Category: Harmonic Origins of the World

Angkor Wat: Observatory of the Moon and Sun

above: Front side of the main complex by Kheng Vungvuthy for Wikipedia

In her book on Angkor Wat, the Cambodian Hindu-style temple complex, Eleanor Mannikka found an architectural unit in use, of 10/7 feet, a cubit of 20/21 feet (itself an outlier of the Roman module of 24/25 feet, at 125/126 of the 0.96 root Roman foot).

She began to find counted lengths of this unit, as symbols of the astronomical periods (such as 27 29 33) and of the great Yuga time periods proposed within Vedic mythology. Hence Mannikka’s title of Angkor Wat: Time, Space, and Kingship (1996). Whilst the temple was built by the Khymer’s greatest king, their foundation myth indicates the kingly line was adopted by a matriarchal goddess tradition.

Numerically Symbolic Monuments

Interpreting a monument using its metrology can be contentious. For example, in the megalithic period the established position has been that there was no metrological tradition and, to be found proposing one can cause your work to be ignored if not exiled from peer-reviewed journals, as was eventually the case with Prof. Alexander Thom.

At Teotihuacan, Japanese professor Saburo Sugiyama proposed an architectural unit of 83 centimeters was used, since the monumental complex would then clearly have numbers of these units corresponding to significant celestial periods, as if periods had been counted out within the City: the eclipse half year of 173 days at the Moon Pyramid, the Tzolkin of 260 days at the Sun Pyramid, and the Venus synod of 584 days at the Quetzalcoatl pyramid’s compound. More such day lengths and a well-known harmonic matrix were also seen in my Harmonic Origins of the World.

Astronomical counting within Teotihuacan (adapted from fig. 8.9)

Sugiyama did not reply to my message that his Teotihuacan Measuring Unit of 0.83 meters was the 2.72 foot length of Thom’s megalithic yard, implying some connection between Olmec/Maya Mexico and megalithic Europe. This was probably not welcome. Wikipedia’s editors of the “Megalithic Yard” page also objected to my mentioning this since it was I that had noticed this correspondence.

Over a 20 year period, Eleanor Mannikka found a numbers that were symbolic** or actual long counts of the solar and lunar years. In her thesis, these numbers were embodied as a ritual background for visiting pilgrims, whose steps corresponded to numbers – the megalithic yard being a metrological step of 2.5 feet. Her eventual counts emerged by a protocol that skipped thresholds, ran beyond, or started before a threshold, the counts were being human walkways but also excellent surfaces for doing accurate metrology.

**Her rule-based system that revealed numbers may well be a later function of the eventual monument, made to correspond with the numbers found in Hindu epic stories, since these are lavishly illustrated within extensive bas-reliefs, visible to pilgrims, depicting major Hindu myths. Statues of the gods punctuate the building’s many walkways to express the Indian practice of parikrama, of circumnavigating holy sites (such as around Mount Kailash or the great dome of Sanchi).

The Temple as AN Observatory

The symbolic use of numbers could only have become established through cosmic measurement in which astronomy (before our own) counted the actual numbers of days or months between repeating cycles of celestial alignment, and the differences and ratios between these. That is, ancient symbolic numbers originated in the Sky, where number-laden events measured in days or months generate whole numbers that were only then held to be sacred. One might think Angkor Wat too recent to have been constructed to suit this ancient sort of astronomical work. But the temple’s explicit orientation, to the west, was suited to just that. This made the temple perfect for observing and counting all sorts of time-counts, repeating measurements made millennia before using megalithic monuments.

That is, Angkor Wat is a current-era megalithic monument to the sky gods, these illustrated using the famous tableau of Vedic and later Indian myths.

The sun and moon set to the west**, each having a maximum range north or south of west. The sun at winter and summer solstice defines a fixed range within the solar year, depending on the latitude of a given site. In contrast, the Moon ranges over the horizon when setting over one orbital period of 27 1/3rd days. However, the moons orbit is skew to the sun’s path (ecliptic) so that the moon rises above and below, except at its nodes where eclipses can take place. These nodes move backwards so that the moon’s range on the horizon expands and contracts over 18.618 solar years.

