proof in the minds of many of the high, (even cosmic) consciousness of the designers of the edifice; an incredible circumstance which would be hard to believe were it not for the many amazing features of the building’s design and construction and location, which few structures in the world can match.

 The Greeks, Egyptian, Hebrew and the Indian cultures all considered this concept to be important, and termed this centrality a navel  – or ‘omphalos’ – whether of the region or the country or the ‘world’. This manner of basing all proportions and dimensions of the churches, etc on a geometrically derived centre-point ensured that the ‘inner’ geometry of the building(s) resonated to the ever-evolving proportions of the non-rational √3, (or other like roots) rather than the ‘lifeless’ full integer proportions, such as 4/7 or 5/8 , which are fully definable.

Such an attitude may seem arbitrary or without sound basis, but it is worth remembering that these key tenets of architecture were applied by the Sumerian, Egyptian, Indian, Judaic, Greek and Roman civilizations…

Also, as we shall see, these square roots are closely entwined with many other mathematical forms and functions, such as Pi and Phi  in a way that few others are…indicating potentially the main reason for their usage.

So Gordon Strachan explains that like Vitruvius, (and as Feng Shui proponents in China still do), the architects would locate the navel-point of the site – often directly over a running subterranean water-course, which provides a connection with natural ‘infinite’ forces in like manner to the irrational roots in maths – and then design the building outwards to the desired orientation.

In doing so their designers were placing them upon significant energy points within the country itself – or in each nation/area’s own ‘omphalos’ where energy flowed upwards from the Earth’s energy fields and into the building – thus connecting the Earth, the temple  and its occupants in significant ways.

  Curiously, the location of the navel was often determined with reference to two central features; firstly the usages of the church builders of the past; the first Christian church at Chartres c6th century Ad for example was built over the remains of what had been a Temple of Dionysus –  and that Chartres was a centre for worship/veneration of Mary in the first millennium; so indeed was a sacred site for already more than a 1000 years even by the year 1256. Notre-Dame de Paris, just 50 miles away, (said to have been a Temple of Diana in Roman times, then of St Denis), was consecrated in 1260, making these two very much products of the same time and possibly craftsmen and builders.

The siting of the church, or temple was most likely – perhaps from the period of the first neolithic sites across Europe and the world – very specifically aimed at a certain place upon the Earth’s surface, relative to the country and the entire globe – meaning there was possibly some form of consciousness of a world-wide energy grid at some level of design…

In the case of Chartres, it is a fact that directly beneath the navel of the site, the central crossing of the church, there is an underground chamber or ‘grotto’, through which runs – a stream. In fact it is considered that even in pre-Christian times the druids  believed Chartres to be a highly special site, giving worship in the underground grotto that held the ‘sacred spring’.( website) Within this grotto since the cathedral was built has been kept one of the widely known ancient carved-wood icons, one of the ‘vierges noire’, the Black Madonnas, possibly so named because of their surface appearance. The placement in this ‘arbor’ indicates how central to the original architects the siting of the church over an underground water course was held to be. (See left – Black Madonna painting of Czestochowa, Poland).

The reasons are debated in many areas of study, but it is possible to conjecture the following analogy –  As the transcendent square roots and numbers are held to be connected to higher dimensions because of their method of ‘unfolding’ ie in their infinite nature, likewise an underground water course adds energy to a site in a way explainable in terms of the natural flow of energies through the earth’ surface via ‘dragon’ lines…

So in a way a water course brings a sense of energy flowing  to a site or special building, in a continuous way. Rather as the umbilical cord does between the mother and the child within. Hence the term ‘Omphalos’/’umbilicus’ for such central sites in Egypt, Greece, India, antiquity-era Europe and so on… 

  NB – from a Gurdjieff perspective we find in the birth of a child from its parents the beginning of a ‘new octave’ – a doubling of the life-energies from each of it’s parents. And so the central phenomenon of the (umbilical) cord – feeding the baby in the womb is highly descriptive of the uniting and creation of new octaves of life…

Incredibly the Black Madonna in the shrine beneath the cathedral, the grotto with the running water source, is called Notre-Dame de Sous-Terre; ‘Our Lady of Under the Earth’ personifies many of the themes discussed in this and other chapters – as shown in the Etymology section, the first deity who was a hybrid ‘man/fish’ or serpent was Enki/Ea, ‘he who loved water’. Much the same as Oannes,  the ‘hybrid’ Annedoti* leader described by the priest and historian Berossus in 3rd century Bce Babylon.(* ’the repulsive ones’) One of the main symbolic epithets of Enki is Lord of the South; meanings attached to this include; the southern hemisphere of predominantly oceans; mines and metallurgy; the depths of the earth and the underworld; as ‘the tripartite division of the world to Enlil, Anu and Enki represents mind, heart and stomach in Sumerian mythology, ‘south’ represents the (subconscious and reptilian) centres of the stomach, concerned with survival, reproduction and skills of crafts/technology; and the ‘abyss’, the ‘ap-su’, which represents both the (cosmic) telluric forces within the earth, and those of deep space also, which is ‘the abyss’ in it’s fullest meaning. So the incredible part is not the close links between the Black Madonnna, the underground grotto, the water source and the cathedral overhead. (like the ‘well of seven’ Bethel in the Bible, an omphalos/ higher dimensional gateway located/sanctified by a marker stone/ edifice), but that the word ‘sud’ meaning ‘South’ has links to words used today like subterranean, subconscious, and sub-marine (symbolized in the Bible by Tiamat, and other ‘sea-monsters’ which link to the reptilian subconscious among other things). The ‘underworld’ is a region overseen by some of the Sumerian deities including Enki, Ningishzidu and Dumuzi/ Tammuz, and while dark, is not symbolic of evil or ‘the devil’ as such, but a place where the waters of ‘the abyss’ which fertilize the earth are stored. A concept replicated in the Foundation Stone of the Temple at Jerusalem, placing the church over the centre-point of the Earth where the waters of the abyss well-up.

  The darkness is that of the subconscious and instinctive centres – as is supported by the fact of the subterranean forces present in underground caves etc linked to the Temple of Apollo at Delphos, Greece; home of the Pythia, the female ‘seer’ or priestess who foretold destinies and saw the essential characters of those consulting the oracle. The (hidden) healing energies present within the subconscious of the instinctive centre are also represented by the Black Madonna; and the reverence for the Madonnas and for Mary in Christianity around the world show the female nurturing aspects of divinity are sought out by women, and men, of all types. And of course, the Black Madonnas are very nearly always depicted holding their newborn child in their arms. Interestingly, many of the earliest Madonnas, or representations of the Blessed Virgin Mary were called ‘Our Lady of the Pillar’, dating back as early as the 4th century. While various explanations exist, it may have some inherent connection to the already mentioned Egyptian city of Pillars; Heliopolis, ‘city of the Sun’ – considering the Black Madonnas may be seen to be linked in nature with the powers of the Sun, explaining the words of the female narrator of the Song of Solomon; “I am black, but comely”. These themes may be seen as uniting the subterranean conduits (as at Chartres ‘grotto’) with the solar energies of the world which flow through them; the ‘marriage of male and female energies’.

So Chartres absolutely enshrines many of the themes discussed in these pages, on many levels; and is also one of the finest expressions of sacred architecture in Europe, if not the world; according to Strachan, (and many other writers) a sense of the sublime permeates every detail of the cathedral.

One related point is the relationship of the oak tree to sacred sites, and myths of many cultures (such as the Judeo-Christian, Indian,Chinese, Siberian/shamanic, druidic Celtic, Greek, Roman, Meso-American, Norse, Baltic, and Slavic cultures) related to the ‘world tree’/axis/Yggdrasil; the curiosity is the relationship between the oak – underground water (of life) – sacred sites (symbolized by ‘foundation’ stones) – and (in the Tarot) the lightning bolt or (in the Bible) the fire of the heavens…

There are some telling instances in the books of the Bible, where these relationships are linked metaphorically; so there are multiple instances in which can be seen;  water – trees – stones – sacred sites – angels/ the heavens.

This subject has been studied by authors such as Sir Norman Lockyer regarding the ubiquity of sacred springs, streams, and wells throughout the ancient world. Often he notes, these springs and groves had a revered tree upon which small ribbons or pieces of paper or cloth were hung, as prayers or marks of worship .

In emphasizing the presence of a stone the Bible’s writers were perhaps alluding to the presence of sacred buildings located at these ‘umbilical’ sites throughout the world. The square/cubic nature of the stone, and the axial nature of the tree in the narratives of the Old Testament link the material plane with both the energies represented by water, and by the ‘fire’ of the angels.

So with Jacob, dreaming of a ladder to heaven, it is marked by the stone he sleeps with his head upon. After a night of dreams and visions, he calls the place Beth-el ; the ‘house of God’.

GENESIS  28;11-19

(and Jacob)…lighted upon a certain place, and tarried there all night; and he took of the  stones of that place, and put them for his pillows. And he dreamed, and behold a ladder set up on the earth; and the top of it reached to heaven; and behold the angels of God ascending and descending on it. And behold, the LORD stood above it, and said, I am the LORD God of Abraham thy father, and  the God of Isaac; the land whereon thou liest, to thee will I give it… And Jacob awakened and he said, Surely the LORD is in this place; and I knew it not. And he was afraid, and said, How dreadful is this place! This is none other but the the house of God, and this is the gate of heaven. And Jacob rose up early in the morning, and took  the stone that he had put for his pillows, and set it up for a pillar, and poured oil upon the top of it. And he called the name of that place Beth-el…

 GENESIS 35.13-15

And God went up from him in the place where he talked with him. And Jacob set up a pillar in the place where He talked with him, even a pillar of stone; and he poured a drink offering thereon, and he poured oil thereon. And Jacob called the name of the place where God spake with him, Beth-el.

  Just prior to this;


And he built there an altar, and called the place El-beth-el; because God there appeared unto him… But Deborah was buried there, and she was buried beneath Beth-el under an oak; and the name of it was Allon-Buccoth (the Oak of  weeping/ remembrance). And God appeared unto Jacob again…and blessed him.

  There are several other examples repeating the metaphorical themes involved;

* JOSHUA 24;26

And Joshua wrote these words in the book of the law of God, and took a great stone, and set it up there under an oak, that was by the sanctuary of the LORD.

* JUDGES 6.11. And there came an angel of the LORD, and sat under an oak which was in Ophrah. 12. And the angel of the LORD appeared unto him (Gideon), and said unto him, The LORD is with thee, thou mighty man of valour.

 * ELIJAH / angel / juniper tree (1Kings 19.4).    

The use of the metaphor for the world tree/ axis/ spine in the Bible of the oak tree is made even more curious by the fact, proposed by Thomas Saunders and others that oak trees are hit by lightning in a far higher proportion than there are oaks to other trees. This is possibly explained by the fact that oak trees having a higher water-level within themselves than most tree types, and grow better in soils near to water-courses, whether surface or subterranean.

A point of interest is the semantic relation of the oak tree, as mentioned, to the ‘world tree’ or ‘axis of the world’… indeed oak and axis (and ox, as creature of strong backbone) have the same etymological root. And from there it is a straightforward step to state the allegorical intent of linking the physical world to the human sphere – where as we shall see, the spine or backbone plays a central role in uniting the spheres of energy and matter – or the spirit and body as it may be said…


 In the etymology section also this book examines the linguistic connections between oak/ox/axis (all representing the strength of the spine/ backbone/ world-tree), and the octave (representing the doubling of energies within the self, through the 7 energy centres called the chakras), and so on. And modern anatomy confirms the importance of this part of the body, as the lower rear part of the skull is the Occipital bone, covering the occipital lobe of the brain, at the point where the brain stem enters the cranium. As the law of seven demands, the occipital bone is 1 of 7 bones that fuse together through adolescence to form the skull in the adult human!

 Chinese Tai Chi wisdom likewise labels the part of the Microcosmic Orbit at the back of the head and the top of the spine the Occipital Gate. And as the Egypt chapter will show, there are clear indications within the internal structures/ geometries of the Great Pyramid that the designers knew of the (symbolic?) importance of this point; and that likewise, Giza’s equivalent geographic position was by no means an accident, but comes at the exact point where the ‘spine’ of the Nile blossoms into the ‘brain-stem’ of the Nile-delta…

Etymology sites state the word ‘occiput’ is formed from the two Latin stems ‘ob‘; behind, and ‘caput’; head. This site obviously considers the etymology given as being more accurate. So for example, the Latin word for bone – oss – is considered to have evolved from the S/ABA and PIE languages basis ak*– from which ox, axis, oak and octave all derived. Thus meaning bones are seen as ‘points of arising/origin/ centre/ axes of’ the body upon which muscles, tendons, skin etc are situated. PIE shows the links between Sanskrit of India, and European languages, including Germanic, Greek, Baltic, Slavic, Celtic.

* ‘to do; to place; to make something with’ ie origin/ source/ framework/point of growth.

That bones are seen as a connecting axis, is reflected in Cos# interpretation of the number 8, (Latin – octa); for it is situated exactly between the number of the ‘divine spheres’  (12) and that of the material world (4).

In this manner sacred architecture has the example of St Pauls cathedral, where the spherical dome is connected to the rectangular body of the church by an (internal) octagonal structure; one apparently so important that Sir Christopher Wren took great pains to ensure was constructed correctly. (See Adrian Gilbert – Building a New Jerusalem).

In physiological terms the spine is the key information superhighway through which all senses and inner stimuli are channelled, In the form of electrical nerve impulses to the centres of the brain – which in return govern the vast complex of tasks which must be sent down the spine en route to every single cell in the body…

This miracle of complex functions and constant signals occupies an amount of impulses per second estimated to be c.100 million every second of our lives… so whereas we can go 3 weeks without food, 3 days without water, and approximately 3 minutes without air, we cannot go 3 seconds without ‘impressions’, as Gurdjieff called instinctive-centre stimuli or information.

(If we invert this ‘mnemonic’, we can begin to see the links between ancient ‘wisdom/ philosophy’, and modern psychology/ physiology.

 For we find > food   – EARTH

                     water  – WATER  

                     breath – AIR   

         nervous system – FIRE.

So the spine occupies a central often overlooked role in our life – and bears equivalence to the axis of the ‘world tree’, which is rooted in the physical earth, and reaches to the heavens.

As we shall see later, Chinese philosophy and martial arts placed great emphasis from the earliest beginnings on these concepts – so the words Tai Chi themselves can be interpreted as ‘supreme heaven-earth pole’. (Likewise ‘duran-ki’, one of Sumer’s central cities, meant ‘supreme heaven-earth bond’, highlighting the links between Sumer and China again).

It is not coincidental that the (ancient) practices of Tai Chi aim to always strengthen the practitioner’s spine, through gentle exercises, good posture conditioning, stretching, conscious awareness, massage, acupuncture, and so on. The spine is seen to be central to our ability to create, and receive the cosmic energies of life. To support this point of the desirability of perfecting the stance and posture is the Chinese saying;  ‘A temple is built from the foundations upwards.’

SIDE-BAR;  And this of course refers equally to how TCMA build the ‘temple of the self’, the body; the Chinese model of self-development differs essentially from those long-prevalent in Western nations. Western notions of strength have often focussed on building the muscles of the upper torso, the arms and shoulders and chest. The posture of chin up, shoulders back, chest out, stomach in and legs straight may give the appearance of strength, but is virtually the opposite of what TCMA consider to be good practices.

