The Vase of Life: Post 6
The Vesica Piscis and 2:1
The book that Wagui gave me back in New York in 1984, as already mentioned, was called “The Temple in Man”. The basic premise of this little book – designed to introduce Schwaller’s more monumental tome: “The Temple of Man” – was that the ancient Egyptians had laid out the famous Temple of Luxor in accordance with extremely learned knowledge of the principles of human anatomy and the ka that they referred to as the “double”. This book reinforced the tradition passed on by Vitruvius and illustrated by Leonardo da Vinci, that Greek temples were proportioned 2:1, in imitation of human anatomy. In essence, Schwaller was suggesting that Luxor was a crucible that described man’s genesis, his phases of becoming and his return to the source as an accomplished spiritual being once the ka had been purified and transformed. The feeling, then, was that the geometry I was pursuing extended to the ka and had to bridge the physical and spiritual worlds. I used the geometry as a motivation to study some of those principles which had also been inspired by my unexpected 1980 dream, a dream that had its own internal choreography and had occurred at a specific point in a major life cycle known to astrologers from all traditions as the progressed New Moon when something new is born from the death of the previous cycle.
One more book introduced me to another indispensable piece of the puzzle. John Michell’s “New View Over Atlantis” (1969, 1973, 1983) was the first place where I saw an interesting introduction to the geometrical construction known as the Vesica Piscis, where the perimeters of two overlapping circles pass through each other’s centers, forming a ‘fish-shaped’ Vesica between them. The Vesica Piscis proved to be of particular importance for drawing the principles of the female which I started with first simply because we are all gestated and born from the female. Without an introduction to the Vesica Piscis I would have been guessing for a very long time.
If op is “1”, “nm”, “fg” and “eh” are 1.73205…, which is the square root of 3.
Yet again, at the heart of this drawing, we see the Prime 2:1 rectangle, ABCD. Note there are 3, equal 2:1 rectangles comprising the 2:3 rectangle, IJKL. The two circles overlap within the Prime 2:1, ABCD, creating a sense of both insulation and remarkable stability or equilibrium. A root-3 rectangle, ‘efgh’, frames the Vesica itself, measured by both long and short axes (‘mn’ & ‘op’). The Vesica Piscis can easily be constructed by the Prime 2:1 by alternately placing the compass point on o and p. With the compass on o, open it to p and swing a circle. Then with the compass on p, open it to o and make another circle. The vesica in the middle, with vertical axis nm, will produce a length that is the Square Root of 3 or 1.732050808…. another fundamental example of an irrational or incommensurable value, such as 2.236068…, 1.61803398….1.273239545…., 1.236067984… My own initial hunch about these irrationals – values that plunge toward the microcosm or toward the curvature of space-time – was that they also pointed to a process. The number was automatically including time as a factor. When insulated like this Vesica, the root of 3 seemed to point to an internal process. But just as important was the realization that the Vesica was being formed within the Prime 2:1 which keeps it connected to outer processes. The 2:1 is the ultimate mediator between inner and outer; small and large; top/bottom; back/front, above/below. It’s like a generator. From it all sorts of functional geometrical forms are being constructed. Instinctively I started to search for where in our anatomy a process might manifest itself through a form as fundamental as this Vesica Piscis.
The perimeter of a 2:1 rectangle consists of 6 of its own widths. A hexagon with radius 1 (and side 1), shown below as QRNSTM (containing 6 small equilateral triangles and 3 large ones), has the same perimeter as the 2:1, ABCD. The Vesica Piscis itself is intimately linked to hexagonal geometry and the square root of 3. For this reason it is a bridge between two worlds: the inorganic mineral world of crystals and the world of biological life. There are 7 divisions
of crystals with 32 subdivisions. Hexagonal crystals constitute one division. An important example of a hexagonal crystal is calcium with its 20 protons and 20 neutrons. Calcium is critical for the structural composition of our skeletal system and contributes to the regulation of the heartbeat. At an even subtler level calcium seems to be essential in facilitating the production of GnRH and LH that bring about ovulation. It is critical in the act of fertilization of an ovum by a sperm. It also seems to be essential in the way that the diffused chromatin material coagulates into chromosomes and self-replicates during mitosis. It also seems to be important in the sol/gel cycle within the cytoplasm of each cell that may contribute to quantum insulation allowing cytoplasmic microtubules to participate in what quantum theory calls “quantum coherence” (see “Toward a Science of Consciousness”, Penrose, Hameroff, 1996, 1998). Meanwhile Carbon-12 likes to form its organic compounds by using hexagonal geometry in its bonding patterns. It is so versatile that it acts as if intermediate between a metal and a nonmetal. Carbon combines into organic compounds 1.5 times more than the inorganic compounds of all other elements combined! Living molecules are closely linked to hexagonal geometry. This suggests a functional compatibility between the Vesica Piscis and the hexagonal architecture that it is a part of.