**Looking west is very convenient since the sun or moon approach the horizon rather than suddenly appearing as they do in the east.

As a consequence, there are seven key points on the western horizon, the maximum standstill to north and south, the minimum standstill to north and south, the solstice extremes of the sun in summer (North) and winter (South), plus the equinox sunrise**. It is possible to calculate these alignments for the virtually flat terrain of Cambodia as in Figure 2.

**The Equinox sunset is a very exact point to measure since the sun appears to move rapidly on the horizon, between sunsets.

Figure 2 The alignments of Sun and Moon to the west (Left) around 1000 CE at the latitude of Angkor Wat using the Processing.org framework.

The notion of alignments seems to throw light upon the highly specific elements of Angkor Wat (see figure 3), if these alignments were viewed from the north eastern and south eastern corners of the raised temple enclosure.

Figure 3 Viewing the alignments of Sun and Moon, to the west (on Left), from the eastern corners.

There is a natural north-south symmetry, where the alignments to the solstice cross in the pream cruciform (see figure 4). The punctuation of the towers of the temple, seen from the eastern corners, would provide landmarks to calibrate the movement of (a) the sun in the year and (b) the moon within the lunar orbit, as the 18.6 year nodal movement expands and contracts the lunar range.

Figure 4 The Alignments seen within the plan of the temple complex.

The cruciform terrace outside the walls and nine fold cruciform within, could relate to the crossings of alignment and the periodicity of these cycles which would be countable in days using units of length.

The maximum moon alignments near 1000 BCE were 30 north and south or west, and one can plot those alignments over a flat Cambodia to the boundaries with Thailand which are, in contrast, significantly mountainous (see dark green areas at end of yellow alignments in Figure 5.

Figure 5 Google Earth view of the mountains at the end of both maximum moon alignments.

Parallels with the Megalithic near Carnac

The basic idea of such an observatory is a stone square instead of a stone circle. Alignments can be built-in, between back-sight observation points and fore-sight marker stones, marking the horizon location of an extreme event such as solstice. An observatory location can also look to an horizon event for which a distinct natural feature exists on the horizon, from that location. The stone perimeters of Carnac, called cromlechs, are various shapes but at Kerlescan, the cromlech is a rounded square, where the western perimeter is concave towards the east. That is, it faced rising events on the eastern horizon instead of setting events to the west.

Figure 6 Alexander Thom’s survey of the Kerlescan cromlech.

Otherwise, the “setup” is conducive to the observation of the sun and moon possible at Angkor Wat. Below I show how the observatory could work for the epoch 4000 BCE. The red lines are solar extremes and green lines are lunar maximum and minimum extremes. Equinoctial events at Spring and Autumn complete the inherently seven-fold nature of such phenomena.

Figure 7 Possible use of the Kerlescan cromlech, as an observatory facing east rather than west (at Angkor Wat).

Developmental Roots below 6

Square roots turn out to have a strange relationship to the fundaments of the world. The square root of 2, found as the diagonal of a unit square, and the square root of 3 of the diametric across a cube; these are the simplest expressions of two and three dimensions, in area and volume. This can be shown graphically as:

The first two roots “open up” the possibilities of
three-dimensional space.
Continue reading “Developmental Roots below 6”

Music, part 1: Ancient and Modern

We would know nothing of music were it not that somewhere, between the ear and our perceptions, what we actually hear (the differences between different frequencies of sound, that is, different tones) is heard as equivalent musical intervals (such as fifths, thirds, tones, semitones, etc), of the same size, even when the pitch range of the tones are different. This is not how musical strings work, where intervals of the same size get smaller as the pitch at which tones occur, grows larger. On the frets of a guitar for instance, if one plays the same intervals in a different key, the same musical structure, melodic and harmonic, is perfectly transposed, but the frets are spaced differently.

The key is that human hearing is logarithmic and is based upon the number two {2}, the “first” interval of all, of doubling. This can only mean that the whole of the possibilities for music are integral to human nature. But this miraculous gift of music, in our very being, is rarely seen to be that but, rather, because of the ubiquity of music, especially in the modern world, the perception of music is not appreciated as, effectively, a spiritual gift.