As said, the Chinese/Tai Chi model encourages growth from the ground up. Why? Firstly, it is to ensure the proponent is well-rooted, as our connection with the ground begins from the feet upwards. There are energy centres for Qi within the crown of the head, the palm of the hands, and the soles of the feet (called ‘bubbling wells’ in the feet). So every moment we are absorbing earth energies into our system via these centres; and putting used/ refined energies back out into the ground. The well trained conditioned Tai Chi adherent thus aims to be as ‘well-rooted’ on the ground as  he or she can be. This ensures optimal energetic exchange, for health, plus in combat situations/performance of martial arts ‘forms’ the person has a much more complete ‘connection to’ the ground. And this is a reality that is achievable to martial artists with years of practice and growth.

 Further emphasis on the strength of the legs comes from keeping the knees slightly bent at most times, the shoulders down, and the chest relaxed and ‘open’. All these things enable a much lower centre-of-gravity, as well as much stronger legs (hips/ thighs/ knees/ calves/ ankles/ feet/ toes !) The spine is held as straight as possible over this lower stance, while the shoulders are marginally curved forwards (rather like a coat hanger.) From this breathing is open (the chest/head pulled back Western model reduces breathing capacity/ability) and deep and relaxed (while the straight spine helps maintain a free solar plexus area for freer breathing at the top of the stomach…)

The relaxed breathing and stomach, at all times enable movement in all directions, the stomach being the centre-of-gravity in each of the 3 axes of movement. And not coincidentally, the stomach (centre of gravity of the body) is where the foremost energy centre of the body is situated, the Dan Tien, meaning the Elixir (Qi) Field (centre).

 The body is thus ideally (well) rooted , strong in all parts of the legs and hips, with stomach chest and shoulders relaxed and ‘open’  and chin slightly down.

From this, the muscles and ligaments etc are relaxed and ready for action, and likewise all the energetic pathways of the body are free-flowing and open, meaning potential is optimal and ready.

 And contrary to older Western notions of muscular effort and tension, the body has the greatest power (and health) when it is performing in a state of (steady) relaxation. Even punches become stronger (counter-intuitively) when the arm and hand are as relaxed as possible…

Thus affirming the saying – ‘Working without strain is not the same as working without effort.’

 Hence a lower stance – and a strong ‘foundation’ – is continually strived for in Tai Chi, because of the benefits of good posture, energetically and physically. More on this later in the year.

To begin with the location of the Great Pyramid, and the entire site at Giza, this is located at N 29.979°N… in other words the site is little more than a mile from the 30th parallel. Giza is based on a layer of bedrock which highly unusually is of level disposition; and while being particularly thin is still the ideal foundation for the siting of the Pyramid complex, which occupies several miles square, providing a stable and strong foundation for the complex which has proved to offer protection from the dangers of earthquake, subsidence, or erosion.

(There is one ‘hillock’ or ‘imperfection’ in this level bedrock, to be found in about a 30 ft rise of the rock towards one corner of the site. And the builders of the Pyramids used this as the base for the pyramid of Khafre itself, turning a ‘problem’ into a solution, namely in anchoring the pyramid firmly upon and around this rise…) As Graham Hancock notes, in the building of this pyramid to the four cardinal points of the compass, to high degrees of accuracy, both in terms of orientation and side lengths, the most efficient way of preparing the proposed walls and stone courses would have been to put cord lengths crosswise from opposing corners, ie in an X style of layout….

But quite how they could have accurately done this with a large mound, to the height of a modern house in the centre of the site, remains as one of any number of unexplained achievements in the building of the pyramids.

To return to the geography of Giza, two points arise in relation to this latitude. Firstly, as many observers have noticed, the site at Giza occupies a unique position in the map of the world – and for this reason of its physical centrality, as well as its immense stores of knowledge contained within its design, features and construction –  many have argued from the 19th.c onwards that Giza would be the best place for which the ‘zero’ degree line of Longitude for the planet should be sited –  rather than Greenwich England, which has been the ‘prime meridian’ since 1884.

    And as Colin Wilson and fellow researcher Rand Flem De’Ath argued convincingly, when this line is sited upon Giza, an entire network of sacred sites around the planet are found to be in harmonic alignment with Giza – indicating the existence of a planet wide energy grid which disseminates it’s (life)forces in mathematically resonant/ meaningful ways…

An interesting corollary of this is the following point; upon dividing  an equilateral triangle as in Ad Triangulum, if we draw lines through each apex to the midpoint of the base we have a mid point which extends in 6 directions. This can be called the midpoint of the 3 dimensions, and further to this, may be viewed as the ‘seventh point’ which unifying point of the three dimensions.

The geometries of the cube, as contained within the hexagon (a), is the keys of St Peter (b) which are effectively identical in outline. Similar geometries can be found on the markings of many ambulances in the world (c).

 The symbol for medical aid is based on the Rod of Asclepius within the star figure (c), and the Caduceus (d), two serpents twined around a central axis. This geometry is a longstanding representation of the expression of energy through matter… (the central axis may be read as being the spine, with the cross-over points representing energy-centres or chakras, and the two wings being the two hemispheres of the brain. The top of the staff is symbolic of the ‘third eye’, or the pineal gland).

And of course, the caduceus was first invented or drawn not by our civilization, but dates back to Greek and Egyptian usages, particularly of ‘asclepions’ – places of healing. Asclepius, the Greek mythical deity of healing who was the ‘son of Apollo’ and a mortal woman, and delivered by ‘Caesarian’ in myth, was represented by the ‘rod of Asclepius’, a staff held vertically with a serpent entwined around it; much like the Nehash created by Moses in the Bible. The Caduceus created also probably by the Greeks in the first centuries Ad, is the emblem of Hermes/ Mercury, and relates to astronomy, alchemy, and commerce/ communication. Previous, and possibly related symbolic representations of two serpents intertwining around a central axis similar to these exist in Sumerian, and Chinese history; Ningishzida in Sumer, FuXi and NaKua in China.

The second point regarding the siting of the pyramids practically on top of the 30th parallel North, is the following piece of information; this is the relation of the latitude (the number of degrees angle from the Equator at which the site is placed.) to the Speed of Light. For the latitude of the Giza site to the degree of accuracy given is an exact match ;

⇦  The apex of the Great Pyramid, Giza lies at 29.9792°N / 31.1342°E

                               and the speed of light = 299,792,458 m/second.

Many people consider this an inevitable coincidence, considering the immense number of coordinates created by the network of pyramids across Giza and Egypt… but it can be argued that considering the geometry involved in the design of sites, at the ‘navel-point’ of a nation indeed mid-point of the continents of the world, and the complex geometries within the pyramids, argue well for a (uniquely) high level of consciousness and cosmic knowledge and intelligence – in the siting and designing (and building) of these incredible works of technology and art.

There are a great many theories concerning the encoding of advanced intelligence and science within the measurements of the pyramids, particularly the Great Pyramid. While finding many points of interest, and potential study in these details of measure and proportion, it is a subject best approached with a sceptical and open mind; the Victorian astronomer and scientist who studied the Gt Pyramid with the most accurate scientific measures known at that time, Charles Piazzi Smith, found it needful to caution against finding more meaning in the said measurements than reason could satisfactorily prove -while it is clear that there are many aspects of the Great Pyramid, and Giza which are effectively beyond all current understanding of science. See the Egypt section for more details on some of the incredible features built into the Great Pyramid and complex of Giza.

 So,to return to our map of the world’s landmass, we can trace the centrality of the site of Giza using this ‘centre of the 3 dimensions’ symbol – the centre of the cube; the Egyptian creation of which bears some indication that the designers of the pyramids and the complex at Giza knew the geometries involved in energy transmission, even within organic life, and the human frame – and of the geography of the Earth… while the Great Pyramid itself was built within a geometric complex oriented precisely to the cardinal points of the compass, to the rising of the Sun, and to certain constellations and stars, and in proportion to the measurements of the Earth, particularly  1 ; 43,200.

    The amazement of some observers and students at the massive amounts of cosmic intelligence  and wisdom contained within the Great Pyramid site is surely to be forgiven, despite causing some to see more and more incredible ‘meanings’ within the geometry of the building(s) and the site… it is an amazing work of science, art and cosmic consciousness, the full understanding of which has most surely not yet been found…

  There are a couple of points regarding the siting of Giza on the 30th parallel which are significant.

Firstly, if we examine measurements of latitude and so on, we find the following –

               EQUATOR;     Dia. 7960 m            Circ.mce    24,883 m

                                     ”  12814 km                  ”      40,258 km                                                

                     3O° N ;   Dia.  6845 m.           Circ.mce    21,506 m

                                    ”  11,089 km                   ”      34,600.2 km

From this a couple of interesting facts arise. Firstly, the figure of the circumference of the world at 30 degrees North is of interest, at 34,600.2 kilometres length; we have seen how 3456 is a cos# value related to 432, 864, 1732 and so on. In the Egypt section we see how the Grand Gallery, the (upper) ‘spine’ of the inner structure if divided into four levels; basement (instinctive body) / ground level (stomach/ physical level). Queen’s Chamber (heart/ emotions level), and King’s Chamber (head/ mind level)… and the measures of the Gallery are significant too, so that it’s maximum height is 288 inches, or 34.56 feet. Another example of the Cube symbolism examined already in relation to Giza’s position at the centre of the world is reflected in the Grand Gallery’ ‘mi-fa’ line on it’s wall (designed with seven levels representing the octave)/ the centre axis of the Pyramid/ the North facing star shaft all crossing each other in the same pattern, at what is clearly a key symbolic point within the entire edifice; as can be seen, the centre line passes from the basement area up through the exact centre of the Queen’s Chamber and it’s apex to the point 61.7 inches (symbolic of the Phi value of 61.8 units) from where the Grand Gallery levels off for the Ante Chamber and King’s Chamber.

This value of 345/346 is not just contained within the key measures of the pyramid but crops up in numerous places; indeed, due to spherical geometry, where at the equator,1 degree of Longitude has a length of 69.1 miles/° (24,883/360) (so that 1/720 of the Earth’s Eq.circumference = 34.56 miles, itself 4 x 8.64 – thus connecting all to the 432 / 864 harmonic), at the latitude of 30°, a degree of longitude measures almost precisely 60 miles per degree.

In fact it is 21,506/360 = 59.9 miles/°. Which thanks to the sexagesimal system created by the Sumerian civilization means that 1 minute of longitude therefore equals exactly 1 mile…

Another point to note in this respect is that in kilometres, if we divide the equatorial circumference of the Earth by 360, we obtain a figure that = 111.2 km/° (!) this value again! And curiously the Escape Velocity of the Earth, at  11.186 km/sec;  also close to half √5 (2.236/ 2 = 1.11 81), and the fraction therefore used to obtain PHI of 161.8 as shown in the

Squared Circle section.

One of the most significant measures in the Vesica, is as mentioned, that the height of the mid-line is 86.6% of the height of the circle(s), ie √3/2. As mentioned, the Vesica contains two triangles of equal length on all sides, and equal to each other, and it is upon this the Ad Triangulum system is based.

From this diagram the following measures can be obtained, if the diameter (AM) of the circle = 100, and the radius (DM) = 50 units.;

AB = 50

AE = 25

AC = 28.8

AD = 50

AM = 100

AL =  86.6

AG = 25

Other measures include; GC = CE = 14.4, CD = AC = 28.8, BF = 86.6 (ie, 6 x 14.43), CL = 57.7, GE = ED = 25, BE = 43.3 as half of that, and BN = 100. E to the point on the line between L and M, (or M and N) = 4.5 of the 6 triangles of 14.4 comprising 86.6 units; therefore this = 64.8; an interesting value as 360/ 64.8 = 5.555, or 360 / 32.4  11.111, etc. The one and a half triangles thus equal just over 21.6 units, another cos#, as 216.

It can be seen that as AG = 25 = 86.6% of AC or AE, as a height: hypotenuse ratio of each equilateral triangle, so a triangle of 12.5 units height has a hypotenuse of 14.4 3 units, ie CE.

The hexagon of the circle divides into exactly 6 triangles (equal in size to the two within the Vesica; each of these triangles has 12 sub-triangles within it (12 half-triangles and 6 full ones), making a total of 72 for the full circle.

Each large triangle in the Vesica ABD, BDL, LDM, MDN etc)  can be divided with lines from each corner to the mid-point of the opposite line (BCE, GCD, etc). The midpoint where these 3 lines meet (C) is not at the centre of the triangle; in fact it is at a point where each line is divided into 1/3 and 2/3 sections. So as the height of the triangle is 43.3, it is short work to calculate the length of these divisions as; (GC)1/3 = 14.43, and (CD) 2/3 = 28.86 units. This is the basic unit of the hexagon, and the triangles which form it; and 14.4 and 28.8 are key numbers in the cosmic# field (or 144/288). That each unit is 14.43 is a reflection of the difference between the 86.6% length of the Vesica to the circles, and the 864 harmonic which is also a key harmonic.

Having noted how the circle has a diameter of 100, and therefore a circumference of 314.159 units, this makes each of the six sections of the circle circumference = 52.36 (ie, AB via the circumference, rather than AGB); and 52.36 is the exact measurement of the Egyptian Royal Cubit in centimetres (or 20.6 inches). (And as π/6 = 0.5236 x5 = 2.618 = φ² )…

Showing the basis for the ER Cubit’s length, linking it to both Pi, Phi and the hexagonal and duodecimal geometries we are studying here.

The ratio of 86.6 :100 is closely related to the 864 value which is expressed in the diameter of the Sun, the length of the day (86,400 seconds in 24 hours) and so on. In fact the area between 864 and 866 is a key ‘grey area’ in cosmic# in which various values can inter-link.

With regard to the height length of the hypotenuse of each triangle of 12.5 units height, the 14.43 building block is related essentially to the 144,000 value given in the Bible as the number of inhabitants of the Heavenly City. This figure is an expression of the ‘divine’ character of the number 12, (or ‘duodecimal’ geometries).

Another example from the Bible is the figure 288, which is associated with the Creative energies of the universe ; several writers/researchers in cos# related subjects have asserted the significance of 288, as a harmonic which represents the ‘emanation’ of the Logos, the Divine Word into the universe. So this number is linked to the pulse-beat of the universe; in the Old Testament this is represented by the following symbolism; on the day of the opening of the Temple of Jerusalem built by Solomon, (in 1Chronicles25.7) there are 288 singers (representing the powers of sound and frequencies at the heart of creation). When their voices and music resonate within the cubic geometries of the Holy-of-Holies the room is filled by a cloud (which is called the ‘Glory of God’). The Hebrew religion has extensive understandings of geometries and numerical frequencies/ relationships within reality. And the cube of the sacred space of the Temple is like the hexagon of the ‘Earth’ circle we are studying – viewed in 3D the hexagon represents a cube; (of material reality) upon which rests the circle ‘of the Heavens’.

This is not just the view of Hebrew philosophers; the holy shrine of Islam, the Dome on the Rock is exactly built according to these concepts – in the place where Solomon’s Temple was said to have been built before it was destroyed in the era of Christ by the Romans. And as we have seen, this site is a higher-dimensional gate-way, a navel-point connecting the heavens and earth. So the value of 288 is connected to the cosmic energies which well up through the earth at this precise place on the Earth’s surface.

The writer G.Strachan studies the importance of this value and examines the work of Ernest Maclain; both assess the values of 72, 144, and 288 within the Bible as representing the octaves of the different sections of the Temple, with 288 being the ‘primal pulse of the Universe’, or the ‘Aum’; and the geometric proportions of the Temple’s three main areas represent these three values. Strachan also mentions that the average pulse beat of a human is 72 beats per minute… ie every 60 seconds, (meaning the rate per second is 1.2, or 6 beats per 5 seconds).

The ‘heavenly number’ 12 is  seen in the Introduction section, has been the number expressing totality for the houses of the Zodiac, the 12 tribes of Israel, the months of the year, and so on. The ‘heavenly’ nature of 12 has been also observed in Sumerian mythology (12 planets), Hebrew/Christianity, Greek (Plato and the Platonic Solids, concerning the geometric nature of atoms, or physical reality), and many other cultures.