Note 1.5 is 3/2, the same proportion that frames the Vesica construction above (IJKL) and completely suspends or insulates both Vesica and Hexagon. Male blood pressure is best set at 120/80 which is the ratio 3/2. This seems to have a bearing on where the thorax is situated in the standing adult. It happens to be 3/2 x Unity or 75% of the height! It is quite fascinating to note that there seems to be a bit of flex or stretch between the full 2:1 and the hexagon depicted above. In some crystals, an alternating cycle between compression and decompression will generate piezoelectricity at a subtle scale. In cells such a current is critical for the storage and transmission of information that will become anything from memory, to thought, to muscular response! But it may also play a role in subconscious processing of information that will become conscious. When tubulin protein forms into microtubules they behave a lot like crystals do. The subunits are dipole structures that change their physical conformation as they process information according to their ‘switching times’. More ahead in another section.
The King’s Chamber offers a Clue
By this point, drawing the 2:1, ABCD, had become for me a prime ritual. Below I plugged in values for the King’s Chamber in the Great Pyramid, probably the most famous 2:1 construction on Earth. I had become interested in it quickly, not unlike my interest in Chartres. The KC was like a granite magnet. The attraction was inexplicable, especially in view of the fact that when Abdullah Al Mamun, the son of caliph Harun Al-Rashid, finally succeeded in 820 A.D. with a veritable army of helpers, in smashing his way into the Great Pyramid in search of everything from treasure, to maps, to tables of the celestial and terrestrial spheres, he ended up finding the King’s Chamber to be entirely empty of everything except a lidless, empty dark granite coffer. No maps, no treasure, no mummy, nothing. Al Mamun claims he had a dream in which Aristotle appeared to him and this drove him to break into the Great Pyramid. Now Aristotle was one who apparently knew that the Egyptians had accurately measured the Earth with a measure of 40,000,000 which of course could only be the meter. Al Mamun was looking for a more accurate size for the Earth than the one the Arabs had measured north of Baghdad to advance the Arabic interest in navigation. The mention of Aristotle was not at all a meaningless one. Aristotle may have inherited some information from Herodotus or others who had traveled extensively in Egypt just prior to classical times.
The more clues gathered, the more I gradually came to believe that the KC was the consummation of human understanding of holism: what holism is and what it implies. In this sense it was the ultimate gate between material process and spiritual liberation. I came to see holism not merely as a human ideal, something invented by the human mind – a psychological striving alone – but rather, it was beginning to appear to me as if it were the basis itself and that we were all the off-spring of that basis! That it was founded upon the principles of the double square struck me as no meaningless coincidence. The geometry had become like a great picture puzzle. Many of the pieces appeared missing or very well hidden. I had to develop a nose for finding them. The King’s Chamber was a piece of the puzzle that appeared early. But like great seeds it was slow to grow into consciousness. Not being blessed with patience at birth, I had to come to understand that patience is like an enzyme: it stimulates growth in understanding. Without a strong root system understanding will break down under stress. The enzyme of patience must stimulate and then allow the roots to take hold. One of the rituals in the life of a cell is the biosynthesis of enzymes!
With the King’s Chamber I had to learn the difference between presumption and reality. This was a ritual all by itself. The puzzle of the King’s Chamber was incomplete until I began to familiarize myself with the Pyramid Texts which ought to accompany all considerations about the King’s Chamber like a manual. Otherwise each wave of explorers simply project upon this empty, beautifully-crafted 17 x 34 x 19 foot room, anything they want to find. The Pyramid Texts do not merely remind us why the King’s Chamber was built, but also how it was to be used. One observation I latched onto early as I started to read about the King’s Chamber is that the angle of the passageways was consistently that of the diagonal of a 2:1 rectangle: namely ~ 26.56°. This angle includes the ascent up the Grand Gallery to the Great Step. Then, when one gets through the intermediate chamber and reaches the apotheosis of the Great Pyramid’s architecture, one discovers that he has entered a marvelously proportioned 2:1 rectangular room – completely enclosed in red granite. So much precision was built into the Great Pyramid that I believed immediately that all design that pertained to the 2:1 construction was intentional and highly informed, and by no means just a matter of convenience the way we moderns manufacture plywood for example. At times it gave me chills: the scale, the workmanship, the magnitude, and the consistent presence of the dynamics of the double square. The double square had become the sole means of access; the sole starting point for me to lay down the geometrical principles of our anatomy. So as I learned about the Great Pyramid I was also learning about man; and vice versa. It took many more years for me to realize that this reciprocity was the secret of the Egyptian neter called Maāt and the basis of truth. In Egypt Maāt and “truth” were one and the same. The science of the pyramid was not a detached science. It was not merely an objective pragmatism. The ancient Egyptians were as concerned with the human spirit as they were with the human body. This has been difficult for us to accept.