Music is often received as a product like cheese, in that it is to be eaten but, to see how this cheese is made from milk requires us to see, from its appearance as a phenomenon, what music perception is made up of . Where does music come from?

Normally a part of musicology, that subject is full of logical ambiguities, confusing terminology, unresolved opinions, and so on. Those who don’t fully understand the role of number in making music work, concentrate on musical structures without seeing that numbers must be the only origin of music.

The ancient explanation of music was that everything comes out of the number one {1}, so that octaves appear with the number two {2/1}, fifths from three {3/2}, fourths from four {4/3}, thirds from five {5/4} and minor thirds from six {6/5}. Note that, (a) the interval names refer to the order of resulting note within an octave, (b) that intervals are whole number ratios differing by one and that, (c) the musical phenomenon comes out of one {1}, and not out of zero {0}, which is a non-number invented for base ten arithmetic where ten {10} is one ten and no units.

Another miracle appears, in that the ordinal numbers {1 2 3 4 5 6 7 8 9 etc.} naturally create, through their successiveness, all the larger intervals before the seventh number {1 2 3 4 5 6 7} leaving the next three {8 9 10} to create two types of tone {9/8 10/9} and a semitone {16/15} thereafter {11 12 13 14 15 16}: by avoiding all those numbers whose factors are not the first three primes {2 3 5}. Almost the whole potential of western music is therefore built out of the smallest numbers!

This simplicity in numbers has now been obscured, though the structure of music remains in the Equal Temperament form of tuning evolved in the last millennium. By having twelve equal semitones that sum to the number two, we can now transpose melodies between keys (of the keyboard) but we have pretty much lost the idea of scales. Instead, each key is the major diatonic {T T S T T T S} (where T = tone and S = semitone intervals) starting from a different key. The fifth is called dominant and fourth subdominant and the black notes (someway fiendish to learn) required to achieve the major key in all keys but C which is all white keys.

The old church scales are achievable by over ruling the clef with accidental notes, and the reason for different keys sounding different is that they contain aspects of what were the scales. So a pop song, for example, is usually in a scale. “Bus Stop” by the Hollies was in the Locrian scale.

Equal Temperament enabled the Western tradition to create its Classical repertoire but it has made ancient musical theory very distant and has abandoned the exact ratios it used to use since every semitone is identical and irrational. Plato described this kind of solution as the best compromise, where every social class of musical numbers has sacrificed some thing of their former self in order to achieve the riches versatility bestows upon modern musical composition.

To be continued.

Music of the Olmec Heads

Seventeen colossal carved heads are known, each made out of large basalt boulders. The heads shown here, from the city of San Lorenzo [1200-900 BCE], are a distinctive feature of the Olmec civilization of ancient Mesoamerica. In the absence of any evidence, they are thought to be portraits of individual Olmec rulers but here I propose the heads represented musical ratios connected to the ancient Dorian heptachord, natural to tuning by perfect fifths and fourths. In the small Olmec city of Chalcatzingo [900-500BCE] , Olmec knowledge of tuning theory is made clear in Monument 1, of La Reina the Queen (though called El Rey, the King, despite female attire), whose symbolism portrays musical harmony and its relationship to the geocentric planetary world *(see picture at end).

* These mysteries were visible using the ancient tuning theories of Ernest G. McClain, who believed the Maya had received many things from the ancient near east. Chapter Eight of Harmonic Origins of the World was devoted to harmonic culture of the Olmec, the parent culture of later Toltec, Maya, and Aztec civilizations of Mexico.

Monument 5 at Chatcatzinga has the negative shape of two rectangles at right angles to each other, with radiating carved strips framing the shape like waves emanating from the space through which the sky is seen. The rectangles are approximately 3 by 5 square or of a 5 by 5 square with its corner squares removed.

Monument 5 at Chalcatzingo is a framed hollow shape. The multiple squares have been added to show that, if the inner points are a square then the four cardinal cutouts are described by triple squares.

The important to see that the Olmec colossal heads were all formed as a carved down oval shape, that would fit the height to width ratio of a rectangular block. For example, three heads from San Lorenzo appear to have a ratio 4 in height to 3 in width, which in music is the fourth (note) or subdominant of our modern diatonic (major or Ionian) scale.