 In a moment of conjecture, it could be said that for the 12 Tribes of Israel as stated in the Bible, the record maintains the loss of 2 of the tribes in time, namely Manasseh and Ephraim. The curious possible ‘meaning’ contained within this is that the 12 tribes were reduced to 10 – or by a proportion of 12/10, or 6/5…thus indicating the celestial : earth ratios and all the geometries which stem from this interaction. Of what might be termed the ‘gear-wheels’ of the higher and lower dimensions.

As such, it is fair to estimate that the two particular tribes which were ‘lost’ of the twelve – Ephraim and Manasseh – were judged somehow as belonging more to the earthly than the celestial spheres. In Victorian times, many nations and peoples, being closely concerned with biblical matters, attempted to trace the roots of the tribes associated with their early history back to the lands and times of the Old Testament.  So for example, there are theories, not without some degree of possibility, that the tribes of the Scythians became one of the early tribes of Scotland – and the tribe of the Danites became one of the tribes of Ireland – and so on.

Equally, there were study groups in England who claimed the English were the ‘lost tribe of Ephraim’… and likewise some groups in America claimed to similarly owe their patrimony to the lost tribe of Manasseh. How these conclusions were reached is unsure, except perhaps from an interpretive reading of the nature of the blessings to be given to each of the tribes; but both groups had extensive followings, and were taken seriously by their proponents. (see – Adrian Gilbert – )

 Concerned as the Bible was with the encoding of complex knowledge and esoteric/ geometric concepts, it is fair to consider whether this change in the numbers of the tribes of Israel was pointing subtly to the key ‘cosmic’ proportion of 6:5, or  1.2: 1

 The proportion of 10:12 – or 6/5 – or 1:1.2 comes from the conjunction of these ‘heavenly’ with ‘earthly’ number systems. For the ‘earthly’ measurement is done in base-10 maths, as being the most efficient ‘base’ for these physical dimensions. The figure of the 4/3/2/1 pyramid – the tetractys – is the best antique symbol of the ‘base-10’ reality; and significantly, is found in the stela we are already considering, the TREE OF LIFE – for to the right of it, a ‘god’ inserts a triangular/pyramidalsymbol into the neck of a human from behind…into the spine, in other words, at the meeting point (the neck) between body and mind..! This is a stunning image, and one of the most important from the ancient world which we have today. Even in antiquity it was considered as one of the key expressions of cosmic guidance or even creation of human consciousness – and as such, was ‘faithfully’ copied across the Sumerian/ ABA cultures, so that there are literally hundreds of different stelae from different places and times of the pre-Christianity millennia which reproduce this image, with greater or lesser artistic ability or ‘consciousness’…

Thus, to the Sumerians, Hebrews, Egyptians, Greek, and more, the harmonic of 1.2 (which = 6/5) is of mathematical significance. And indeed, we find that the relationship between Pi and Phi is expressed clearly as;

                        Pi / 6   =  0.5236 (E.R.C)

                              x 5 =  2.618  (φ²)

                  ie, π / 1.2 = φ²

           Or 1.2 x 2.618 = 3.14159

  There are more 6/5 harmonics throughout the most significant numbers

obtained in ‘ cosmic’ number, (such as 51.84,61.81 (φ),90,144,173.2,216,360,432, 864,

1728,3456 and so on.)

     So for example the circle quadrant,

    90   x 1.2 = 108  

   108  x 1.2 = 129.6   (see 12,960,000 period of time in Hindu yugas)

129.6  x 1.2 = 155.5   (see circles of 11,14,and 15.55)#

            144 x 1.2 = 172.8     (86.4 x 2)

         –  288 x 1.2 = 345.6

         –    72 x 1.2 = 86.4

         –    20 x 1.2 = 24

         –   24  x 1.2 = 28.8… and so on.

Interestingly, 444.444 / 1.2 – 1.37035999206 (Fine Structure Constant) = 369    (a number we see in other sections to be considered a highly significant cosmic# value, by such people as Nikolai Tesla and Georges Gurdjieff). (444.4444 is also 123.456789 x 1.2 x 3)

  And for 173.2 /1.2 = 144.3 showing the proximity of the two values √3, 1.732/2 – .866,  and .864 x 2 = 1.728; /1.2 = 144 = 12²)

  Further, we have      360 x 1.2 = 432

                                432 x 1.2 = 518.4

  • the angle of the Great Pyramid (51.84°), and 72² (5184),etc.

It has also been argued by many researchers that the Great Pyramid fits proportionately in the Vesica as a triangle. And indeed the dimensions of the Pyramid do enable this; and as the Great Pyramid  is in exact 1: 43,200 ratio with the Earth, this fits nicely with the measurement of the triangle of height 43.2 or 43.3, as half of 86.6, the difference is small.

Anyway, the Great Pyramid does bear more than passing resemblance to one of the inner triangles of the set, as we shall look at in more detail later.


 And indeed if we refer back to our original plan of the Great Pyramid, there are some very interesting similarities. Thus the passages from the entrance to the Grand Gallery and the Kings Chamber – in addition to the ‘inner shafts’ beyond the chamber which reach to the inner edge (but not through the courses of the pyramid) – clearly can be said to resemble the inner division of a triangle into 3 crossing lines which meet at the centre.

And from a slightly different perspective of the same central area of the Vesica more information can be derived; if the triangles comprising the Tree of Life diagram (which we have just seen in their ‘harmonic’ aspects) are extended within a circle of radius equal to the height of each triangle – ie 43.3 units – each triangle of the centre area can be continued to the circumference. In doing so a ‘Star of David’ is formed, instead of the ‘diamond’ of the triangles within just the central zone.  And following the fractal nature of the inner triangles, two ‘inner’ stars can be formed within the centre region of the outer star;

This diagram shows some curious features, related mainly to the Great Pyramid; using this template the inner triangles of the ‘diamond’ can be viewed as closely related to the pattern of the structures inside the Pyramid; from the entrance-way a passage leads downwards, to a point where it levels. A distance along this horizontal passage at the centre-line of the pyramid is a downward pointing space – the ‘bottomless pit’ as it is called occasionally. The passage continues to a point where it stops. Going upwards from the entrance there is the first level where there is a horizontal passage, leading to the Queen’s Chamber. Again, the Queen’s Chamber points upwards (the niche) precisely on the central axis of the pyramid. The Ascending Passage continues upwards, in the Grand Gallery to a point where it reaches the level of the Ante Chamber, then the King’s Chamber. There are some slight differences in how the King’s Chamber etc are designed, in comparison to the template, and yet the upward triangle of the innermost ‘star of David’ does bear resemblance to the ‘Relieving Chambers’ above the King’s Chamber. The ‘star-shafts’ may be equated with the centre-line running from the corner of the base of the largest triangles upwards to the centre of their opposite lines; or equally, the lines running upwards across the top of the innermost triangle… thus providing both Queen’s and King’s shafts to both sides. The nebulae of Orion may be relevant to the inner structure of the Great Pyramid, in the resemblance to the ‘well-shaft’ feature of the pyramid, which runs vertically from the subterranean passage-way through the rock and aggregate up to the level of the Queen’s Chamber. Equally, or even more similarly, the inner (right-hand) lengths of the smallest six pointed star in the diagram may be seen to take a course similar to the well-shaft…

One of the key features (examined in more depth in the Egypt section) is the cross-over point of the Grand Gallery, the central axis, and the North-facing ‘star-shaft’ from the King’s Chamber. Lines on the template also may be likened to the other 3 star-shafts of the Great Pyramid.

In all about 14 or 15 points of similarity exist; certainly a theory worth considering.

And there are further possible theoretical connections to be considered from this idea of the design of the Great Pyramid, and the geometric patterns within the Vesica.

For if the fractal nature of the triangles formed within the Tree of Life geometries (the vertical; horizontal; vertical, pattern etc) is studied, it may be represented as a ‘diamond’ which has a peak, three points along the ‘belt‘ at the middle horizontal axis (being the two corners plus the mid-point), and the peak of the triangle underneath. This pattern can then be repeated as the inner triangles are, in a fractal manner. And there is one constellation in the night-skies of the northern hemisphere which displays similarly ‘geometric symmetries’; and that is Orion.

(left) the inner geometries of the Vesica and the hexagon as related to Orion; ie the main outline is comparable to the points B as Betelgeuse – AED as Orion’s Belt – and F as Rigel.

Firstly, in comparing the Pyramid with the Vesica’s geometries, is what is called the ‘Well-shaft’ in the Pyramid; this can be seen to be the same as the line which ‘bends upwards’ from a point along the horizontal basement passage up to the point where the Ascending Passage becomes the horizontal level of the Queen’s Chamber…

In fact this part of the Pyramid has further metaphorical connections; if the entire inner structure of the pyramid is based upon the ‘geometries’ of Orion, this curved ‘well-shaft’ just outside the central area of Orion’s Belt may be considered to be a representation of one of the nebulae which are located within the region of Orion; regions of vast cosmic gas-clouds which gradually coalesce with time, and the forces of gravity, into planets or stars…

left) Orion, showing how the ‘hunter’ Orion was portrayed in mythology, with bow, and with sword hanging from the belt.

(right) Orion with main stars emphasized. Orion’s sword, hanging from the belt contains several collections of stars; the Orion nebular, (M42), the M43 nebula, the Running Man nebula, and individual stars. The wider clouds of interstellar dust and hydrogen gas molecules that form nebulae are visible around Orion (in the picture left), indeed the M42 Orion nebula within the sword is the largest visible nebula in the night sky. It is also a primary star-making nebula where stars are formed from the vast areas of gas as it coalesces into spherical shape, and then ignites; much like our Sun did five billion years ago..

There are some arguments that the complex of pyramids within Giza are related to Orion. Firstly, the theory from the writers Gilbert and Bauval in the 1990’s which caused such an upset in the world of Egyptian studies… they maintained that firstly, the ground-plan of the three main pyramids at Giza was designed deliberately to reflect the line of the three stars of Orion’s Belt whereby the two larger stars are aligned, and the third is off-set to a small degree so that there is a slight variation from a straight line.

This is by no means unreasonable, especially considering the veneration the Egyptian religion of that era had for Osiris, the deity equivalent to Orion (as Isis was to Sirius). The two authors went further in this, maintaining that the star-shafts of the Great Pyramid were designed at such angles as to have been intentionally pointed to Orion and Sirius at the time the pyramids were built.

This was ‘supported’ by the use of computer apps which can compute the night skies position at any time; so the position of constellations or stars at dawn on the spring equinox – the beginning of the solar year in ancient Egypt – were calculated in this way.

But objections were raised, based on the lack of decisive evidence, as well as the fact that some of the relevant star-shafts are not straight lines, but bend and divert along their routes by considerable degrees. Whether the outward ‘line’ of the shafts is made unascertainable by these inner divergences (due mainly to structural obstacles) is a matter for personal opinion really. But Gilbert and Bauval, despite receiving much academic criticism over the years certainly inspired a massive amount of interest and enthusiasm for the modern awareness of Egypt, and the pyramids and temples built there throughout antiquity.

 Was the constellation of Orion so significant in this era? It is certainly mentioned in the Bible, one of only a few constellations named (along with Arcturus, and the Pleiades). So for instance, it is mentioned in the Book of Job at 38.31 as the Lord speaks of the Creation;

‘Canst thou bind the sweet influences of Pleiades, or loose the bands of Orion?’

(It is interesting to note that even then in pre-Christian times  Orion was viewed as being ‘banded’, or in geometrical form). And this neatness or orderliness, as contained within the symbolism of the Great Pyramid  is part of it’s mythology; indeed the etymology of Orion contains within it’s term the Sumerian/Babylonian word for ‘the light of the Heavens’; ‘Ur-Anna’. The Hebrew word for light is even today ‘Or’. As we see in the Etymology section, many words stem from this meaning of light, including the French word ‘or’ meaning ‘gold’. Likewise Latin ‘orire’ ‘arise’ or ‘originate’; and Latin ‘ora’ meaning ‘cord’ – and so on. Even order, as in the divine dispensation, orchestrate, organic and so on are linked to the life-giving powers of the light of the heavens.

And the Bible has another passage which may be referring to Orion; when YHVH casts Adam and Eve out Eden;

 (Gen 3.23-24)

Therefore the Lord God sent him forth from the Garden of Eden, to till the ground from whence he was taken. So he drove out the man, and he placed at the east of the garden of Eden cherubim, and a flaming sword which turned every way, to keep the way of the Tree of Life.

It is by no means unreasonable to consider that the words here relate as much to a stellar setting as a terrestrial one. Indeed cherubim means ‘winged angel’ or ‘hybrid’, ‘winged beings’…immediately relating the siting of the ‘garden’ to the higher spheres… And the sword which turns may be interpreted in the sense that Orion turns in the night-sky, from being level with the horizon at first before rotating clockwise towards the vertical as the night sky changes…for as the three middle stars are said to represent Orion’s Belt, and the sword hanging from the belt formed by the Orion (M42) Nebula, etc (where stars are formed from vast clouds of interstellar gas), additionally the entire constellation can be viewed as a sword, with a hilt across the centre. And the ‘fiery’ nature of the sword in the biblical reference may in this way indicate the energies of celestial bodies such as those of the sun, or these stars and nebulae.

Additionally, the phrase ‘to keep the way of the Tree of Life’ gives reference not only to Tree of Knowledge in Eden, but the Tree of Life in Sumerian culture we have seen in other parts of this website is intimately connected to World Tree/axis concepts – and with surrounding figures, the relationship of the deities of Sumer to the creation of the completed human, ‘homo sapiens’, the original ‘Adam’.

So Orion’s inner mythology may well be concerned with some central aspects of the Bible, and the Great Pyramid; so that it may have been seen in terms of a ‘heavenly balance’ or ‘axis’ or ‘way’. Interestingly, Orion does lie close to the plane of the Milky Way, and is from Earth, very close to the celestial equator. Indeed this line, extended out from Earth’s equator outwards through space passes through Orion’s Belt itself  (as well as many other constellations, obviously).

(rightOrion, the Celestial Equator, and the ecliptic of the Milky Way).

(above left) the Tree of Life stela, from Nimrud, Assyria, 865-860Bce, with the king Ashurnasirpal beside the Tree, and Shamash the sun deity, wearing the Anunnaki horned head-dress; or a winged Apkallu (one of the Seven Sages) behind him in several versions; from the British Museum. (right) a much older Sumerian or Akkadian precursor version of the same mythic tale.

The Tree of Life as a concept arose first in the earliest civilization, Sumer of course, and passed along throughout the Near East, especially the Mesopotamian cultures, and from there to other people’s in contact with them; The Hebrews had a concept of the Tree of Life, as did then the early Christians; so in the Bible for example, the apocalyptic, mystical, and mystery-school related book of the Revelations of St John of Patmos describes it in the very last chapter of the Bible, at Revelations 22.2;

In the midst of the street of it, and on either side of the river (of Life), was there the tree of life, which bare twelve manner of fruits, and yielded her fruit every month; and the leaves of the tree were for the healing of the nations”.

(right) engraving of the Tree of Life as described in the Bible.

19th centuryWikimedia, PD, from the Phillip Medhurst collection

Also significantly, it is possible to characterise the values of the Vesica’s two inner-triangles and the fractal ones inside them, as follows; see first of the diagrams below.

As we have seen, this double triangle geometry was at the heart of the Gothic ‘school’ of architecture; every building was centred upon a ‘navel-point’, from which the framework of the Ad Triangulum system was then derived, with important features such as central columns/pillars placed exactly upon the lattice-work ground plan. The navel-point would see the altar placed directly over this ‘energetic gateway’…

(below); diagrams of the Vesicawith double triangles along the vertical and horizontal axes of the Vesica Piscis (the ‘Way’ or ‘Tree’ of Life).