The ritual here was to begin to feel the volume of the KC. I learned quickly that if you swung the arc of CG down to the floor you could get a square that was 5.236 x 3/2 on a side or 7.854 m. This square (IJKL) would have the same perimeter as the KC floor (DEFC) and a circle with a diameter of exactly 10 meters. 3/2 is 3/4th the length of a 2:1 rectangle. It’s the diagonal of the end wall. So, 7.854 x 4 = 31.416 = 10.472 x 3 = 5.236 x 6 = 10 x 3.1416…. an excellent value for pi (3.141592654…).
Every time I drew out a 2:1 rectangle, I realized I was drawing the shape of the most famous chamber on Earth. And every time I started a new drawing of the human form I always started with a 2:1 rectangle. And always – despite all simplicity – I felt I was performing a special task, something I could feel in the middle of my chest. It inexplicably felt like Christmas or Easter when something too delightful to measure is reborn. I felt as if it was the greatest keyhole to which I had been given a key. But most importantly, I was discovering a side to myself that had been lost at school. I lost all patience for geometry, arithmetic, anatomy, these basic things – through no real fault of the school. I had been injured severely on a football field when I was 17 and after exactly one month in the hospital I emerged a different man. Nobody thought to get me a tutor to help me catch up. In fact some people saw me as a troubled “loser” after that injury. Nobody was imagining the healing process except the nursing staff at the hospital who exhorted me to go easy on my ribs for many more months. Everything was quickly replaced by social drama; the drama of employment, of relationships, of politics, of corruption; of injustice; of human deception, bigotry, cruelty; the drama of the struggle for survival and how to overcome the alienation that had seeped into the 1960’s and induced a violent revolt from many strata of society. In that drama I lost touch with the basis. While in the hospital visitors kept telling me about the movie “Midnight Cowboy”. As soon as I got out of the hospital I immediately started to search for a new basis. I walked from Wellesley Street in Toronto up to Young and Eglington to see Midnight Cowboy. It was the first thing I did. I sat through it twice. I could barely get out of my seat. Dustin Hoffman, in one blow, altered my whole view of humanity – he reset my emotional compass. The way he interacted moment-by-moment with Jon Voigt transformed my image of all of us strewn across the glossy neon highways and byways of North American alienation. Several years later – after much trial and error – my 1980 dream encouraged and drove me to rediscover the basics that I had abandoned after the trauma on the football field. But I never imagined that the basis would start with something as simple as a double square; a rectangle the length of which is twice its width.
Ritual: A Necessary Discipline
I had avoided ritual my whole life. I hated it. I rebelled against all forms of indoctrination – to my own peril. But that changed unexpectedly. I learned to tolerate yoga from 1977 to 1980. It had become essential in order to handle the overwhelming constellation of the archetypes when they started to ‘erupt’ in the summer of 1976 in New York City. In one dream that summer, in particular, I was shown the categorical difference between suppression and repression. This was some dream – completely unexpected! My real self had been repressed beyond all recognition. The dream blew the cap off the repression as if it were the top of a volcano. The lava was the rage I had held down for decades. By holding it down I had blocked the development of a truthful self. My cynicism was also repressed and was assuming many masks. Yoga was one critical way back to authenticity, for want of a better term. It helped to guide the chaos of my whole internal life as it was progressively shaken up and ‘reduced to ash’ from 1976 to 1980. The breathing and correct posture became my support when everything else had fallen away: everything except the steadfastness of a few relationships and the body and mind the breathing was animating. I also recorded as much as I could in my journal. I wrote down every dream in full. No matter how tired I was, I wrote out the whole dream. The dreams felt instinctively like medicine. I made sure that I ingested the entire dose that Nature was dispensing into my psyche with each dream. I was convinced that a conservation law was at work and that the psychic energy of each dream had to be caught in the hermetic flask. Every drop. Every nuance. This was pure, but deep, instinct. Nobody could have taught me this. It was something I had to feel. Something about the dreams themselves awakened the instinct. I was never cynical about dreams. They always ignited my inner attention and left me wondering, curious, deeply interested.