Even narrower is the fourth head at San Lorenzo, whose height is three to a width of two. This is the ratio of the perfect fifth, so called as the fifth note of the major scale.

And finally (for this short study), the ratio 6/5 can be seen in Head 9 of San Lorenzo and also at La Venta’s Monument 1 (below).

MUSICAL RATIOS

If the heads were conceived in this way, the different ratios apply when seen face on. The corners of the heads were probably rounded out from a supplied slab with the correct ratio between height and width. The corners would then round-out to form helmets and chins and the face added.

And as a group, the six heads sit within in a hierarchy of whole number ratios, each between two small numbers, different by one. At San Lorenzo, Head 4 looks higher status than Head 9 and this is because of its ratio 3/2 (a musical fifth or cubit), relative to the 6/5 of Head 9. We now call the fifth note dominant while the fourths (Heads 1, 5 and 8) are called subdominant. These two are the foundation stones of Plato’s World Soul {6 8 9 12}, within a low number octave {6 12} then having three main intervals {4/3 9/8 4/3}* where 4/3 times 9/8 equals 3/2, the dominant fifth.

*Harmonic numbers, more or less responsible for musical harmony, divide only by the first three primes {2 3 5} so that the numbers between six and twelve can only support four harmonic numbers {8 9 10}

San Lorenzo existed between 1200 to 900 BCE, and in the ancient Near East there are no clear statements for primacy of the octave {2/1}, nor was it apparent in practical musical instruments before the 1st Millennium BCE, according to Richard Dumbrill: Music was largely five noted (pentatonic) and sometimes nine-noted (enneadic) with two players. However, the eight notes of the octave could instead be arrived at, in practice, by the ear, using only fifths and fourths to fill out the six inner tones of a single octave; starting from the highest and lowest tones (identical sounding notes differing by 2/1). A single musical scale results from a harp tuned in this way: the ancient heptachord: it had two somewhat dissonant semitone (called “leftovers” in Greek), intervals seen between E-F and B-C on our keyboards (with no black note between). Our D would then be “do“, and the symmetrical scale we today call Dorian.

The order of the Dorian scale is tone, semitone, tone, tone, tone, semitone, tone {T S T T T S T} and the early intervals of the Dorian {9/8 S 6/5 4/3 3/2} are the ratios also found in these Olmec Heads*. The ancient heptachord** could therefore have inspired the Olmec Heads to follow the natural order tuned by fourths and fifths.

*I did not consciously select these images of Heads but rather, around 2017, they were easily found on the web. Only this week did I root out my work on the heads and put them in order of relative width.

**here updated to the use of all three early prime numbers {2 3 5} and hence part of Just Intonation in which the two semitones are stretched at the expense of two tones of 9/8 to become 10/9, a change of 81/80.
(The Babylonians used all three of these tones in their harmonic numbers.)

To understand these intervals as numbers required the difference between two string lengths be divided into the lengths of the two strings, this giving the ratio of the Head in question. The intervals of the heptachord would become known and the same ratios achieved within the Heads, carved out as blocks cut out into the very simple rectangular ratios, made of multiple squares.

The rectangular ratio of Head 4, expressed within multiple squares as 3 by 2.

The early numbers have this power, to define these early musical ratios {2/1 3/2 4/3 5/4 6/5}, which are the large musical tones {octave fifth fourth major-third minor-third}. These ratios are also very simple rectangular geometries which, combined with cosmological ideas based around planetary resonance, would have quite simply allowed Heads to be carved as the intervals they represented. The intervals would then have both a planetary and musical significance in the Olmec religion and state structure.

Frontispiece to Part Three of Harmonic Origins of the World: War in Heaven
The seven caves of Chicomoztoc, from which arose the Aztec, Olmec and
other Nahuatl-speaking peoples of Mexico. The seven tribes or rivers of the old world are here seven wombs, resembling the octaves of different modal scales, and perhaps including two who make war and sacrifice to overturn/redeem/re-create the world.