(second row); diagram of the measures of these inner triangles, in terms of height, hypotenuse, and width.

(second row); photograph of the Sumerian Tree of Life symbol, from ancient Nimrud, (Calhu of the Bible), Assyria, c.860Bce

                  43.3                      50                                25
                    25                                     28.8               14.4
                   14.4                                     16.66                8.33
                    8.33                   9.64                4.82
                    4.82                   5.55                2.77
                    2.77                   3.21                1.60

In the diagram, the proportions are as follows;

a. √3/ 2: 1    0.866 : 1

b. 2 : 1        1.0 : 0.5

c. 1 : 2        

d. 1 : √3 /2   1 : 0.866

e. 3 ; 2         1.5 :

f. 1 : √3        1 : 1.732

g. 1 : √3       1 : 1.732

h. 1 : √3           

k. 2 √3 : 1        3.46 : 1

 Noteworthy are several of the ratios and measures which come out of this diagram; for example, the 43.3 and 86.6 % values connect the table closely to the 864/ 432 harmonics of the sun, and cosmic#, as would be expected from results stemming from the square root of 3.

The 28.8 is the same value as the Bible’s number of singers performing sacred music at the consecration of the Temple of Jerusalem; in the cubic Holy-of-holies, at the geometric centre-point of the room (located precisely over the Eben Shettiyah, the Foundation Stone in the basement, placed over the ‘abyss’ to keep the ‘waters’ in abeyance, ie the navel-point of the chamber, at the navel point of the temple, at the navel point of Jerusalem, at the navel point of the world, a thick smoke filled the chamber upon the tonal vibrations of the 120 musicians and 288 singers; described in the Bible as the ‘Glory of God’…

Additionally, 288 (related to 72 and 144) is also believed by some writers and researchers, such as Ernest Maclain, Gordon Strachan, etc, to be the ‘vibrational frequency’ of the universe, according to ‘sacred number’… Moreover, in keeping with the perennial geometries of the hexagon in cosmic #, we see that the values of 1/6 (ie 0.1666), and it’s fractions such as 8.33 play a central role in the Vesica’s double-triangles / Tree of Life. It is curious that the same values exactly occur in the Squared Circle, as we have seen, as well as elsewhere. The value of 5.55. as half of 11.11, and quarter of 22.22 and related values of the square root of 1.2346789 is likewise noticeable…

 To hypothesize that the Sumerian and Assyrian Tree of Life was based upon the table’s values is a breath-taking conclusion; and yet the similarities are inescapable. One conclusion which is clearly possible while remaining only speculation, is that the ‘inspiration’ for the Sumerian/ Akkadian symbol was indeed ‘higher dimensional’.

In addition to the obvious points of interest, several other points worth mention occur ; 

A study of the possible meanings of the Tree of Life stelae gives a fascinating set of possible connections; of the ‘gods’ giving mankind wisdom (see the ‘god’ placing a pine cone (Phi patterned) toward the brain – the pineal gland? – of the human.

And noticeably, the deity Shamash (the god of the Sun) is placing the pine-cone/phi symbol/tetractys/pyramid symbol into the point of the spine where the brain and the body – or energy and matter –  meet, the Occipital ‘gate’. (The midpoint between the heavens (12) and the Earth (4) is 8; showing the significance of the Octave… ) Thus the stelae’s symbolism links the human body, or more specifically the human spine and brain-stem to the world tree/axis, or ladder, upon which energies of the heavens ascend and descend. So this narrative may be concerned with some action or procedure or course of events in pre-history which finally gave mankind divine status, as ‘homo sapiens’, marking the ascension to becoming spiritual beings; as beings ‘made in the image of God’, or as macrocosms of the universe.

And as a last point, this significance of the spine has some deeper reverberations through history than might be expected; virtually all life-forms on Earth, in the animal/ fish/ reptile/ bird and human classes share the same body form based upon the axis of the spine; and have done so since the Pre-Cambrian Explosion around 574 million years ago, when virtually all the major phyla of animals start appearing in the animal record.

And this 10-25 million years period, a tiny fraction of history, oversaw the introduction of the most stable body-form Nature has ever devised; a head, with organs of perception and breathing, a chest for heat and lungs, a stomach for digesting food and internal organs, two arms with hands and legs with feet, and areas for excretory and reproductive organs at the base of the trunk. All organised along the line of the spine; thus seeing the organisation of all of the advanced creatures of earth as ‘chordata’ – creatures made of  a chord, or spine. Not only is it an essential framework upon which all growth is made possible; but in the simplest terms it represents a cord – a length of string upon which ‘notes’ can be played of varying frequencies; as the length of a string is halved the note is doubled in frequency, creating a new octave.  Both Pythagoras and the Chinese experimented with  stringed instruments studying this aspect of reality; Pythagoras with a one-stringed instrument, the Chinese with the ‘gu-qin’ circa 500Bc, of similar if not greater sophistication.

(left) A carved stone representation of the ‘sacred geometer’ Pythagoras, with his stringed research instrument among the figures at the right bay of the western entrance at Chartres cathedral.)

     The chordata body shape enables the arrangement of organs, and energy centres in like manner – as Eastern theology states the seven chakras of the body are arranged (towards the front of the body) from crown of the head to the base of the trunk (see right for diagram of the seven energy centres / chakras ) aligned along the axis of the spine. So we are made in the universal form of a (vertical) axis, with symmetries on both sides of the body – much like the symbols of antiquity shown, and much like the symmetries of the Tree of Life, from the Sumerian and related cultures. A fact which was surely part of the consciousness of the creators of the symbol.

In the next section concerned with the geometries of cosmic number, we shall take a look at the nature of the measures of the planets of the solar system as they incorporate the principles of the geometries concerned.


i. Life’s material and energetic nature; it’s susceptibility to such factors in arising within this solar system.

 As we noted in passing in looking at the squared circle, the Earth and the Moon are harmonically linked to each other, and the Sun, and other planets, in their sizes, and orbital distances and periods, to an extent which to an impartial observer indicates the action of some unspecified cosmic laws upon these bodies… it might be ascribed to gravity and left at that; and yet this section will examine the possibility that the actual sizes of the Sun, the Earth and the Moon are energetically related, and even derived – ie, as expressions of higher cosmic laws and principles, concluded from the arising and evolution of organic life, these three planets embody together some of the deepest harmonic mathematic relationships possible.

(left) a diagram from Sir Thomas Diggett’s 1559 book on the celestial order of the planets around the Sun, which Diggett asserted was derived from Pythagorean theory…

In the 18th and 19th centuries astronomers and mathematicians attempted to formulate a law descriptive of the apparently ‘well-ordered’ nature of the solar system;  the Titius/Bodes law theorised the planets of the solar system followed a pattern of distance from the sun which is based upon an approximate doubling of distances from the nearest planet (Mercury) outwards; but this is considered to be a hypothesis that is not proven at best, and has largely been discarded since the 19th century and yet, there may well be some highly significant relationships between just the three main planets, the Earth, the Sun and the Moon, in particular, and further, between their sizes and the energetic imperatives indicated by some cos# values; at least in the case of the main planets of this solar system, (ie. the Sun , Earth and Moon) which has proven it’s ability to create and sustain life (then able to develop the intelligence required to understand this reality) over billions of years. So it is fair to question whether there are features of the system which are unique to this (miraculous) process, indeed are essential to it; indicating as such an uncategorised relationship between the ‘forces of Life’ and consciousness in the universe, energetic imperatives of reality related to light, gravity, the structure of the atom and other such factors, and the measures and proportions of the solar system planets and their orbits… for by some combination of these unique circumstances conscious life has evolved, and continues to evolve upon Earth; as to whether non-material life or consciousness exists upon other planes, or other planets of the universe, the stars, the Milky Way or the solar system we inhabit, the only accurate answer that can be given at this time is, “No evidence of organic life has been found on any other planet or moon in the solar system; other than that, we simply do not know”…

Such a world-view or ‘paradigm’ would lend support to the consonance of all the different levels of reality, as (again) asserted by the works of antiquity; providing not an alternative to modern science so much as a corollary from which understanding can be furthered; ultimately the aim of any system of understanding.

But to look at the question of how material Life arose and then survived on Earth within this solar system – and only here – at least in material terms, the list of circumstances on Earth that have been shown to be necessary for life to survive for billions of years has been shown by science to be detailed and long.

For example, factors scientists have argued may have been essential for life to flourish here include; the distance of the Earth from the Sun, in the optimal region where the Earth receives enough heat for life to survive, but not so much that it becomes inimical to life. Scientists call this the ‘Goldilocks zone’, and use the broad parameters of it to assess whether bodies orbiting stars in distant constellations might potentially be home to any forms of life, of presumably carbon-based biochemistry, as all life on earth is, (ie between the limits of heat and cold that life has proved able to survive here on Earth).

Secondly the Sun is unusual in it’s isolation; most suns have two or three stars in a star system, for example the nearest system to our planet, Alpha Centauri consists of three stars, A B and C. Without a stable system of day and night life would have struggled to evolve at all, depending as it does on broadly equal diurnal periods for energy reception, and activity (light), and rest and processing of the energies which will be required in the next 24 hour cycle (night). As A.N.Dodd in ‘Interactions between Circadian Clocks and Photosynthesis’ (2015) outlines, plants that require a night cycle would not survive brighter nights, because certain critical plant biochemistry only occurs at night. As can obviously be seen, humans too are organised to survive this way, requiring the sleep and relaxation of the night hours while the body regulates and attends to it’s energy systems, and enumerates the energies to be used the next day (from the previous day’s exertions).

The length of a planet’s year (the time it takes to orbit the sun one complete circle in respect to it’s starting point among the stars), and of a day (the time it takes to rotate 360 degrees on it’s axis while on it’s orbital pathway) vary among the planets of the solar system. The length of the day, the 12 solar months, and the year appear to be essential to life on this planet, and are unique to Earth; so Venus for example is not only the hottest planet, but unlike most of the solar system’s planets it rotates in the opposite direction to the Sun, and also takes longer to turn one complete circle on it’s axis (225 Earth days) than it does to complete one full orbit of the Sun in relation to other stars (223 days) – making it’s year shorter than it’s day! Obviously Earth has required the stability of the full solar year, as well as the four seasons to provide a rhythm of life throughout the millennia; life has grown and adapted to the seasonal rhythms of the year in myriad ways. Agriculture also is dependent upon the procession of the seasons, though even before then, hunter-gatherer groups relied on the seasons to provide different food-sources at different times of the year.

The seasons have always resulted from the angle of the earth’s poles in relation to the plane of the sun, being as such that the earth is tilted at an angle of 23.5 degrees. Without the seasons it is easy to predict that biological diversity would have never evolved so widely, thus endangering the entire biosphere when certain conditions of heat or cold prevailed. Diversity of life and ecological conditions has always ensured greater survival of more of the species in all biological classes.

The presence of an atmosphere is also essential – life can only exist where the atmosphere protects the planet surface from harmful cosmic ultra-violet rays and other particles, such as pour from the centre of the galaxy towards our solar system –as well as maintaining the required (complex) balance of elements in the atmosphere, such as the oxygen, carbon monoxide and dioxide levels, and so on. Scientists believe the central inner metallic core of Earth has enabled the creation of a magnetic field around the planet to protect the evolving atmosphere since life began 3.6 billion years ago. Mars is believed to have lost it’s atmosphere at an early point in it’s history, leading to the evaporation of all water molecules on it’s surface. Beneath that there may or may not exist evidence of rudimentary forms of life.  The metallic elements present within the earth’s geology, not unique to earth but not ubiquitous throughout planetary bodies either, are products of the repeated transmutation of elements within stellar and solar supernovae cycles. These elements and compounds have also been essential to the complexities of the evolution of organic life; the human body for instance contains numerous small but essential amounts of such metals, including calcium, potassium, sodium, magnesium, zinc, copper, lithium, cobalt and more.

(Regarding the ‘unique’ atmosphere of earth in regard to supporting organic life, it is possible to say that it is not only the earth which has an ‘atmosphere’ ; but even the massive area of space within the solar system does too; which protects it, and us, from harmful emissions pouring through space from the galactic centre, composed of a high-density cluster of billions of stars, all emitting high levels of harmful sub-atomic particles. Likewise the expanding solar system’s ‘atmosphere’ results from the solar wind, a brew of solar particles and energies pouring outwards from the sun’s surface at more than one million miles an hour in a gravitationally charged field. The forward motion of the sun’s movement through space creates a similar ‘bow-wave’ of the solar bubble, and a tail behind, both of which protect against the incoming galactic energies of the interstellar medium, in what is known as the ‘heliosphere’.  

This extends well beyond the inner planets of the solar system we have usually considered in the past to be the extent of its area, ie out past Neptune (at around 1.8 billion miles/19Au, ie 19 x 93 million miles). The heliosphere extends out past Pluto, at 3.6 billion miles/39Au  the Kuiper Belt (3 – 4.5 billion miles/30 – 50Au), a further 12 or so billion miles/160Au to a broad region called the Heliopause, where the forces of the solar wind and the incoming interstellar medium meet in ‘parity’, at a point called the ‘termination shock’. (Although the meeting point is also part determined by ‘particle pressure and magnetic field energy density’ – wikipedia quoting NASA and John Hopkins University publications on the heliosphere).

This boundary line is effectively the beginning of interstellar space. As mentioned, at the ‘bow-wave’ in front of the system ‘upwind’ the distance is around 7 – 9 billion miles/80 – 100 Au, while ‘downwind’ the point may reach double that; showing the pathway of the solar system as it orbits the centre of the galaxy, and indicating that the solar system does have some sort of atmosphere which is shaped by the Sun’s movement through space. But the relevant point is that harmful cosmic particles from sidereal space and the centre of the Milky Way are limited by the outgoing solar wind (and the sun’s gravitational field) up until this point. . . as the Voyager I and II space probes showed when passing out of the solar system from around 2010 onwards; V1 measuring forty-fold increases in harmful plasma density as it left the solar system, at 119Au (11 billion miles), in June 2012).

So the concept of an intermediate ‘Goldilocks’ zone may even actually apply not only within the solar system, but also to the galactic system we inhabit; as the solar system is situated at a point on the Orion arm of the Milky Way, (see left) approximately 26,000 light years away from the centre of the galaxy, it has been able to evolve away from the high pressure environment of the central regions of the galaxy, where billions of stars are clustered at its heart – a point already mentioned with regard to the necessary balance of light and dark for organic life, as well as the distance needed for protection from solar particles harmful to life.

 Additional magnetic and gravitational forces at work in the centre of the galaxy would be almost certainly equally harmful to life as we know it.

The solar system orbits the galaxy centre at a rate of one complete orbit every 225 million years or so; this is called the Great, or Galactic Year. Some have put forward the theory that the regions of the Milky Way the sun and the solar system pass near to in it’s orbit of the centre of the galaxy as it turns in space has created epochs on earth either less or more supportive to life; so possibly providing one explanation for the extinction of the dinosaurs. While the effects of different galactic regions upon the evolution of life as the Orion arm rotates the centre are unknown, it can be surmised by the sensitivity of organic life to many forms of cosmic energy that the stability of our galactic position has been a central factor in the long evolution of life on earth. So indeed protecting the solar system from other regions within the neighbouring arms of the galaxy which contain supernovae, gravitational instabilities, and radiation which could affect organic life.