This disciplined ritual was eventually accompanied by the ritual of art – sometimes very simple; at other times overly complex – in which I allowed order to ‘spontaneously’ organize a chaos that I saw as the beginning of things. The chaos was providing energy for a self-organization that took different forms. At specific stages I used only pen and ink on natural papers. I was trying to feel out a dynamic balance between two forces: centripetal and its centrifugal reaction to the confinement of an energy. The relationship between macroscopic gestures and quantized patterns that formed within them, affected the way I did yoga. Prior to yoga I had forced everything. I did not value or understand gentleness. I had no patience for the way Nature grew and built itself up from the small to the large. I wanted everything immediately. In other words I was in a non-stop panic. The only purpose I had was passion. But the passion lacked pattern and order. I had no sense of building a life. After the football injury I felt I had nothing; no foundation, no trajectory, no support.
By repetition and inclusion, eventually these rituals accumulated energy. With each accumulation the nucleus was altered and influenced. Slowly it transformed until I realized that only a few archetypes held together the transitory sequences of appearances – important at the moment, but not permanent in the long run.
It was as if a new nucleus were giving birth to itself. I felt like a mother hen, listening and brooding. I was offering heat, concentration, attention to a ‘yolk’ that eventually hatched into a revelation of the self, providing a new foundation stone for me to keep going. The ‘lotus bloom’, its final product, took the form of a dream – to my continued surprise. I had no idea what to expect. There were sign posts but the suspense was too deep to have concrete expectations. The dream possessed the numinous feeling that somehow it contained everything, or that somehow everything was implied in it. That was a remarkable feeling. It contained a spiritual root, not merely a physical root. I recall clearly the time when I believed everything was ‘physical’. I thought the spirit was merely an illusion or worse: a delusion. This dream completely overthrew that conceit. I experienced directly the dramatic difference between body and spirit. Moreover, it was experienced as a metamorphosis – a sudden transformation and expansion from physical form and sensation to spirit state. There was orientation of consciousness but no ‘body’. The ‘perfume’ of this experience has never gone away: for better or worse. This extraordinary revelation of the inner paradigm of the self, which in my case assumed the forms of many religions as if coalesced into a single, harmonious whole, compelled me to look at the geometry with fresh eyes, not as a school boy. It was not indoctrination. It was self-evident discovery: seeing the ‘givens’ in a more natural way.
The Vesica Piscis and Female Anatomy
Back in 1985 I was so taken by the Vesica Piscis that I immediately tried to find out how it could be applied to our anatomy but needed to get the 2:1 rectangle and its close cousin, the golden rectangle, working first. The 1:2 was evident immediately in the Vesica Paradigm. (a) The width of the Vesica (EF) which separates the centers of the two circles, the Root-3 rectangles (QRST, RNTM, etc.) and Prime 2:1 (ABCD), as well as each side of the hexagon (QRNSTM) seen in the image repeated below, are all equal and can be called ‘Unity’. An important part of the ritual of geometry is the acceptance of unity. Everything is relative to it. It is one’s module and it seems that it behaves both as a particular unity
while yet also imitating Unity as a universal image. Unity is a bit enigmatic, suggesting that in fact it is sunk in a greater whole as is the imaginative mind that is contemplating such images. Each unity is itself a fraction of a greater unity, while yet acting as a representative of that greater unity in the way it behaves.
(b) The diameters of the two interpenetrating circles, (GF and EH, AD and BC), the height of the 2:1 rectangle (ABCD), the height of the 2:3 rectangle (JK), the span across the hexagon (e.g. MN), the sum of two equilaterals side-by-side and the diagonal of the Root-3 rectangle, QRST, are all 2.
(c) The long axis of the Vesica (OP) and the height of the rectangle QRST that contains it, are both the square root of 3. For this reason it is wholly contained within the 2:3 rectangle, IJKL, and also within the two overlapping circles – except exactly where they first meet! Those are the ‘mouths’ of the Vesica. This suggests a level of insulation. It is capable of incubation and gestation. Psychologically, it seems to imply the potential for what could be called an inner life; or the cultivation of a soul. But also it could refer to inner physiological processes capable of homeostasis. Hexagonal geometry, with the Vesica Piscis, seem to lend themselves to this type of self-regulation. Also note that the 3 large equilateral triangles, MRS, QNT, NTQ, all have sides equal to the root of 3. The difference between the perimeters of these equilaterals and the perimeter of the hexagon that contains them is, in the standing adult, the measure from the vocal cords to the crown of the head. I’ll come back to this.