A Musical Cosmogenesis

Everything in music comes out of the number one, the vibrating string, which is then modified in length to create an interval. Two strings at right angles, held within a framework such as Monument 5 (if other things like tension, material, etc.were the same) would generate intervals between “pure” tones. However Monument 5 is not probably symbolic but rather, it was probably laid flat like a grand piano (see top illustration). Wooden posts could hold fixings, to make a framework for one (or more) musical strings of different length, at right angles to a reference string. This would be a duo-chord or potentially a cross-strung harp. Within the four inner points of Monument 5 is a square notionally side length. In the image of Monument 1, and variations in height and width from the number ONE were visualized in stone as emanating waves of sound.

The highest numbers lead to the smallest ratio of 6/5 then the 6/5 ratio of Head 9 can be placed with five squares between the inner points and the 3/2 ratio of Head 2 then fills the vertical space left open within Chalcatzingo’s Monument 5.

Monument 5’s horizontal gap can embrace the denominator of a Head’s ratio (as notionally equal to ONE) so that the inner points define a square side ONE, and the full vertical dimension then embraces the 3/2 ratio of the tallest, that of Head 2.

It may well be that this monument was carved for use in tuning experiments and was then erected at Chalcatzingo to celebrate later centuries of progress in tuning theory since the San Lorenzo Heads were made. By the time of Chalcatzingo, musical theory appears to have advanced, to generate the seven different scales of Just intonation (hence the seven caves of origin above), whose smallest limiting number must then be 2880 (or 4 x 720), the number presented (as if in a thought bubble) upon the head of a royal female harmonist (La Reina), see below. She is shown seeing the tones created by that number, now supporting two symmetrical tritones. The lunar eclipse year was also shown above her head (that is, in her mind) as the newly appeared number 1875, at that limit. This latter story probably dates around 600 BCE. This, and much more besides, can be found in my Harmonic Origins of the World, Chapter Eight: Quetzcoatl’s Brave New World.

Figure 5.8 Picture of an ancient female harmonist realizing the matrix for 144 x 20 = 2880. If we tilt our tone circle so that the harmonist is D and her cave is the octave, then the octave is an arc from bottom to top, of the limit. Above and below form two tetrachords to A and D, separated by a middle tritone pain, a-flat and g-sharp. Art by by Michael D Coe, 1965: permission given.

Starcut Diagram: geometry to define tuning

This is a re-posting of an article thought lost, deriving in part from Malcolm Stewart’s Starcut Diagram. The long awaited 2nd edition Sacred Geometry of the Starcut Diagram has now been published by Inner Traditions. Before this, Ernest McClain had been working on tuning via Gothic master Honnecourt’s Diagram of a Man (fig. 2), which is effectively a double square version of the starcut diagram.

The square is the simplest of two dimensional structures to draw, giving access to many fundamental values; for example the unit square has the diagonal length equal to the square root of two which, compared to the unit side length, forms the perfect tritone of 1.414 in our decimal fractional notation (figure 1 left). If the diagonal is brought down to overlay a side then one has the beginning of an ancient series of root derivations usually viewed within the context of a double square, a context often found in Egyptian sacred art where “the stretching of the rope” was used to layout temples and square grids were used to express complex relationships, a technique Schwaller de Lubitz termed Canevas (1998). Harmonically the double square expresses octave doubling (figure 1 right).

Figure 1 left: The doubling of the square side equal 360 units and right: The double square as naturally expressing the ordinal square roots of early integers.

Musical strings have whole number lengths, in ratio to one another, to form intervals between strings and this gives geometry a closer affinity to tuning theory than the use of arithmetic to calculate the ratios within a given octave range. The musicology inferred for the ancient world by Ernest G. McClain in his Myth of Invariance (1976) was calculational rather than geometrical, but in later work McClain (Bibal 2012-13) was very interested in whatever could work (such as folding paper) but was especially interested in the rare surviving notebook of 13th century artist Villard de Honnecourt, whose sketches employed rectilinear frameworks within which cathedrals, their detailing, human and other figures could be drawn.

“I believe we have overlooked Honnecourt as a prime example of what Neugebauer meant in claiming Mesopotamian geometry to approach Renaissance levels illustrated in Descartes. If Honnecourt is 13th c. then he seems more likely to be preserving the ancient picture, not anticipating the new one.”