In addition to the already mentioned specific conditions necessary for the evolution of life upon the planet Earth, another factor is the existence of the Moon, and it’s specific size and distance. This is believed to be 100 per cent necessary for the growth and survival of life due to reasons of magnetic/ gravitational fields existing between the two, (thus the Moon acts to stabilise the Earth’s orbit of the Sun slowing down the Earth’s rotation rate; without the Moon the planet would have shorter days, with resultant drops in the temperature of the Earth). It also creates through gravity the effects of tides within the ocean of the world – something which may have been an essential stimulus to early organic life, which began in the seas, by fertilising coast lines and then receding with the tide, to let early plants and organisms develop and breathe on land. Even today the tides and seasons and lunar cycle are all inextricably linked to the processes of life on Earth. Many if not all life-forms on Earth have body clocks tuned to the cycles of the moon, and the mentioned 24 hour diurnal cycle of the sun.

Some observers maintain that the Moon is ‘the perfect size, shape, density and distance’ in relation to both Sun and Earth for life to exist. . ! And scientists also argue the Moon’s ‘gravitational influence helps to ensure the stability of Earth’s spin axis’ tilt angle’ of 23.5° (the angle of ‘obliquity’) of the planet to the Sun – meaning there has been stability in climate and seasonal terms throughout history. (Ward and Brownlee, Rare Earth; Why Complex Life is Uncommon in the Universe, 2000). Without the constant pull of the Moon the tilt angle would wander in response to the gravitational pulls of the Sun and Jupiter.

The cosmology of the first book of the Bible reflects this reality closely;

And God said, Let there be lights in the firmament of heaven to divide

the day from the night; and let them be for signs, and for seasons, and

for days, and for years.

And God made two great lights; the greater light to rule the day, and

the lesser light to rule the night; he made the stars also.

And God set them in the firmament of the heaven to give light upon the

earth … and to divide the light from the darkness: and God saw that it

was good.

And God said, Let the waters bring forth abundantly the moving creature

that hath life, and fowl that may fly above the earth in the open firmament

of heaven. . .

So God created man in his own image, in the image of God created he him; male and female created he them. . .And God said, Behold, I have given you every herb bearing seed, which is upon the face of the earth, and every tree,

in the which is the fruit of a tree yielding seed. . .”   (Genesis 1.14-29)

The emphasis on seeds appears to highlight the unique ability of organic life to create images of itself from it’s own flesh; as indeed, science can take the smallest sample of any living thing’s DNA and make a new version of it from it.  The image of the planets circling through the heavens, amid the ordered courses of the stars in the vast spaces of the universe has from the earliest eras given mankind a sense of the harmonic forces which underlie the physical planes of the universe.

Indeed, in an interesting example of the synchronicity of many religious texts and tenets of antiquity, the ancient Taoist Chinese beliefs (said to be brought from the Heavens by celestial emissaries such as FuXi and his wife NaKua among others) regarding yin and yang closely parallel this biblical passage; from the text ‘The Spring and Autumn Annals of Master Lu Buwei”;

The origins of music lie far back in time. It arises out of proportion and is rooted in the Great One. The Great One gives rise to the two poles; the two poles give rise to the powers of darkness and light. The powers of darkness and light undergo change; the one ascends into the heights, the other sinks into the depths; heaving and surging they combine to form bodies. If they are divided they unite themselves again; if they are united they divide themselves again. That is the eternal way of heaven. Heaven and earth are engaged in a cycle. Every ending is followed by a new beginning; every extreme is followed by a return. Everything is coordinated with everything else. Sun, moon and stars move in part quickly, part slowly. Sun and Moon do not agree in the time which they need to complete their path. The four seasons succeed each other. They bring heat and cold, shortness and length, softness and hardness. That from which all beings arise and in which they have their origin is the Great One; that whereby they form and perfect themselves is the duality of darkness and light. As soon as the seed germs start to stir, they coagulate into a form. The bodily shape belongs to the world of space, and everything special has a sound. The sound arises out of harmony. Harmony arises out of relatedness. Harmony and relatedness are the roots from which music, established by the kings, arose.”

(Quote taken from David Tame,p.39, The Secret Power of Music. Lu BuWei was a merchant, political adviser, chancellor, and regent of the Qin state, and following the banishment and exile of his later years, a philosopher and poet until his death, circa 239Bce).

The 8 or 9 planets of the inner solar system are spaced around the Sun in what are highly predictable and ordered orbits, which mostly follow the same direction of movement around the Sun, upon virtually the same plane, (the ‘ecliptic’) for the most part, thus forming a ‘disc’ basically, it has been an aim of scientific study since the 16th century to find any simple harmonic relationships between the different planets’ orbits. The law (or rule) of Titius-Bode (from 1715 and 1766 respectively) states that with each successive planet moving outwards from the Sun, these occur at predictable distances, measured by the ‘semi-major axis’, which means the distance from the planet to it’s maximum orbital distance around the sun along the relevant longest diameter of it’s orbit. The theory stated that the spacings followed the general doubling – as found within the octave – beginning from Mercury outward, so that Earth is twice as far from Mercury as is Venus; likewise Mars is twice as far from Mercury than Earth is, and so on. The Greek mathematician educated in Egypt, Pythagoras, based his studies around both the courses of the planets in the heavens, as well as the musical mathematics of the halving or doubling of energies within the octave. David Tame argues that Egyptian drawings of the sun’s rays reaching down to earth (Akhenaten’s symbol, as well as the Sumerian/Assyrian/ Egyptian winged sun disk shown below) were representations of these groups of tones, and as such were most often shown in groups of 7, or 12 or 13 etc. . .(Tame, p.212). Here at this website we also interpret the same depictions as follows; firstly, where the rays are drawn with serpent’s heads at the  end was to encode the complex symbology associated with the ‘serpent’  or ‘lion’ energies of the sun, as well as indicate the pathways (the ‘dragon lines’) the solar energies take through the earth’s surface as part of life’s processes. And additionally the winged sun disk similarly points visually to the importance of the different geometric relationships created by the various square roots of 2, 3, 5 and so on, values which underlie many aspects of sacred geometry and cosmic number.

(left); Akhenaten and Nefertiti bas-relief, with sun’s rays as serpents. (centre) Egyptian winged sun disc, alike in respects to the Sumerian winged disc with a deity in it above the Tree of Life. (right); table of the square roots of 2, 3 and 5. In this A = 111.81, (or √5/2 ; 1.118) which equals Phi of 161.8 if 50 (0.5) units are added. B= 41.41 (√2), and C= 2.236, the √5.

The ‘music’ Pythagoras deemed to be at the basis of the creation of matter within the material universe was called by him the ‘Music of the Spheres’; as with the Chinese text quoted above, the belief was that each celestial body had it’s own position and level within the whole creation, and contributed it’s own characteristic tone. This tone is not one which can be heard, but is sound nonetheless – another way of describing the tone might be to say each planet has it’s own individual vibrational frequency; created presumably by it’s size, density, materials, energies within the system, and so on. What might be called their own distinctive ‘heartbeat’. This conceptual view is symbolized by (but not restricted to) the movements of the planets through space in their connected pathways; and each planet’s particular ‘note’ is subject  to differences in it’s tone according to the time of the day, of the year, the inter-related positions of the planets, phases of the sun, and such – much as the human heartbeat varies periodically throughout the day and night (and other time periods). This ‘music of the heavens’ is the equivalent of the ‘Word’ as described by St John in the Bible, the outpouring of energies from the source of creation in harmonic patterns, creating octaves of notes (each with their own inner-octaves) throughout the levels of reality; as Ouspensky stated, this points to the significance of the Law of Seven in all studies of reality.

 Something which is found to be relevant in cosmic energies such as light, sound, the periodic table of chemical elements, biological life (in each organism’s open-ness to different dimensions’ energies), etcetera. . .

To return to the inner harmonics proposed by some scientists to personify the spacing of the celestial bodies of the solar system, this ratio has been revised to be somewhere between 1.7275, and 1.728-1.732 by some 20th century astronomers; this is of relevance because of both values connections to cosmic number; firstly, 1.7275 x 2  = 3.45 5, and secondly, 1.728 is the result of 864 x 2, 432 x 4,  3456/ 2, and so on. And 1.732 is, of course, the square root of 3, another key value in cosmic number, present crucially within the axis of the Vesica Piscis, in the ratio of 2: 1.732, or 1 : 0.866 for the circles’ diameter: central axis length. These two related values, of 86.4 and 86.6 relate to the harmonics of light and the sun, (which has a diameter of 865,000 miles), as well as time (there being 86,400 seconds in a day) – as we shall look at later in this section.

But for the moment, the original theory as stated by Titius and Bodes was as follows;

PlanetPerfect OctavesMean Orbits 
Asteroids8approx. 8 
Neptune64 x 1.5 = 9698.9 

Between Mars and Jupiter astronomers had found no planet at that time in the 18th century; in 1801 G.Piazzi discovered the planetoid Ceres, of diameter 480 miles exactly at this distance. Further discoveries soon after established the existence of the ‘asteroid belt’ along this orbital plane, indicating the existence once of a planet at this harmonic ‘station’ which had been subsequently destroyed. This is something the Babylonians, successors to the Sumerians, and originators of the ‘ancient science’ of ‘astrology’ , (something quite different to modern versions)asserted happened in pre-history, in their ‘Myth of Marduk and Tiamat’. Tiamat was a planet which entered the solar system without ‘divine permission’, without a ‘destiny’ as said. In the Bible Tiamat is used to refer to an ‘ancient sea-monster’, of the ‘depths’; as we have noted in the Sumerian section in the references to the deity Enki as the Lord of the Deeps the Ab-zu, there is a meaning as much related to the heavens as to the depths of the seas or the earth. That noted, the mythic tale tells of the deity/ planet Marduk swinging round on its orbit straight into Tiamat with devastating effect, destroying the intruder, and thus creating the asteroid belt. (Marduk is conjectured to have been representative of the earth, and the collision created a fracture in the earth’s surface that created what is now the Pacific Ocean).

Around the end of the 18th century the discovery of the planets Uranus Neptune and Pluto lent further support, as Uranus and Pluto have mean orbits ‘extremely close to the exact distances necessary to complete two further octaves’ (David Tame, The Secret Power of Music, p.238) – Neptune meanwhile occupies a position almost exactly one-half of the distance between the two other planets, which some have considered to be significant somehow.

Another way astronomers have formulated Bodes Law is as follows; semi-major axis = 4 + x.

where x = 0, 3, 6, 12, 24, 48, 96, 192, 384, 768, so that each step apart from the first, from Sun-Mercury, sees the distance double.

Now this is interesting for the following reason; in his book covering the main points of Gurdjieff’s system of self-development, Ouspensky reproduces on p.79 (in what can be a confusing explanation!) the manifestation of the three forces of reality, active, passive and neutralizing down the various seven levels of the Ray of Creation; in other words how many ‘laws of reality’ each dimension becomes subject to as energy becomes further immersed in matter with each successive level.  So at the level of the Absolute, the sum total of everything existing, Ouspensky states that ‘since by it’s nature everything in the Absolute constitutes one whole the three forces also constitute one whole. . .they possess full independent will, full consciousness, full understanding of themselves and of everything they do. The idea of the unity of the three forces in the Absolute forms the basis of many ancient teachings – the Trinity – Brahma Vishnu and Siva, and so on’.

As these conscious forces divide, the ‘will of the Absolute creates the worlds of the (sidereal/All galaxies) level, and governs their creative work, in which a mechanical element makes it’s appearance’. The three divided forces of the second level meet together in each of these worlds, creating new worlds of the third order (galaxy level). And so on; and the total number of laws, and the number by which each level is referred to is effectively the same as the Bodes Law progression;

                                  1, 3, 6, 12, 24, 48, 96,  at which the Ray of Creation ends the octave at the lowest point, the Moon, which as the ‘densest point’ of the Ray is most constrained by the ‘laws of the cosmos’. The numbers obtained are used to describe each particular level of the cosmos’ energies also, for instance in the food diagram of the human body (p.181-197/ 319-325) which describes the daily intake of three levels of energies*; head (‘impressions’, or sensory stimuli) chest (breathing), and stomach (food). In this way the human body contains and uses/ interchanges the energies of every level of the cosmos, in varying degrees, which is why esoteric belief has always stated that man is a microcosm of the universe. And points to how changes at the level of All Galaxies/ sidereal space (Si), or the Milky Way (La) and so on, distant though they are from the level we inhabit, can affect the psyche and physiology of the human being.

*(This division of the human body according to the Law of Three is reflected coincidentally (!) by the ‘law of threes’ used by medical/ survival experts; broadly speaking, the human body can survive three weeks without food; three days without water; three minutes without air; and three seconds without impressions (ie the messages of the nervous system by which the body functions/ energies flowing from Nature, p181 In Search. . .) Presumably also, the additional level, of water, is equivalent to the non-instinctive centres subsumed within the stomach centre, which consists actually of three centres itself through the workings of the Law of Three at all levels of reality; the instinctive, sexual, and social sub-centres. These form the ‘centre-of-gravity’ of human life and can be adequate for some people’s inner lives, according to Gurdjieff and Ouspensky, in again, broad terms. . .)

And if every breath takes 3 seconds, then it is an easy calculation to say that the average human being takes 86,400/3 breaths a day; which turns out to be 28,800; something we will see shortly is an intrinsic part of the Bible’s cosmic number symbolism regarding music and the Temple of Jerusalem (288), the inner proportions of the Vesica Piscis (28.8), and is maintained by some researchers to be the ‘Pulse Beat of the Cosmos (288) – Strachan, Maclain, Tame, Cathie etc.

 While it is difficult to say whether the two progressions are related, it is interesting to note that both are concerned with the cosmic interactions between energy and matter with relation to distance from the ‘centre’. To find the mathematical progression of the Ray of Creation repeated in the spacings of the bodies of the solar system, having postulated life’s rising on earth requires harmonic relationships within the solar system’s bodies regarding distance, and size, does seem to indicate some sort of energetic imperative is at work here, shaping material reality.

As can be seen, from this and from revised workings in the 20th century, some form of significant relationship appears to exist in the distances of the planets, whether at doubled ‘octave’ distances, or the alternative figures mentioned, proposed by Mary Blagg in 1913 to be around the 1.7275 mark, or D.E.Richardson in 1954 and others, to be 1.728 – 1.73 instead.

It may be just a feature of orbital resonance, whereby their gravitational fields exert an influence on surrounding bodies, which has caused the planets to find spacings so mathematically meaningful; and astronomers have estimated that the Titius-Bode laws may apply to other solar planetary systems. So the possibility that any system of planets will eventually gravitate into well-spaced orbits is not scientifically unacceptable, indeed many scientists regard it as likely in such systems, so that ‘results from simulations of planetary formation support the idea that a randomly chosen stable planetary system will likely satisfy a Titius-Bode law”. (wikipedia).

 But the premise that this solar system is especially open to energetic laws – which have allowed the development of life on Earth – makes the significance of the regularity of the system’s main planets extremely interesting, whatever the reasons for their orderliness.

For considering the overall structure and well-ordered nature of the solar system and its planets, as well as the fact that life has arisen upon our planet for around 3.6 billion years, it is likely that some form of positive harmonic relationship existing within the system has enabled life to grow in synchronisation with the physics of sunlight.

    The apparent uniqueness of material life as seen on Earth would therefore seem to be connected to the physics of the Sun, the planets, and light. Having noted that there is no proven discernible relationship between the different planets’ orbits, well-spaced though they are, it might be of use to recall that in the Ray of Creation described by Gurdjieff and Ouspensky, the steps or levels of energy existing (within the solar system) downflowing into matter are – (Do- Si -La) Sol – Fa – Mi – Re ; or  Sun: planets: earth: moon. (the table/ Ray of Creation showing the full Octave or Seven ‘notes’ is shown below);

D0 – the Absolute / All Galaxies

SI –  Sidereal space

LA – Lactae, the Milky Way

SOL – the Sun.