(d) The diagonal of the Prime 2:1 is the square root of 5.
Below I have shown how the Prime 2:1, ABCD, can evolve from a root-3 rectangle or vice versa. Here you see “2” as the diagonal of the root-3 rectangle and the length of the 2:1 ABCD. What they share is unity: their common root. In the case of the King’s Chamber, Unity is 10 cubits or 5.236 meters which I’ve written in here as a module.
It is quite an interesting exercise to contemplate the evolution of measures that arise between 1 and 2. One key is the right angle (e.g. DEC, ADC, etc). I should mention that in the Big Dream that I am always referring to, the right angle was very pronounced throughout the entire dream including the mystical part of the event. It seemed to characterize the relationship between magnetism and electrical energy in what we call electromagnetism. The right angle seems to be at the heart of self-organization at a very subtle level of our makeup. It likely plays a key role in the architecture of the genome and the way specific parts of it will rotate their position affecting both adaptation and behavior. The image above evolves when 2 is dropped from an original right angle (as a 2:1 rectangle) to the opposite right angle or side and becomes a diagonal (DC). Below you can see what happens when the square root of 2, the diagonal of a square, is dropped down. The new diagonal of the root of 2 rectangle becomes the square root of 3 and its longest diagonal becomes 2 or the root of 4. These progressions occur naturally from within the 2:1 rectangle.
Now what happens is the diagonal of the base (root of 3) is turned into a right angle and becomes the volume below with 2 as the new diagonal. The root of 5 now becomes the longest diagonal of the volume.
Note that this box contains 4 root functions. 3 of the 4 behave similarly in that they are all irrational numbers: 1.4142135…; 1.7320508…; 2.236067978…. The 4th behaves differently because it is a whole number, 2. That same ‘2’ will be the height of the human figure that starts as a 2:1 rectangle. In this progression, the root that behaves differently is the 3rd one. Coincidentally in the Egyptian Djed Column, they usually depicted 4 Capitals atop the Djed. The 3rd of the 4 capitals usually portrayed the two eyes that protect the seed.
The new long side above becomes the root of 3. The diagonal rectangle (HBCE) within the volume, restores the Prime 2:1, with a diagonal that is the square root of 5. So you have the root of 2, the root of 3, the root of 4 and the root of 5 as a progression within a very well-proportioned volume. I renamed the 2:1 diagonal HBCE to show it is a new development as well as a restoration. This root-3 box is a very attractive volume. It plays an important role in determining critical elevations at Chartres. The cathedral would be quite different without it. Note that the angle CED is 60° which is simply the inverse tangent of 1.732050808….
The Root of 3 also acts as a bridge from hexagonal to cubic geometry through volume. It measures the transformation from a square plane to the volume of a cube. We can see this below where the Root of 3 is the common denominator between the diagonal of a Root of 2 rectangle and the volumetric diagonal of a cube.
And as we have seen, it is also the long axis of a Vesica Piscis, demonstrating a behavioral affinity between volume and plane. Under a pressure variation cycle, such as tidal gravity and other forms of movement, piezoelectric and ferroelectric current can be stimulated in biological material that is behaving like a crystal. By compression/decompression cycles the electrons change their positions relative to one another. When the material reverts to a previous position or conformation, current flows as the electrons also fall back to their initial positions. This current, if stored in patterns, could be a deep layer of biological memory as well as the substratum for sub-conscious processing capable of becoming conscious in rapid intervals – contributing to the experience of a stream-of-consciousness that makes life what it is. The fact that unity as a module is so ubiquitous throughout diverse structures of one’s anatomy; and the fact that progressions like the square root of 3 show up in such diverse – but interrelated – geometrical architectures, suggests that one compression/decompression wave might affect a large area of crystal-like biological matter. The current moved, or stimulated, is not merely electrons, but strengths of photons (the ‘exchange’ particles), which mediate the electromagnetic force, are also being stimulated. Photons are massless, move effectively at the speed of light, and have no anti-particle. They must play a significant role in the nature of mind and the way the senses function as an ensemble in acts of consciousness, states of appreciation, emotional states when we are profoundly moved, etc.