This draws one into significant earlier traditions of sacred art in Egypt (Canevas) and in Indian temple and statue design, and to Renaissance paintings (see end quote) in which composition was based upon geometrical ideas such as symmetry, divisions into squares and alignments to diagonals. Figure 2 shows one of Honnecourt’s highly stylised sketches of a man, using a technique still in use by a 20th century heraldic artist.

Ernest McClain, Bibal Group: 18/03/2012

Figure 2 The Honnecourt Man employing a geometrical canon.

The six units, to the shoulders of the man, can be divided to form a double square, the lower square for the legs and the upper one for the torso. The upper square is then a region of octave doubling. McClain had apparently seen a rare and more explicit version of this arrangement and, from memory, attempted a reconstruction from first principles (figure 3), which he shared with his Bibal colleagues, including myself.

Figure 3 McClain’s final picture of the Honnecourt Man, its implied Monochord of intervals and their reciprocals.

To achieve a tuning framework, the central crossing point had been moved downwards by half a unit, in a double square of side length three. On the right this is ½ of a string length when the rectangle is taken to define the body of a monochord. McClain was a master of the monochord since his days studying Pythagorean tuning. Perhaps his greatest insight was the fact that the diagonal lines, in crossing, were inadvertently performing calculations and providing the ratios between string lengths forming musical intervals.

Since the active region for octave studies is the region of doubling, the top square is of primary interest. At the time I was also interested in multiple squares and the Egyptian Canevas (de Lubitz. 1998. Chapter 8) since these have special properties and were evidently known as early as the fifth millennium BC (see Heath 2014, chapter two) by the megalith builders of Carnac. In my own redrawing of McClain’s diagram (figure 4) multiple squares are to be seen within the top square. This revealed that projective geometry was to be found as these radiant lines, of the sort seen in the perspective of three dimensions when drawn in two dimensions.

Figure 4 Redrawing McClain to show multiple squares, and how a numerical octave limit of 360 is seen creating lengths similar to those found in his harmonic mountains.

Returning to this matter, a recently developed technique of populating a single square provides a mechanism for studying what happens within such a square when “starcut”.

Figure 5 left: Malcolm Stewart’s 2nd edition book cover introducing right: the Starcut Diagram, applicable to the top square of Honnecourt’s octave model .

Malcolm Stewart’s diagram is a powerful way of using a single square to achieve many geometrical results and, in our case, it is a minimalist version that could have more lines emanating from the corners and more intermediate points dividing the squares sides, to which the radiant lines can then travel. Adding more divisions along the sides of the starcut is like multiplying the limiting number of a musical matrix, for example twice as many raises by an octave.

A computer program was developed within the Processing framework to increase the divisions of the sides and draw the resulting radiants. A limit of 720 was used since this defines Just intonation of scales and 720 has been identified in many ancient texts as having been a significant limiting number in antiquity. Since McClain was finding elements of octave tuning within a two-square geometry, my aim was to see if the crossing points between radiants of a single square (starcut) defined tones in the just scales possible to 360:720. This appears to be the case (figure 6) though most of the required tone numbers appear along the central vertical division and it is only at the locations nearest to D that eb to f and C to c# that only appear “off axis”. The pattern of the tones then forms an interesting invariant pattern.

Figure 6 Computer generated radiants for a starcut diagram with sides divided into six.

Figure 7 http://HarmonicExplorer.org showing the tone circle and harmonic mountain (matrix) for limit 720, the “calendar constant” of 360 days and nights.

Each of the radiant crossing points represents the diagonal of an M by N rectangle and so the rational “calculation” of a given tone, through the crossing of radiants, is a result of the differences from D (equal to either 360 or 720) to the tone number concerned (figure 8).

Figure 8 How the tone numbers are calculated via geometrical coincidence of cartesian radiants which are rational in their shorter side length at the value of a Just tone number

It is therefore no miracle that the tone numbers for Just intonation can be found at some crossing points and, once these are located on this diagram, those locations could have been remembered as a system for working out Just tone numbers.

Bibliography

Heath, Richard.