FA –Fatae, the Fates, or planets

MI – Microcosmus, the Earth

RE – Rhea/ the Moon

DO – Absolute matter / inertia.

So consideration of particularly the three planets of the Sun, the Earth, and the Moon (as representatives of different discrete ‘levels of reality’), while not ignoring the significance of the other planets of the system, may well be of value in attempting to understand the dynamics of the solar system’s harmonic relationships existing between these three bodies, something we will see is well beyond the laws of chance or coincidence.

In addition to this point, it is possible to consider things a little further thus. . .  that from the formation of the Sun onwards and outwards, that it is possible that not only the distance, but the actual size of these planets including the Sun, or the ratios between, them are relevant factors in the creation of circumstances preferable for life.

From a Western science viewpoint, the perspective might be that having found a system with the right proportions and relationships such as ours displays, life then arose as a natural circumstance. Eastern, or ‘esoteric’ philosophy might agree with the basics of that view, while also maintaining that life, as a manifestation of the highest energies of the cosmos may have created the circumstances necessary for it’s arising, according to mathematical principles inherent in both energy and matter; ie higher dimensional energies shaped the material dimensions of the solar system in order to be able to evolve further into matter – this is much the view of the Bible also in depicting the creation of the sun, the stars, the earth and the moon, according to God’s will, and through the actions of his Word, or logos. Which of the two views is right – or perhaps both are – is a matter of opinion ultimately.

So bearing in mind that the circumstances necessary for life to develop and evolve on Earth were so ‘individually unique’ in this solar system, does this possibility show in the size and relationships between these three key bodies?


There is no physical evidence for any kind of biological life on any other planet ever studied by mankind – admittedly the minutest of fractions of the potential number of galaxies, stars, planets, and moons within the observable universe; our galaxy, the Milky Way, for example consists of around 100-150 billion stars, each with numerous planets and moons within orbit of them; and likewise there are billions of similar sized galaxies within the universe; so a staggering number of possible equivalents to Earth exist. The number of planets being studied via astronomy or even through the activities of space travelling probes certainly make this an incredibly eventful and exciting time for the sciences of astronomy, physics, biology and so on.*

* (One related subject of interest is that of how to enable humans to withstand the rigours and inherent problems of decades-long space travel, if missions are to be sent beyond the boundaries of the solar system, towards the nearest stars (4.37ly being the nearest, Alpha Proxima/ Centauri, ie 26,000 billion miles). It is possible that the solution to decades of relentless space-travel is most probably going to be solved by advances in the area of android technology, along the lines of self-repairing (teams of) robots skilful enough to be able to effect tests/ repairs on any damaged or faulty sections of the craft – as videos of robots able to run obstacle courses and do back-flips and dancing would indicate are possible – and be able to ‘switch off’ and recharge for 23 hours per 24, and fulfil functions and communicate with Earth the active hours… the problems and resources-demands of needing to feed, protect the health of, and provide a breathable atmosphere every second of the journey are in this scenario much reduced.

The main question is whether these systems of artificial guidance etc can be recharged with available resources from within the craft or local environment of space for the periods of travel which would be measured by decades, considering the vast distances of space, even within the solar system. But the idea of humans spending more than a year or two year on a small space craft is, realistically, unlikely for the foreseeable future.

It is puzzling that the Sumerian Epic of Gilgamesh, from 2800Bce or earlier, has the prescient inclusion in one of the first mythic works of literature in history, not one, but two, maybe even three ‘conscious/ non-human’ beings which could be called ‘robots’; there is the machine-like ‘terror’ the Guardian of the Forest, Humbaba/ Huwawa, who Gilgamesh and his friend Enkidu fight and destroy, thus displeasing it’s creator, the Lord of Command (of Earth) Enlil; and there is the ‘Wild Bull of Heaven’ (sent down by Anu from the ‘heavens’ to punish them for this plus disrespecting his daughter, Inanna), for which many possible interpretations exist, (including the ‘mighty’ Sumerian/ Assyrian deity Ninurta, (believed to have been the inspiration for the first king of Babylon Nimrud, who was a ‘mighty hunter before the Lord’, and a bloodline- great grandson – descendent of Noah, in Genesis10.8-10). Ninurta is actually addressed as ‘the Wild Bull of Heaven’ by one of his fellow Anunnaki in the myth ‘The Return of Ninurta to Nibru’, lending support to the idea the Bull of Heaven is clearly not a normal living bull but representative of something higher (while the epithet is also used in regard to Enlil in one or two texts);  and there is Puzur-Amurru, the ‘cybernaut ’ (or guide) who Utnapishtim entrusts the guidance of the Ark to when the Deluge begins and the hatches of the craft are sealed, (the ‘helmsman’ is possibly a metaphor for the separation which exists between the deepest instinctive levels of the subconscious, and the conscious mind – ie in everyday life the instinctive functions are ‘given over’ to a separate part of the brain which is highly difficult to influence or change; and in survival situations an injured or traumatized person who is unable to think coherently (or even remember the date, where they are, even who they are…) may still act purely on ‘instinct’ and hence survive – this is explored in more depth in the Gilgamesh section). Puzur -Amurru means in one interpretation ‘Pilot of the West’; perhaps giving meaning to his name in terms of the difference throughout history whereby the West has symbolized technological or material knowledge, and the East spiritual or inner-knowledge).

It also points to the idea that nearly all depictions of ‘lower’ functions, or beings – such as Enkidu – are representations of people sui generis whose inner being is at the level of the ‘mechanical’ lower levels of physical/ instinctive being. (Hence the lapse into insults and negativity the ‘wild-man’ experiences when struck down with illness as a sentence of death by Enlil and the council of the gods). Humbaba too, as such, represents in one sense the lowest state of people who are ‘slaves’ of (negative) impulses, appetites and inner-dynamics they have no awareness or control of… he too reverts to insults, and threats during the fight with Gilgamesh and Enkidu.

 As we see in the Gilgamesh section, the deity Enki, who helped Utnapishtim survive (whose name means ‘of the fish’ from PIE ‘peysk ’, ‘piscine’ etc;  as with the deities depicted as or with serpents, or with fish scales etcetera throughout antiquity, meaning essentially ‘of the reptilian centre’) was the Lord of the South, or Ab-zu (subterranean waters/ depths).

Enki as such represented of the three main gods of Sumer the stomach level of the body, the instinctive, physical centre concerned with survival, as well as genetics and reproduction etc (hence the entwined serpents of Enki, Ningishzida etc of Sumer are identical to the intertwined helices of DNA – with which both deities were concerned in their separate ways with the creation and fertility of organic life ; as well as the Egyptian and modern emblems of serpents connected with medical matters). As with the ‘pythonesses’ of the Delphic Oracle, Enki/Ea represented the intuitive wisdom of the instinctive centre, also known as the ‘reptilian centre’ in the broad division of the areas of the brain into 1, 2 or 3 brained systems; reptilian, mammalian, and human, a division symbolized in the Sumerian mythology of the Anuna whereby the three areas of the Earth with relation to the night skies were divided between the father Anu and two sons Enlil and Enki – Enlil (Lord of the Skies/ the northern constellations), Anu (Lord of the Equatorial constellations), and Enki (Lord of the southern constellations/ the Ab-zu). This threefold metaphor can be extended to mind ; heart; and body, a division reflected in the actions and roles of the three primary (male) deities of the Anuna. Enlil’s emotionless dedication to upholding the standards and laws of the heavens in his dealings with mankind certainly reflect his ‘impartiality’ in this respect, for example in decreeing mankind not be warned of the Flood beforehand – a facet of his ‘impersonal’ role which caused considerable upset to the many other deities of the heavenly tribe, with their loyalties and attachments to humanity and the peoples of Sumer, especially in the areas for which they were the individual ruling deities, such being the case in many cities and settlements. As such it may have been a lesson as to the dangers of relying on/ developing the powers of the intellect without emotional or instinctive balance…

This allocation of roles extending from the heavens, to areas of the world, to aspects of the human being, not only reflects the multi-levelled nature of Sumerian/ Anunnaki symbolism, but also goes some way to explaining the equation of Enki with the Serpent in Genesis in what is a continuation most probably of the Sumerian mythology derived from cosmic wisdom…

And symbolic of the tensions between Enlil and Enki as shown in their actions in the story of the Flood in Gilgamesh. This role of Enki may thus be perhaps why the serpent of Eden (Enki/ Oannes) is punished for ‘giving’ humans the knowledge that would enable them to ‘be as gods’ (Genesis 3.5).  Enki, and his Babylonian version Oannes were both deities concerned with giving humanity wisdom and technology, ie the establishment of civilization. Related to this duality is the widely read and quoted Hebrew text from the first or second century Bce, the Book of Enoch, which takes considerable time describing the various forms of divination, healing, and technology, such as metal-working, and the making of ‘swords, and knives, and shields, and breastplates. . . and bracelets, and ornaments and antimony’ etcetera that the ‘sons of the gods’ the Nephilim passed to mankind, furthering their sins. (see 1Enoch, chapter VIII, Charles version). In chapter IX the archangels of the Lord condemn the Nephilim for this, judging them as having ‘taught all unrighteousness on earth and revealed the eternal secrets which were preserved in heaven, which men were striving to learn’ – as well as mating with  human women and creating a race of giants possessed of insatiable appetites and a wish to dominate the world, so that the ‘complaints of men rose to the heavens’, stating the ‘whole earth has been filled with blood and unrighteousness’ and ‘lawless deeds’. . .

And despite the lord YHVH deciding to create the Deluge, to end the subjugation of humanity by the Nephilim, and to wipe away the sinful works of the giants, or ‘mighty men’, whose lineages had begun to oppress the world (Genesis 6.5). . . yet their bloodlines did survive the Deluge, as is depicted in encoded metaphors and narratives within the Bible, such as Nimrud being the Sumerian descendent of Noah. And in related manner, this is why the construction built by the same Nimrud – who according to Josephus and other writers in antiquity was the ruler of Babylon and descendant of the line of the nephilim/ Anunnaki – the Tower of Babel was destroyed by God, followed immediately by the ‘confusion of the tongues’ of the peoples of the world (Genesis 11.1-9), to effectively therefore ‘limit their wisdom’.

These narratives of mythology were themselves taken partly from Sumerian and Babylonian sources by the Hebrew scribes of the Bible, most probably during the Babylonian Capture from 597Bce-539Bce, and included and re-worked their original encodings of higher consciousness into new forms for the Hebrew religion. For more on this see; the Bible, Gilgamesh/ Sumer, and Etymology sections. . !

All else said, it is also a remarkable circumstance that the Epic of Gilgamesh, this earliest of works of art, from an era before virtually any machines whatsoever existing, circa 2800Bce, raises the concepts of celestial/ human hybrids, creations of the gods; proto-humans of great strength being given consciousness and civilization; deities intervening to protect humans such as Gilgamesh, Enkidu and Utnapishtim against celestial beings/ events; beings created by the gods for non-human purposes/agendas (Huwawa the robot, the ‘scorpion-men’ at the gate of the Divine Mountain, Puzur-Amurru the ‘cybernaut’/helmsman, and so on); cosmic beings such as Siduri and Urshanabi; humans such as Utnapishtim being granted immortality after the Flood; a boat / craft which carries it’s owner through the reaches of space to find the ‘Far-away’ (possibly the ‘afterlife’, or ‘underworld’) located at the ‘confluence of the rivers’, a place of clear celestial meaning; and so on. Moreover that these early myths of Sumer laid the outlines of a wisdom concerning the workings of the heavens on Earth that founded the basis of many of the cultures and civilizations of mankind throughout the Near East, and then ‘western civilization’.

 So this is a complicated and strange narrative in the era of the gods  – the Anunnaki – establishing their societies on earth; and the epic’s early origins, in tandem with its themes of certain humans attaining the highest possible levels of consciousness, fit into all the other similar indications from the period that higher forces played a central role in the establishment of human civilization – both in Sumer, and elsewhere across the continent of Asia.

To study the three bodies of the Sun the Earth and the Moon from a point of view incorporating both traditional and modern perspectives requires a non-scientific framework which asserts that all the parts of the solar system, and indeed the universe, are connected in essential ways.

The view of the Universe as a living, evolving, inter-related  system of separate but connected, interacting parts is encapsulated in the symbol (given by Ouspensky from Gurdjieff’s teaching) whereby the material universe is likened to a Tree of Creation (the totality), in which the Absolute (Do) is the totality of everything existing; the anchoring roots beneath the soil are the ‘abyss’ of unmaterialised energy, the main trunk is the aggregation of galaxies (Si), from which the larger branches stem into the level of galaxies, such as our Milky Way (La); from which boughs develop of individual stars, ie. suns, such as our own (Sol), with it’s complete solar system; the planets (Fa); the Earth (Mi) and the moon (Re) – and thus to the end-point of the Ray (Do), beyond which energy plays no part, being encased entirely in matter. . .the Moon in this symbol therefore is the place of densest matter, and least energy along the continuum from top to bottom; a place of lifelessness slowly receiving the energies from Earth with which it will hopefully grow in the long term future across the aeons. This helps explain the role of the moon – and perhaps all moons – in being in magnetic/ gravitational relationship to their ‘parent’ planet.

So in the what might be called pre-science cosmology – but can also be called a ‘parallel  conceptual framework’, and completely compatible with the findings of modern science – the Earth and it’s Moon sit at the end of a branch on the Tree of Creation, far removed  from the purer energetic processes of the larger sections, and higher dimensions they represent also in their unique character.

With regard to examining the relationship of two of the lowest levels of the Ray, namely Earth and it’s moon, putting aside for a moment the complex details of the energetic relationship described by Ouspensky and Gurdjieff, it is possible to simply compare the sizes of the two, and in doing so create a geometrical symbol which highlights some of the energetic forces at work in the creation of the two planets.

This diagram, first brought to attention by John Michell in his book City of Revelation, (after Plato perhaps?) offers a close up view of the related sizes of the two bodies.


EG = 3960 miles

CG = 7920 “

DF = 2160 “

EF = 1080 “

EH = 5040 “

BH = 10080 “

CD = 2880 “  

EJ  = 6120 “

AB = 1080 / BC = 1080 / CD = 2880 / DE = 1080 / EF = 1080 / FG = 2880 / GH = 1080 /

HJ = 1080.

Beginning with just the inner circle, the Earth at first, we can obtain a square of equal diameter, and a circle of equal circumference to that.

The radius /diameter of the Earth being 3960/7920 miles, a square of equal diameter will = 4 x 7920m = 31680 miles.

For a circle to have equal circumference to this square, it must have a radius of 5042 m. (circumference/2 π ).

And incredibly, as the Moon is of radius 1080/ diameter 2160m, a circle drawn through its centre from Earth’s centre will have radius of

5040 miles – just two miles difference..!

    =>   earth radius + moon radius  =    3960 + 1080 = 5040 miles.

And further to this, is the presence of Phi proportions within the geometry of the two bodies; a triangle drawn of the Earths radius, from the centre of Earth to the moon’s centre, and joining these 2 lines, forms a 90° right angle triangle has the following measurements:

     WT²+ ST² = WS²

    earth radius² + earth/moon radius² = hypotenuse²     ->   3960² + 5040² = 6410²

The proportions of this triangle perhaps afford us a possible clue as to one of the energetic imperatives affecting the size of each planet, and their orbital rates and distances; the proportions of the golden ratio, Phi . . .