So I will repeat this principle. The fact that unity appears in so many places in a complex geometry; and the fact that progressions from unity, like the root of 3, also appear in diverse biological architectures, suggests that many parts of our make-up are being affected by the same stimuli. Everything is contributing to consciousness. Unity is being preserved throughout as if it were an architectural module. This is significant because at the quantum scale only minute amounts of energy are needed to effect a difference capable of registering ‘memory’ or information. Once you have volume it is possible to store dynamic information that can even self-replicate. Some of our most important knowledge may well be holographic and through shared resonances offers the experience of synchronicity which greatly enhances the feeling of belonging in a meaningful way to a greater whole – even if often unconscious. The square root of 3 is ‘measuring’ a common resonance preserved as an energy paquet or module across a variety of states which may alternate rhythmically producing not only life’s vast variety of qualities but also our deep inner sense of a universal musicality.
I will repeat, in the image below we see how a Root-3 rectangular volume expresses all the root functions: Root-2, Root-3, Root-4 (i.e.2) and Root-5, when Unity is the Prime square at each end. By making Unity 10 cubits we can see how this volume engenders the exact size and shape of the floor and ceiling of the King’s Chamber, here seen as the diagonal rectangle, HBCE. The floor and ceiling of the King’s Chamber are 5.236 x 10.472 meters. The height is half its own diagonal and is 5.854 meters or 11.18 cubits.
So, the long axis of the Vesica (PO) is an irrational number: 1.7320508…., the square root of 3. The diagonal of the Root 3 rectangle is 2 (QS), a whole number. The diagonal of the
Prime 2:1 (AC) is the irrational root of 5 or 2.236068… while its height is 2 (BC). What we have is an alternation within one image between rational and irrational, inside and outside; right angle and diagonal. The inside of the rational 2:1 is irrational; and the inside of the irrational Root-3 rectangle is ‘rational‘. What the two share, however, is Unity as the width and on each side of the hexagonal lattice (e.g. MQ, QR, RN, etc). It will be important then to see what role an upright Vesica might play in our anatomy, since its vertical axis is an irrational number (1.73205…), while its horizontal axis is the same as the width (Unity) of the Prime 1:2 rectangle and all the sides of the hexagon and the width of the Root-3 rectangle! Infinite and finite are operating together!!! The infinite values that fall off toward the infinitesimal may refer to process + curvature, which produces a cycle, feedback loops that return information, or energy, back into the system, which when it self-regulates results in rhythm. A cycle can induce a pulse: a wave of energy that keeps regenerating itself. One immediately thinks of the centrality of the auricular and ventricular nodes of the heart.
I should also mention that each curved half of the Vesica (O thru F to P and O thru E to P) represents 120 Degrees of the opposite circle. 120 Degrees is a ‘trine’ in astrology, often associated with perfection, ease of activity, harmony and success. Taken together the two sides make up 2/3 of a whole circle, while each side is 1/3 of a whole circle. The Temple of Solomon also used the division 1/3 and 2/3. The Holy of Holies was 1/3 the whole, while the 2:1 rectangle of the main court was 2/3 of the whole. One principle, two different applications. The width of this very special Vesica or vase, is 1/3rd the whole structure (2 interpenetrating circles span ‘3’), giving the Vesica some affinity with the “Holy of Holies”.
2:1 as the Seed
We just saw the role played by Unity, AB, in the images above. In the quite prolific image below, that Robert Lawlor calls “the polygonal world” in his very comprehensive 1982 work called “Sacred Geometry”, you can see that every single side of each polygon spawned by the commanding 2:1 rectangle equals Unity (EF, AB, QR, etc.)! The importance of the width of a 2:1 rectangle will also be evident when we get to the canon of the female. Lawlor sees the Vesica as the ‘seed’ in this image, but I have altered the image by completing the 2:1
rectangle (ABCD) and since it is the way to construct the Vesica and the Root of 3 rectangle, it strikes me that the 2:1 is the ‘acorn’ from which the entire glorious tree of life springs, including the polygonal world depicted in the illustration (see Sacred Geometry, p. 34, Thames and Hudson Publ). Everything comes from the 2:1 and everything returns to it. It is the crucible. The 2:1 supports the interpenetrating circles within the 2:3 rectangle (IJGH), that in turn suspends the Vesica within the hexagon (not shown here). As stated more than once already, the Vesica appears to be insulated and this insulation may be one key to both life and consciousness. Eventually they develop a relationship of vital, reciprocal support.
