Gurdjieff’s Diagram of Everything Living

first created: 28 October 2017

Gurdjieff first presented his ideas to groups in pre-revolutionary Russia. Amongst his carefully chosen students it was the habit to reconstruct talks and diagrams as much as possible, an endeavour that gave us a textbook of Gurdjieff’s ideas called In Search of the Miraculous (P.D. Ouspensky, 1950). This early form of the teaching was radically revised and extended by Gurdjieff, now as an author, during the 1920s, producing All and Everything whose part one was Beelzebub’sTales to his Grandson (G.I. Gurdjieff, 1950). Prior to drawing this diagram just after February 1917, Gurdjieff had been presenting ideas about transformation of energies, human and cosmic, using the musical theory surrounding the octave of eight notes. The Diagram of Everything Living was “still another system of classification… in an altogether different ratio of octaves… [that] leads us beyond the limits of what we call ‘living beings’ both higher [and lower] than living beings. It deals not with individuals but with classes in a very wide sense.”


Figure 1 The Diagram of Everything Living
Continue reading “Gurdjieff’s Diagram of Everything Living”

Numbers of a Living Planet: Preface

The image above is Kurma avatara of Vishnu, below Mount Mandara, with Vasuki wrapped around it, during Samudra Manthana, the churning of the ocean of milk. ca 1870. Wikipedia.

  1. Preface
  2. Primacy of low whole numbers
  3. Why numbers manifest living planets
  4. Numbers, Constants and Phenomenology
  5. Phenomenology as an Act of Will

Please enjoy the text below which is ©2023 Richard Heath: all rights reserved.

It is impossible to talk of a creation outside of the time and space of Existence, though from it, other dimensions can be inferred such as an “Eternity” visible in the invariances of numbers and structures. It is this higher dimensionality that leads to

  1. The recurrence of celestial time periods,
  2. The mental powers to recognise manifested patterns,
  3. The use of spatial geometries of alignment,
  4. The numerate counting of time,
  5. A phenomenology which is neither factual nor imaginary.

The quantification and qualification of Existence, adequately conducted, reveals harmonious structures within time and space, especially in the spacetime of our planetary system, when this system is as seen from our planet. The harmonious nature of our planetary system helped the late stone age to develop a large numerical and geometrical model of the world through counting astronomical recurrences. This model, which shaped ancient texts, implies that solar systems may have an inherent intelligence which makes them harmonious.

Harmony in a planetary system must therefore employ invariances already present in the number field, by exploiting the recurrent orbital interactions between planets and large Moons, this in a connected set of three-body problems. Before our exact sciences and instruments, prehistoric naked-eye astronomers could understand the planetary world by counting the duration of planetary time cycles: the subject my books explore. Through counted lengths of time, the megalithic age came to understand the invariances of the number field and so evolve an early and distinct type of numeracy. This numeracy lived on as the basis for the ancient Mysteries of the early civilizations, embodied in their Temples and in the Pythagorean approach to ordinal numbers and geometries, expressing the “number field” in two or three dimensions, areas and volumes. (see Sacred Geometry: Language of the Angels for an introduction to this)

That is, this early human numeracy naturally manifests within the maths governing rotational systems, this involving key transcendental* constants such as π, these regulating what is actually possible, mathematically, within dynamic planetary systems that are gravitational attractors of each other: these constants include pi {π}, √-1 {i}, e, and phi {φ}.  The first three { π, √-1, e} are surprisingly well-organized rotational frameworks making the behaviour of vectors relatively simple using geometry. For example, the lunar year of twelve lunar months has become strongly resonant with the two outer gas giants, Jupiter and Saturn. The Golden Mean (or Phi {φ})1 can be approximated by orbital ratios between planets through exploiting the Fibonacci number series2, most visibly in the orbital recurrence of Venus and the Earth, seen in the 8/5 {1.6} relationship of its synod* to the solar year. Phi φ is also expressed in living forms of growth, since growth is often based upon the present size of a living body and what it has previously eaten.  Fibonacci ratios are ideally suited to creating the “strange attractors” which can create stable patterns out of otherwise chaotic orbital interactions.

1 My use of curly braces is borrowed from a stricter world of set notation. It offers an ability to place groups of numbers, symbols and other non-grammatical element next to their grammatical context.

2 The series reinvented by Fibonacci uses addition of two previous number to create the next number. His version of that algorithm is {0 1 1 2 3 5 8 13 21 34 55 and so on}. These numbers are found within natural form of life, where such numbers can be generated from two previous states or when two counter rotating spirals of seeds will fill the surface of an egg shape with maximum packing. More on this later.