 A quick breakdown of the figures gives us these ratios –

                             3960  –    5040   –  6410m

     1  :          1.27   :  1.618     

ie,        1  :         √φ      :      φ

Another point to recall, we saw these proportions when looking at the  angles and dimensions of the Great Pyramid, in terms of it’s base radius,  height and length of sloping side. Again, providing some indication of the sophisticated geometry / (cosmic) consciousness of the builders/designers, as we examine in the Great Pyramid and Giza section.

Back to the diagram, to measure the small right angle triangle formed from the radius edge of the Moon, to the radius enclosing square of the Earth – taking the three lines; ZY = 2160; XY = 2880, using Pythagoras’ Right Angle Triangle theorem; then ZX  = 3600.

  ie;     2160²+ 2880² = 3600²       (ie. ZY² + XY² = ZX²)

– hence forming a perfect           3      :        4       :    5    Pythagorean triangle with a base unit of 720.

(All 3 distances, incidentally, are ‘cosmic numbers’, ie. have particular significance, in the canon of ‘sacred geometry’, across cultures. . . the value 288 as we have seen is said by writers such as Ernest McClain, and Gordon Strachan, and Bruce Cathie, to represent the value of the ‘pulse-beat’of the universe, it’s primal creative emanation; which is ‘encoded’ into the Bible as the number of singers and musicians of king David (1Chronicles25.7) present in the Holy of Holies of the Temple of Jerusalem on the day of it’s inauguration (thus indicating the application of harmonic ratios of notes within sacred music to the ‘music’ of the heavens); due to the cubic geometries of the innermost chamber of the Temple, which focus energies at the centre of the cube, as well as the resonances of the 288 harmonic, and the presence of the mysterious Ark of the Covenant, the chamber is filled with a thick cloud of smoke; what the Bible terms the ‘glory of the Lord’ (2Chronicles 5.12-13/2 Chronicles7.1-3).

And as we saw in the section on the Vesica Piscis, and the Ad Triangulum, (and the Sumerian Tree of Life), the value 28.8 units is double the basic 14.4, and each third of every line in the two triangles of the Star of David measures 28.8 units, giving 86.6 in full – the percentage of this line to the diameter of the ‘Earth’ circle, and one which connects it with the 864/865 geometries of the Sun, as well as time, such as there being 86,400 seconds in 24 hours; while as noted, one breath per three seconds mean the average human takes 28,800 breaths a day.

Additionally, 3600 is the square of 60, as well as 360 x 10…thus linking the degrees of circles and spheres to time, via the sexagesimal system; 3600 being the number of seconds in each hour – as well as the number of ‘arc’ seconds in each degree of the circle;

 – each degree being divided up into 60 minutes, with each minute containing 60 arc-seconds… it was also said in Sumerian mythology to be the period of one year, or orbit around the sun, of the home planet of the Anunnaki, called Niburu. The Niburu ‘year’ was therefore 3600 years long, and was termed a ‘sars’. The rule of the deities of the Anunnaki on earth before and during the creation of mankind and civilization was measured therefore in these periods of ‘sars’, as the ancient King Lists of Sumer show.

 It is for this reason, plus the functionality of related base-12 systems of counting in dividing up the periods and geometries of the heavens, that the sexagesimal system is therefore so perfect, in uniting the measurements of time and space, on many levels; as ‘gear wheels’ in other words, which link the different ‘rates of flow’ in different ‘levels’ of scale or ‘dimensions of energy’; 12-fold in the heavens and decimal on earth in particular providing example; so in the Bible for instance both forms are interleaved, a fact which may be symbolized by the 12 tribes of Israel being reduced by the loss of Manasseh and Ephraim to only ten. As we see in the Vesica section, the relationship of 1:1.2 or 10:12 is central within the subject of cosmic number; and as proof of that the close relationships between many cosmic# values are within the proportions of 1:1.2. See the Notes and Numbers section for many examples of this.

 That the (effectively) first ‘full’ civilization (of mankind’s long journey into settled agrarian communities, of towns, cities, with divisions of labour resulting from concentrated crop production – and all the ensuing developments of mankind) created the sexagesimal system, is one of the best indicators of the higher consciousnesses which guided mankind’s earliest innovations; that ‘coincidentally’ told the people of Sumer, and onwards, they had come ‘from the heavens’, in order to give mankind the wisdom of civilization, and ‘guide the development of mankind’.

Further to this, as we shall examine later, the artworks of the Sumerian/ ABA civilizations are of an incredibly high quality of production; and conception, containing as they do both incredible artistic craftsmanship, and conscious understanding and use of sophisticated concepts and symbols still  relevant today . . . As such these factors likewise indicate ‘higher consciousnesses’ at work in the guidance of early mankind; as we shall see in a while.

 To return to matters at hand – if we examine all the dimensions obtained by this simple but informative geometric analysis of the Earth and the Moon,we can look at a table of the measurements’ harmonic relation to two central factors – 72, and 360.

                                          72           360


                  1080              15            3       

                  2160              30            6       

                  2880              40            8        

                  3600              50            10             

                  3960              55 *         11 *

                  5040              70 *         14 *

                  6120              85            17

                  7920             110*         22*

                  9000             125           25

                  10080           140*         28*

                  12240           170           34

Firstly of note, is the quite incredible synchronicity of these figures, in the fact that all these various combinations of aspects of the Earth and the moon are whole number divisors of both 72 and 360, two of the most significant values in the geometries of ‘spherical’ objects, and space. . .

for example, in the study of the measurements of the Precession cycle – we find that for the Earth to travel 1° of it’s full period takes 72 years (actually the figure is just under that, at about 71.6 years)

 As a fifth of a complete circle the number 72 also occupies a central role in the geometric evolution of Phi ; a feature of organic life in both the human body – hence why ‘5’ is the ‘number of man’ in it’s resemblance to a pentagonal star; as well as in plants in the patterns of their branches, stems, leaves and seeds. This use of Phi in plants (and possibly in the human body) in inherent growth patterns helps to ensure the most efficient ‘allocation’ of biological matter with regard to sunlight, or perhaps the energies of life with regard to the body. . . so for a plant as it grows upwards spirally will try to avoid placing leaves directly over each other, for obvious reasons. But as the leaf pattern grows it will eventually return to the same point, with the optimal distance or number of leaves between the two. And the number between them is very often a Fibonacci number, ie 2,3,5, 8 and so on, (while each full cycle can be equated to an octave, in its return to an identical point).

For example, the pear tree will space it’s leaves in a spiral with 8 between each pair of identical leaves; this growth pattern, called phyllotaxis highlights the mathematical and geometrical nature of the inner laws of biological matter, and the relationship to the rhythms of music and the octave, themselves based in the mathematics of reality.

The pentagon, composing of 5 equally spaced points has internal angles of 72°, and phi proportions throughout;

From this internal geometry of the pentagon come Fibonacci spirals, based upon the division of 360° by Phi , namely 137.508°(angle b in diagram left). So there are many plants – pine cones, pineapples, sunflower seeds, acorn cups, cacti etc – which exhibit the Fibonacci numbers within their growth patterns;

 – the resulting repeating growth patterns in plants follow a Phi spiral from the centre outwards, Phi being shown in the sequence of numbers;  0,1,2,3,5,8,13,21,34,55,89, which the plants exhibit in the number of spirals running in both directions.

 Related to this closely are the geometries of the Flower of Life, a hexagonal symbol containing

Depictions of the Flower of Life have been found from Egyptian sites such as the Temple of Osiris at Abydos, as well as in Assyrian works dating from 645 Bce. One of the first western artists to study Fibonacci numbers and phi was Leonardo da Vinci, who spent much time examining Phi relationships within nature and the human body. Some of his research notes on Phi spirals and the Flower of Life are shown (right). Another example from the Middle Ages is from Chartres cathedral, one of the best examples of ‘sacred architecture’ in Europe ever constructed, built from circa 1194 – 1220 Ad. This is the Rose Window, designed with Flower of Life geometries of the hexagon, and the dodecahedron (12 sided figure).

The musical pattern of ‘the cycle of fifths’ in the production of (musical) octaves is closely related to the patterns organic life follows in it’s growth, pointing to the receptivity of life to the harmonious ‘tones’ of energy found at all levels.

Another example of the importance of the Golden Angle within the geometries of the world’s energy fields is that if you divide 360 by Phi (1.618033) the result is 222.4  – or by Phi² (2.618033) which gives 137.508, exactly the Golden Angle (thus the two fractions add up to 360). As we see in the Ley-lines of England section, if Giza, at 31.13°E, (the geometric and energetic centre of the continents/land-masses of the world) is taken as the world’s prime meridian, then 137.508 degrees to the west of this lies both the US nuclear laboratory of Los Alamos, New Mexico where the world’s first nuclear bomb was designed; and a few hundred miles south on the same longitude, still in New Mexico, is found the Trinity site, location of the detonation of the world’s first atomic bomb, in 1945. So the placings of which may hold some form of inner significance in relation to the planet’s energy-fields, as we consider in the Leylines of England section.

Of note also is the fact that it is not only the Earth and moon which are related somehow to the harmonics of 72 and 360; the Sun also shows such proportionality to these key values, as we will look at shortly.

The table examining the propensity of the Earth/Moon radii/diameters/proportions to resolve into harmonics of 72 and 360 also show perfectly the significance of the divisions/harmonics of 22 and 7 we saw are part of the ‘inner’ geometry  of the Giza complex and Great Pyramid;

(left) the use of 7/ 11 ratios within the GREAT PYRAMID.             

We have seen in other sections how the dimensions of the earth are able to be considered in terms of 2, 3, 5, 7. and 11, the first five prime numbers, and essential aspects of cosmic#, particularly as the ‘generative’ square roots of 2 3 and 5.

So with regard to the diameter of the Earth/Moon circle in the diagram from Michell, we find the following;

2 x 3 x 5 x 11 x 5280 (feet in a mile) / 31680 = 55.

While 31680 / 5280 = 6, which x 55 = 330, which = 2 x 3 x 5 x 11…

31680 /  2 x 3 x 5 x 11 (330) = 96; the level of the moon in Ouspensky’s table of the number of laws each level is subject to; for the levels from Do-si-la-sol-fa-mi-re,  ie 1 – 3 – 6 – 12 – 24 – 48 – 96, a strange coincidence; while the sun is exactly 330,000 : 1 to the earth in terms of mass.

So in other words 31680 /11= 2880, and /22 =1440, /44 = 720, and so on. And curiously 31680/ 2π gives us the original figure of 5,042 miles; which divided by 44 = 114.59; the highly significant and interesting cos# value obtained by a number of means, such as 360/3.14159 (360/π); 141.4/ 1.234 (√2/1.234), 127.3/ 1.111 (√φ / √1.23456789);  90 x 1.273 (360/4 x √φ); 38.19 (1-φ) x 3, and so on.

So some complex relationships do appear to exist between the ‘building blocks’ of the first five prime numbers, and the measures of the earth and moon as well as the numbers indicated by Ouspensky to be intrinsic to the levels of the Ray of Creation in relation to the two bodies concerned; numbers we have already seen are reproduced in the harmonics of the solar system in the proportions of the planets orbital distances as they circuit the Sun.

This harmonic based character appears to be supported by the figures of Earth’s diameter – when divided thus by the highly significant harmonic of 11, some cosmic# values result;

                                            7920 / 55  = 144

                                                     / 22  = 360

                                                     /110 = 72

So already from this study of the distances of the size of the Earth and the Moon, in many aspects, we have discovered the physical dimensions of these two planets are closely related to the ‘energetic’ dynamics, or guiding forces – of Pi – Phi – Pythagorean 345 triangles –  the Golden Angle – Fibonacci sequences – octave ‘fifths’ – and ‘spherical geometries’ of 72° and 360° – Gurdjieff and Ouspensky’s Ray of Creation – sacred number proportions and amounts, as well as the importance of the geodetic measure the mile, from Egypt onwards – as found within the proportions of the Great Pyramid, and the Giza complex !

So as a further example of the inter-linked proportions and ratios that crop up in this way,

The measurement of Earth’s radius raises a peculiar feature; for

if 10 x 360 = 3600, then Eq/radius of 3960 = 11 x 360

                    Likewise the diameter of 7920 = 22 x 360

If the diameter is multiplied by Pi (π) to give the circumference, that gives 22 x Pi = 69.11. (And 11 x Pi = 34.56). And where have we seen the  figure 69.11? It is the distance in miles of 1° of the Earth’s circumference, at the equator ! – ie 360 x 22 x π gives a result of 69.12 miles/per ° of equatorial circumference. . .

And 360 x 11 x π = 34.56 miles per half of one degree, or 1/720 of the earth’s circumference; (matching the proportions of the Earth:Moon 3:4:5 triangle composed of multiples of 720; 2160, 2880 and 3600 miles, as mentioned above, all cos#).

This fact again gives support to the centrality of 11 and 22 units in the subject of Cos#,  and further, the design and dimensions of the Giza complex, and the Great Pyramid; especially considering the apparent pains the designers and builders of the Pyramid took to make it proportionately 1/43,200–43,300 of the Polar Radius of the world.

And the ‘compatibility’  of the planetary dimensions with the fractions which form Pi,  and the sexagesimal circle, gives indication of there being some resonance between material sizes, and energetic principles… in ‘special cases’ such as this solar system, (which has proven requisite for the creation of Life), has been formed embodying such relationships within their material dimensions.


There are some incredibly interesting features to the Earth-Moon geometries, as various researchers ( such as John Michell, Richard Heath, Robin Heath etcetera) have uncovered (in the footsteps of Plato, the Greeks, and the Egyptians in antiquity); so as John Favrineau has noted in his ‘Book of Coincidences’, the two figures of 5040miles (Earth radius + Moon radius) and 7920miles (Earths diameter) can be described as follows-

          1 x 2 x 3 x 4 x 5 x 6 x 7  =       5040     =   7 x 8 x 9 x 10

                                                       +    7920    (8 x 9 x 10 x 11)             


                                                       =  12 960   = 6 x 6 x 6 x 6 x 10

                                                       =  36² x 10 = 360²/ 10

                            Also 3600 / 12960 = 0.277777; which x 3.6 = 1 (!)

We saw in the Gilgamesh section the importance ascribed by the character Urshanabi, the (celestial) ‘boatman’ to the number of 120; when Gilgamesh destroys the ability of the boat to reach the ‘far-away’ where Utnapishtim (Noah) resides by ‘smashing the stone things’, and ‘picking the urnu snakes’ he is made to cut down 120 poles for propelling the boat instead.

This may represent as 120 the lowest third of energies, (ie. purely material energies) as 1/3 of 360. The value also has significance in being 1 x 2 x 3 x 4 x 5 as well being divisible by 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24,  30, and 60. When multiplied by 6 it gives 720, then that by 7 gives the radius of the Earth plus the radius of the Moon, at 5040 (miles).

So Earth’s radius /120 = 33, and the Moon’s radius /120 = 9, which break down to 11 x 3 and 3 x 3.

If 11 is divided by 3 the number obtained is 3.66 recurring; a match to 366, the number of sidereal days in a normal Earth orbit; (a sidereal day is the time the Earth takes to turn once on it’s axis with respect to a fixed point in the stars, independently of it’s orbit around the Sun; whereas 365.24 days is the length of time it takes for the Sun to cross the celestial equator moving north, due to the Earth’s rotation ie is defined by the vernal equinox.

And as another example of how the celestial physics of the three planets are closely matched to these significant values, in considering the effect the Moon has on Earth’s orbit of the Sun, it is seen that the centre of the Earth is not at the ‘centre’ of their orbits; the two planets orbit around their barycentre (gravitational centre) in an ellipse approximately 1/81st (1/ 3⁴) the distance of the Moon from Earth – roughly 4,480km in radius every 29 days. And as we have seen, 1/81 (or 10²/9²) = 0.0123456790123 etc.