Note the dominance of the Prime Unity Square, ABFE. It contains 1/2 the Vesica and is 1/2 the Double Square ABCD. We will see this again in the female canon just ahead. The point H will become very important in the human being. Within the Prime Square is the equilateral triangle HFE. This is supported by Unity EF, so EF becomes the ‘root’ of the whole image. Both the apex of the triangle EFH and the ‘mouth’ of the Vesica converge on the point “H”. Note, too, how each new polygon develops by using the radius (unity) of the two circles which compose the Vesica Piscis. So when they grow from the common base or root, EF, their next side is always the radius of one of the circles (for example, Fg, Fc, FB,FH, EH, EA, Ea, Ed, etc.
EF represents 2 points, E and F, which have become separated through assimilation/growth/division. They represent the equator or maximum width of the Vesica. As it pushes outward it meets resistance, placing the Vesica under pressure, to which it responds until some form of ‘hydrostatic equilibrium’ between inside and outside has been achieved. So EF is both Unity as a module of spatial measure and Duality simultaneously, since it implies separation, a width and a pressure! H offers a 3rd point creating the triangle, HEF. It appears insulated. But in the human being, “H” offers an outlet for what has been insulated: a means of exchange between inside and outside; a fundamental alternation.
So EF is Unity or 1. HEF is 3 and 4 is ABFE. What is missing is ‘2’, except we can see that BC is 2. So BC-EF constitutes a T-cross. We start with the 2:1 rectangle. It consists of 2 squares divided in ½ by EF. EF leaps to 3, the equilateral, and everything grows from there. An equilateral triangle, the Grand Trine in astrology, is remarkable in that on the outside it is treated as three 120° angles = 360°. But on the inside it consists of three 60° angles which comprises ½ of 360° = 180°. It is as if the equilateral really had a double life: an inner life that is highly constructive and an outer life that flows well with the outside world.
In any case, each entity is composed of unity for each of its sides. It grows and transforms and changes shape by using the same length, AB or EF. Moreover, if ABFE is 4, then ABCD should be considered 6 where half is submerged like a root in the line of 5 interlocking circles at the bottom; 3 adjacent to one another; 6 radii in all and so the same length across as the perimeter of the Prime 2:1 ABCD. It is AB + BF + FC + CD + DE + EA = 6. It has the same perimeter as the hexagon QRNSTM above and FdefgF in this same image. Meanwhile the two circles that create the Vesica Piscis have a length of 3, which is ½ the perimeter of the Prime 2:1. Note how the extensions of the diagonals of the lower square of the Prime 2:1 (EC and DF) give rise to the sides of the octagon and points of the dodecagon. The extended diagonals of the upper square, ABFE, give rise to points on the octagon and points on the dodecagon.
After the square, 4, comes 5, the pentagon, here marked EabcF. It too grows by using the same side: ab = EF = Unity! The ratio 4:5 determines the Major Third in music. Next is 6, the hexagon, EdefgF. It has one more side than 5 and so they are in the ratio of the Minor third, 6/5. This ratio is used precisely to determine the oblateness of the snake-wrapped disc atop Ra-Horachty shown earlier. Both lengths are among the strings of the harp as well and make an appearance in the internal measures of the throne! After 6 comes 8 and it too grows by using Unity for its sides. After 8 comes 10, the decagon which also uses Unity for each side. And finally we come to the dodecagon with its 12 sides which also uses Unity on each side.
Vesica Piscis and Pacific Pompano
Since the Vesica Piscis gets its name from the shape of a fish, we might expect to find it in the construction of a real fish. I will use the Pacific Pompano as an example.
In fact I did not get to the fish as fast as the anatomy of the human, but eventually I drew up the following image of the Pacific Pompano. Note that in Nature, symmetry is difficult to maintain. Harmony seems to depend on a complementary relationship between dominant and recessive. Something must assert while something else yields. This breaks symmetry which by itself would remain pure antagonism or just rupture, separation. In Egypt antagonistic symmetry is represented by the strife between Horus and Seth that is ‘Judged’ or mediated by Thoth using the principles of Maāt. Thoth ends up reconciling the two combatants, one of the great attributes of ‘Hermes’ in Western subculture, a subculture that bloomed at Chartres.
In this fish, the upper circle is a little larger than the lower. One imagines that it has to do with water pressure. The upper circle is more expanded due to less pressure being exerted on it as it is closer to the surface than the lower part of the fish.