  • 2014. Sacred Number and the Lords of Time. Rochester, VT: Inner Traditions.
  • 2018. Harmonic Origins of the World: Sacred Number at the Source of Creation. Inner Traditions.
  • 2021. Sacred Geometry: Language of the Angels. Inner Traditions.

Lubitz, R.A. Schwaller de.

  • 1998. The Temple of Man: Apet of the South at Luxor. Vermont: Inner Traditions.

McClain, Ernest G. 

  • 1976. The Myth of Invariance: The Origin of the Gods, Mathematics and Music from the Rg Veda to Plato. York Beach, ME: Nicolas Hays.

Stewart, Malcolm.

  • 2022. Sacred Geometry of the Starcut Diagram: The Genesis of Number, Proportion, and Cosmology. Inner Traditions.

Double squares: Venus and the Golden Mean

The humble square, with side length equal to one unit, is like the number one. It’s area is one square unit and, when we add another identical square to one side, the double square appears. Above right the Egyptian Djed column is shown within a double square. The Djed is the rotating earth which the gods and demons have a tug of war over. This is also a key story in the Indian tradition, called The Churning of the Oceans, where the churning creates both the food of the gods (soma) and every wonderful thing that emerges upon the Earth. In this, the double square symbolized the northern and southern hemispheres of the Earth. The anthropomorphic form Djed shown above has elbows indicative of the Double square.

Figure 1 The churning of the ocean (Samudra Manthan in Sanskrit)

The Djed appears to be the general principle of rotation of, and apparent motion around, the earth.

The god Isis is (as a planet) Venus and is shown (fig.2) offering up the sun disk: another Djed is below, with her Ankh symbol of Life atop the Djed, now having female arms . This sun most probably points to the practical year as 365 days which is 5/8 of the Venus synod of 584 days. (This ratio of 1.6 is the sixth note of the octave 1 to 2.)

In figure 2, two female attendants provide the duality which one might take to be her two famous manifestations of (firstly) the brightest Evening Star, as the sun goes down, and then (after that) the brightest Morning Star before the sun rises. Above there is duality again with three baboons either side of the sun, perhaps representing the six visible planets: Moon, Venus and Mercury: Jupiter, Saturn, Mars and their “tug of war”.

Figure 2 The creation of Horus-Ra from out of an ankh with female arms atop a djed. from Budge 1899, also fig. 7.8 of Richard Heath, The Harmonic Origins of the World.

The numbers 5 and 8 are Fibonacci approximations {1 2 3 5 8 13 21 34 …} to the golden mean, a transcendent number {1.618034…} which rational numbers can only approximate. Venus and the Earth have clearly settled into orbits around the sun resonant with Fibonacci ratios since the Venus orbital period (224.701 days) is 8/13 of the solar year. And it is this fact that eventuates in what we see on Earth, namely the manifestations of Venus every 8/5 of a practical year. of 365 days.

Figure 3 The double square, its in-circle and out-circle manifesting golden rectangles around itself.

In this post, I developed a result sent to me, that a square drawn within the upper hemisphere of a circle must define a golden mean rectangle either side from its height of 1 and the remaining radius of 0.618034… and so it can be seen that the divine principle of the Golden Mean emanates from the double square, either side of each square, when the double square is embraced by a circle drawn from its center. Obviously, on Earth and between orbits (of Venus and Earth), the Golden Mean (also called Phi) has to be approximated by whole number ratios but the principle is present within the geometry and its out-circle. Schwaller de Lubicz thought the dynastic Egyptians held the Golden Mean to be “the fundamental scisson” (literally “scissor cut”) in the range one to two and, its reciprocal can be seen to share the portion over 1 (figure 3).

One can see that geometry and the early numbers would have been seen as two aspects of what we call space and time, in which “things” are separate from each other in Existence but somehow conjoined within Eternity. What we call order is in fact an achievement of harmony made possible by the arranging and fitting of parts to form a coherent whole. It is this insight which gave meaning to their study of geometry and numbers from the prehistoric onwards, into the recorded history of early civilizations. The meaning for Life on Earth became encoded within ancient and prehistoric symbols, whose geometrical and numerical language of expression went to the heart of phenomena.