So in all, the inner harmonies present within the size of the Earth and the Moon’s dimensions begin to look anything but accidental, as might be expected of planets which between them and the Sun have enabled the growth of organic life on the Earth; something unique in the universe in material terms, as far as we know. In other words Life is indeed a miracle, one enabled by the geometries and sizes of all three planets concerned. Without these ‘perfect’ harmonics sunlight would not be able to be absorbed by living cells, or the earth, or it’s energies encircle the globe in meridional pathways and ‘energy fields’. The solar system may be defined as ‘perfect’ in it’s geometries, as life has arisen there for over 4.5 billion years without any serious interruptions.

*(see the section on Gilgamesh and the achievements of Sumer for a consideration of the meaning of these; ‘stone things’ may signify crystals of some sort; and ‘urnu snakes’ may signify light-energy or place-specific energy lines, ie solar energies that are derived from the points in the space-time field where they manifest most clearly. As we examine in the Etymology section, Ur-anna meant the ‘light of the heavens’ in Sumerian, Assyrian, Hebrew, Greek and Latin civilizations, leading to the stem of ‘Or-‘ creating words around the theme of the sun’s ‘arising’/origin, the ‘gold’ of sunlight, and the growth of life from that. The first city of history, the Royal city of Ur, said in the Bible to be the birthplace of Abraham, took it’s name from this celestial theme. The word ‘urban’ today likewise stems from this.

One last note concerning the incredible harmony of the numbers comprising the dimensions of the Earth and the Moon when considered together, we note the presence of the cosmic number 1,296 (from different civilizations) often used in calendars, such as that of Hindu sacred texts, denoting cosmic significant measures…

 Yet equally incredible harmonic and geometric measures/proportions come to the fore, when the Earth, and the Moon, both separately and together are compared with the geometric and physical ratios of the solar system and the Sun.

Firstly, the Sun itself has full integer harmonics of 72 and 3600 – with radius 432,000m/diameter 864,000m, the factor of 360 divides into the Suns diameter exactly 2,400 times – the key number in the setting out of the Giza complex, according to some observers (E.Schmidt) – (and 864,000 divided by the factor of 72 gives the result of 12,000 – the heavenly number given in the Bible as the measure of the Divine City, etc etc).

For all 3 related planets, Sun – Earth – Moon, to be divisible by 360, and 1/5th of  that, 72 shows us a clear relationship between the 3 bodies that is mathematically meaningful.

So for instance, as we have seen, the diameter of the Earth divided by that of the Moon, gives us 3.66, while the number of days in a sidereal year is 366…

Likewise, the Sun’s diameter – 864,000 miles – divided by the Moon diam. (2,160 ms) = 400 precisely,

a proportion which is suggestive of the perimeter of the Square/Circle where the square haslength of 100 units.

From which √φ x π = 4 (or 127.3 x 3.14159 = 400)

 It is this ‘coincidence’, that the moon is precisely 400 (!) times nearer to the Earth than the Sun ,and 400 times smaller, that gives rise to the phenomenon of the solar eclipse – whereby the moon, passing in front of the sun, occupies the same amount of space in the sky, thus leaving just the bare outline of the Sun visible – the ‘diamond ring’ effect. . .

The writers Alan Butler & Christopher Knight, who wrote ‘Who built the Moon?’ note there are several 4/400/10,000 relations between the Sun and the Moon. In addition;

  •  the size of the Moon to the Earth is;                 7920/ 2160 = 27.322%
  •   the # of sidereal days in 1 lunar orbit of the Earth                = 27.322 days.
  •  the # of Earth diameters inside Sun diameter

                = 865,000/7920= 109.21

                                                                                    109.21 / 4   = 27.30  (109.21/3.997 = 27.322)

 and; 864 900/ 7920/ 3.14159/ 27.32 = 1.272 360  (~√φ ) !         

–      or alternatively;  864 900/7920/ 3.14159/ 1.272 360 = 27.32

ie. Sun diameter/ Earth diameter/ Pi / √φ = Moon diameter percentage of Earth diameter (as well as Moon orbital period)!

These are incredible figures, with no current explanation.

And as we note in the Vesica Piscis chapter, if you take the Precession cycle – the number of years it takes for the Sun to recess around the earth against the background of the stars through the houses of the zodiac is estimated at 25,920. This is called ‘The Great Year’, ie. one ‘year’ for the sun, constituting it’s 360° circle through the 12 zodiac signs, at c.2160 years per sign.

       And if you divide 864,000 / 33.333   (3 x 11.111) = 25,920

(ie the diameter of the sun / 30 x √1.23456789 = P.   – the precession period of the Sun as seen from Earth in years. 30 is 1/12 of the 360 degrees of a circle, and is therefore the number of degrees in the night-sky each ‘House’ of the Zodiac occupies. As well as being each zodiacal Age in years, 2160 is ‘coincidentally’ the diameter of the Moon in miles…

Similarly, if you divide the diameter of the sun by 33×10⁴;

                                  864,000 / 330,000 = 2.6181    =  φ²  (alt. 86.4/ 33 = 2.6181)

                                   In fact φ² x √φ (2.6181 x 1.272) =  3.33 (or 1.11 x3)

Apparently ( the (gravitational) mass of the Sun is precisely 330,00-333,000 greater than the earth; (from NASA/ Of note in the Bible, is the connection between a king called Nahash, and the number 330,000. This passage occurs in the first book of Samuel (in fact it is 1Samuel 11.1 !);

¹Then Nahash the Ammonite came up and encamped against Jabesh Gilead;

and all the men of Jabesh said unto Nahash, Make a covenant with us, and we

will serve thee. ²And Nahash the Ammonite answered them, On this condition…

that I may thrust out all your right eyes, and lay it for a reproach upon all Israel.

( Upon which the elders asked reasonably for 7 days to think it over – during which rescue arrived in the form of Saul, Israel’s first king) – of note here is that when the (rescuing tribes were counted, the ‘children of Israel were three hundred thousand, and the men of Judah thirty thousand’ (1Samuel11.8 !), making a total of 330,000.

ie (10 x 30) + (1 x 30)  x 1000, or 11 x 3 x 10 000).

 Another connected example is at  Deuteronomy 33.1-3 where Moses blesses the children of Israel in his last speech before his death, using  Sun related references to describe God’s love;

Deuteronomy 33.2;

And he said, The Lord came from Sinai, and rose up from Seir unto them;  he shined forth from mount Paran, and he came with ten thousands of saints; from his right hand went a fiery law for them.

33.3 Yea, he loved the people; all his saints are in thy hand. . . and everyone shall receive of thy words.

(Notice that 333 is also 10,000/ 30, or 1000/3… also the ten thousands reference, relevant as mentioned to the Sun: Moon mechanics in several ways.

See the Bible chapter for a  more detailed discussion of the verses mentioned, as well as the Nahash, Nagas and kingship in Israel, and more numerical encodings of cosmic numbers within chapter and verse numbers, such as 16.18 and so on).

As we have already seen, not only the size of Earth, the Moon and of the Sun are in harmonic resonance with the Cosmic number of 432, but the speed of light itself is: so where

 C = 299.79km/sec,the square root  √2.9979 = 1.731444 /4 = 0.432 861 (Sun radius)

                                                                                                      x2 = 865 722 (Sun diam. = 864-5,000).

in fact 1.731 444 / 4.32 5 (actual radius) = 0.400 333 87. . . raising the sun/moon 400 ratio; the moon being exactly 400 times smaller than the sun, while the sun is exactly 400 times further away from the earth, means that they are the same size when viewed from earth, enabling the occurrence of solar and lunar eclipses.

Additionally, according to Alan Butler and Christopher Knight, the moon is rotating at 400 kilometres a day, and the earth is rotating at exactly 100 times more than this,  at 40,000 km/day. More examples of the many ways in which the three bodies seem to be linked whether viewed in terms of miles or kilometres. And moreover linking again to the 400/π =  √φ (127.3), as we have seen in the analysis of the square and the circle.

So in miles, C = 186,282 m/sec. And the √186,282 = 431.6039.

 This itself raises further support of the amazing inter-relatedness of the Sun, Earth, and Moon’s dimensions, as well as the Speed of Light; having seen that –

√3                          = 1.732

√2.9979 (C)             = 1.731 444 (0.000 555 diff.)

      1.731444/ 4.00333 = .432 5  x 1000² = Sun/radius

      C/ 1.609 = 186,282 m/s


√186,282            =  431.6039

        432 – 431.6039 =  0.3960

       0.3960 x 100²    =  3960 = Earth/radius

 So, 1/1000 the radius of the Sun (from centre to edge)             432

 Minus 1/10000 the radius of the Earth (from centre to edge)       – 0.3960            

 equals the Square Root of Speed of Light !                           =  431.6039

(And less relevantly perhaps ; 186,282 x 86.4% = 160,947. 648 [m:km = 1:1.6093. (Whether the scientists of the French Republic knew of the fact their chosen length for the kilometre in relation to the mile was close to Phi value of 1.618, is uncertain, but an interesting parallel relationship).

All these relationships point to some energetic source for the material dimensions of these three planets; not for all stars and associated planet systems, but in respect of our assessment of this solar system as being – despite being very  ‘lowly’ placed on the Ray of Creation-  a very advanced evolution of energetic principles (for some reason, which is why such systems are – as far as we know – extremely rare in the ‘statistics’ of planetary systems…)

 The Gt Pyramid is situated at approximately – 29.9792°N / (31.134°E) – there are more Speed of Light (C.) relations to some of the most significant numbers we have found in the study of sacred sites, and cosmic phenomena;

186,282/ 3600 = 51.74 = angle of Gt Pyramid to within 0.05 of a degree.

Possibly related, the Escape Velocity of Earth is 11.186 km/s, which may be closely related harmonically to 111.81;     

186 282/ 111.81 = 1666.058 (ie.10⁴/6) or more simply 1/6 = 0.1666

  • considering that Phi is deeply connected with the growth of all organic life on Earth, and to Pi via the 6/5 harmonic,

and the production of the Hexagon from the Vesica Piscis, these relationships clearly point to some deep connections

between Phi, the √5 (2.236), hexagonal geometries of space and matter, and C. of course.

This is seen in the equation; 3.14159 / 6 = 0.5326 (= E.R.C)

                                                                      0.5326  x 5 = 2.618  (= φ²)

   – the 6/5, or 1.2 harmonic relates the decimal 10-based numbers of ‘earth’ to the 12-based geometries of the ‘heavens’.

 All these factors point to the interconnectedness between the planets’ proportions, and the speed of light; (and the position of the Great Pyramid on the Earth’s surface – and it’s transformational, consciously designed structure ) indicate the importance of these geometries, and energy processes in the cycle of the planet’s absorption and refinement of higher energies, for life and energy processes upon the planet. 

Considering these 3 inter-related planets, it is fair to assert the existence of a process which gives rise to their complex harmonics.

We can further examine the geometries of the 3 circles of this diagram:

                 Earths Diameter                        (3960m R)       Inner circle.    (EG)

                 Ea.radius + Moons radius          (5040m R)       Squared circle (EH)

           and Ea.centre – Moons diameter       (6120m R)       Outer circle     (EJ)

In terms of the various circumferences of these circles, and divisions of these circles into degrees and so on.

 The Circumferences are:                miles        /        Km


 (EG)  3960     x 2 π                 24,881              40,033.5                         

 ->√2.99792/4.32 5  = 0.400 335.

   (EH)    5040  x 2 π                  31,667              50,952.2

   (EJ)    6120  x 2 π                  38,453              61,870.8

 There is one more immediate point of interest about these given measures; and that is that, if we divide these circumferences by 86.6%, giving the √3/2, (or 1.732/2) harmonic for circle of radius 1 unit, as seen in the Vesica Piscis, and denoting the midpoint line between centre and edge of circle –  we obtain some quite interesting results ;

(100 x 86.6% = 115.47)

                For circle 1 (I), 24,881m /    115.47   = 215.49 (~216, diameter of the Moon)

                For circle 2 (E)  31,667m /   115.47   = 274.24  (137.12 x 2)

           and for circle 3 (O) 38,453m /   115.47   = 333

One last fact to raise is that the circumference of Earth, 24,881m divided by 432 = 57.6; part of the inner geometries of the Vesica Piscis and so the inner triangles sub-divide the hexagon into lines of length 14.43, 28.86, 43.3, 57.7, 72.17, and 86.6, units.

To recap;

AB = 50

AE = 25

AC = 28.8

AD = 50

AM = 100

AL =  86.6

CL =  57.7

AG = 25

CE = 14.4

Other measures include; GC = CE = 14.43 equals the basic side length of the 72 equilateral triangles which constitute the hexagon;

CD = AC = 28.8 6, BF = 86.6 (ie, 6 x 14.43), GE = ED = 25, BE = 43.3 as half of that, and BN = 100.

It can be seen that as AG = 25 = 86.6% of AC or AE, as a height: hypotenuse ratio of each equilateral triangle, so a triangle of 12.5 units height (half of AG or AE) has a hypotenuse of 14.4 units, ie CE; or as  while AE = 25 height, and GA = 28.8 hypotenuse.

The hexagon of the circle divides into exactly 6 triangles (equal in size to the two within the Vesica; each of these triangles has 12 sub-triangles within it (12 half-triangles and 6 full ones), making a total of 72 for the full circle.

Each large triangle in the Vesica ABD, BDL, LDM, MDN etc)  can be divided with lines from each corner to the mid-point of the opposite line (BCE, GCD, etc). The midpoint where these 3 lines meet (C) is at a point where each line is divided into 1/3 and 2/3 sections. So as the height of the triangle is 43.3, it is short work to calculate the length of these divisions as; (GC)1/3 = 14.43, and (CD) 2/3 = 28.86 units.

This is the basic unit of the hexagon, and the triangles which form it; and 14.4 and 28.8 are key numbers in the cosmic# field (or 144/288). That each unit is 14.43 is a reflection of the small difference between the 86.6% length of the Vesica to the circles, and the 864 harmonic which is also a key cos# value for various reasons, such as there being 86,400 seconds in a day.

To conclude, it is fair to assert that the figures and values examined in this section point to some incredible harmonic relationships existing between the three relevant bodies of the solar system considered – hence the focus of many of the astronomic studies of the eras of antiquity; the Egyptians, the Babylonians, the druids of Celtic Europe, all devising sophisticated architectural ways of incorporating celestial dimensions into earthly buildings; (which took most societies such as modern Europe almost 3000 years before the maths of such sites could be understood to be related to the connected measures of the Earth and the heavens); in such a way that both heavens and earth could be studied in tandem ad infinitum, and furthermore, have their energies conjoined at such sites, placed as they were at points on the Earth’s surface where the geometric features of the energy-grid of the planet enabled the strongly occurring energies to be refined (through the materially and mathematically harmonious structures of each site). Hence why such sites were always considered to be ‘wells’, from which the subterranean energies of life could be drawn up to water the land and the people. (And where often, in what is a meaningful correlation, was to be found the presence of underground water-courses; believed since the earliest times to be conduits of the life-force).  It is no surprise therefore to find that even after thousands of years such sites were chosen as the place where the highest expression of western religion and architecture, the Gothic cathedrals, were to be built – directly over the original sitings.

Something we shall now study in one of the first examples of sacred architecture in the early eras of (pre-) history; the Great Pyramid, and it’s several companion  and satellite pyramids, and the Sphinx, at the site of Giza (centre-point, left); at the centre of the world’s continents as the point at the centre of a cube, or the Vesica, with it’s hexagonal geometries. As we shall see, it is highly probable these geometries were at the heart of both the siting, and the design of the Great Pyramid; again, one of the most mysterious, technologically amazing if not inexplicable creations of mankind’s history.

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