Through universal mathematical laws and constants, rotational and recurrent systems will effectively provide numerical shortcuts* (J.G. Bennett’s null-vectors) expressing Musical or Fibonacci ratios, and without those ratios being available, relationships within existence would be more complex, less synchronous, and truly accidental. Harmonic shortcuts have therefore given the planetary world a simplified mathematics when viewed from the surface of the earth, within the geocentric pattern of time. This synchronicity provided the stone age with a path towards a direct numerical understanding of time through phenomena (that is, a direct visual and countable phenomenology).

In this way, the megalithic cultures of prehistory found that the geocentric planetary system expressed numerical invariances (these already within the number field itself) thus making the time world of the sky unusually harmonious and intelligible. This contrasts with the now-popular modern notion that, while the solar system is a large and impressive structure, its origins come only from the mathematical laws of physics, these forever operating in a mechanical way. That is, the modern way-of-seeing planetary time is heliocentric and causal and this has hidden an ancient view, gained through the megalithic study of the phenomena in the sky using megaliths as large instruments with sightlines to the horizon events of sun and moon, to simply count of time-as-length and, evolve a very basic numeracy based upon numerical lengths (a metrology) and triangular geometries to compare lengths.

Megalithic methods employed the properties of circles, ellipses, squares, rectangles, and right triangles before the analytical geometry of Euclid, Greek math, or ancient near-eastern arithmetic. This was only possible because key parts of the mathematics of complex numbers, for example, are directly visible in the form of the right triangle and unit circle; as the natural form of two vectors: a length at a given angle (or direction) and another length at different angle gives access to ratios. A right triangle can therefore express two vectors of different length and differential angle, and this applies to a pair of average angular rates in the sky, without knowing the math or physics behind it all. If the two vectors are day-counts of time, then the right triangle can study their relationship in a very exact way. Such a triangle may also have been seen as the rectangle that encloses it, making the diagonal (vector), the hypotenuse of the triangular view.

The properties of the imaginary constant i (√-1) represents, through its properties, the rotation of a vector through 90 degrees. It is this that gives the right-angled triangle its trigonometric capacity to represent the relativity of two vector lengths. My early schoolroom discoveries concerning vectors in applied math classes, that right triangles can represent vectors of speed for example, was without any knowledge of the mathematical theory of vectors. This geometry enabled prehistorical astronomy to study the average planetary periods as vectors. That is, rotational vectors enabled the sky to be directly “read”, from the surface of the third planet, through simple day-counting, comparing counts with right triangles, and forming circular geometries of alignment to astronomical events found on the horizon; all without any of our later astronomical instrumentation, maths, or knowledge of physics.

Physics has not yet explained how the time constants between the planets came into a harmonious configuration, because it is unaware that this is the case. The mathematization of Nature, since the Renaissance, has hidden the harmonious view of geocentric planets and all preceding myths, cosmologies and beliefs were swept aside by the heliocentric world view (see Tragic Loss of Geocentric Arts and Sciences, also C.S. Lewis’s The Discarded Image).

The modern approach then emerged, of blind forces, physical laws and dynamic calculations. That is, while the simplifying power of universal constants is fully recognized by modern science (these having made maths simpler) the idea that these simplifications came to be directly reflected in the sky implies some kind of design and hence an intelligence associated with planetary formation.

Furthermore, modern way of seeing things cannot imagine that the megalithic could conducted an astronomy of vectors (using geometrical methods while not understanding why they worked) and that this empowered a simple but effective type of astronomy, without our mathematical or technical knowledge. This is an anachronistic procedural heresy for the history of Science and also for the present model of history, where science for us is the only science possible, evolving out of near-eastern civilization after the stone age ended.

Foundational myths of modern civilization are threatened by the notion that the world is somewhat designed by a higher intelligence. Until these subconscious conflicts of interest are overcome, prehistory will remain the prisoner of modernity where mysteries remain mysteries because we don’t wish to understand.

2. Primacy of low Whole Numbers

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).

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 ascending fourth (note) 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.

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.

Eleven Questions on Sacred Numbers

In 2011, Sacred Number and the Origins of the Universe was nicely re-published in Portuguese by Publisher Pensamento in Brazil. Their press agent contacted my publisher for an email interview from a journalist who posed eleven questions about sacred number.

Interview:

1) Is the universe a mathematical equation? 

If the universe is a creation then it needs to have organizing principles governing its structure. I believe that this structure is governed by what we call sacred numbers. Numbers relative to each other form proportions that in sound are perceived as musical intervals. The universe is more like a set of musical possibilities, making it more dramatic and open-ended than an equation.

Continue reading “Eleven Questions on Sacred Numbers”