This is not the place to comment on all the geometry in this fish – the marvelous engineering feat that it is! It will have to wait. However, note the opening of the mouth right at the extreme left pole of the long axis – the Root-3 axis of the Vesica! This phenomenon will become beautifully evident in the human female. Note here that the complete fish is contained within a prime 2:1 rectangle, labeled 5 x 10 for convenience. But the main body of the fish, minus its fins, is contained within a 4 x 8 rectangle that also contains the Vesica. We will see further importance of the ratio 5/4 ahead when we examine ‘squared-circle’ geometry. It has several, diverse applications and corresponds to the Major 3rd in the diatonic musical scale. The 5:4 ratio is a rich function – a great bridge.
Thoracic Vesica Piscis and Woman
Below, you can find the Prime Square, ABTS, surrounded by EFUV which is 1.236 x 1 (EF x BT). A Vesica Piscis (red) has been inscribed right in the middle of the Prime Square. So now, W and X, a new unity, act like E and F, seen above in the poly construction. WX is 1/3rd of AB or QR! I did not mark H because it’s too crowded, but it falls exactly on the mouth of the female canon. The long axis of this Vesica appears to measure the span between the level of the last lumbar (where we bend at the waist) and the level of the mouth. It strikes me
without any reservation, that this Thoracic Vesica Piscis is a classic example of where vital function and aesthetics converge and act together. Beauty, looked at this way, cannot be construed to be merely in the eye of the beholder, but rather is where nature combines form with function. Note that in the case of the thorax, this beauty, strength, uprightness, hope, vitality, pride, joy, firmness and calm do not arise from the golden section. These attributes are arising from a vertical division of the height into ¼ and ¾ or .5 + 1.5 and a horizontal division of 1/3rd of Unity across, all in the standing adult. Horizontally it is unity divided by 3. Vertically it is 3/4 of the way from the ground to the crown of the head. It is ½ way from the pubic bone – which divides male sexuality from female sexuality – to the crown. In yoga this is the crown chakra. The center of the thorax is the heart chakra.
WX forms the base of an equilateral triangle that has its apex on the mouth. This line WX is also the line along which one places one’s palm in the case of a cardiac arrest. One presses down forcefully right in the middle of this line in order to restart the heart. In view of this fact, it is quite remarkable to see the Prime Equilateral ‘open’ with the Thoracic Vesica on the mouth of the female. One imagines here that two fields of force – two spheres – have united. Their combined energy has caused an amplification in the middle – opening it up like a fish-shaped vertical mouth. In its midst is the heart that oddly resembles clasped hands, as if our actual hands were an unfoldment of what had been in the heart. The cross of life is formed by the two axis of the Thoracic Vesica.
One can see clearly below that the upper branch of a Prime 6-pointed star (ab) establishes the latitude of the Thoracic Vesica. Where the two equilateral triangles intersect is the level of the pubic bone and the flexibility of the upper femur bones that allow the hips to swivel.
The 6-pointed star was one of the first figures to be included. It convinced me of the importance of dividing the 2:1 into quarters. ‘f’ is the complement of ‘c’. ‘R’ is the complement of ‘S’. ‘ab’ is the complement of de. An equilibrium is struck at the pubic bone through the structural ingenuity of the femurs and their heads and the way they support the pelvis. I will show this detailed geometry at another point. It is very rewarding. Initially I found it disconcerting that there were not many obvious points where rational structures like the 6-pointed star, converged with the irrational structures, but Nature is strategic and thematic, so one only has to dig and persevere and these convergences become evident – such as the vertical axis of the Vesica Piscis. One of the most important convergences happens at the mouth.
To repeat, WX is 1/3rd of Unity (AB). We will come upon this identical division within the face – for the nose! What the Thoracic Vesica and nose have in common, is the breath: exchanging what we call atmospheric gases; oxygen and carbon dioxide – a process that starts in the first cells. But the mouth is involved in another type of exchange as well: the trafficking of words and all that words convey. The mouth is also involved in the intake of liquids and solid food. I will show the face later, when other details have been established first.
Thoracic Vesica Piscis: Vase of Vitality, Dignity, Graciousness, Strength
I must emphasize that this placement of the Vesica Piscis inside Unity, the upper square of the Prime 2:1 rectangle, centered on the 3/4th latitude or 1.5 x Unity, has constituted the single most important breakthrough in this
entire body of work that began in New Zealand in 1985. It can be no coincidence that if somebody has experienced cardiac arrest, what you do is pinch the nose, open the throat by tilting the head back, breathe into the mouth until the thorax rises. Then you draw a line across the nipples; place your palm in the dead center between them. Pump vigorously 30 times at a rate of 103 per minute. Then breathe again until the chest rises. Resume pumping in the dead center of the chest. This may well restart the heart or at least keep the blood pumping to avoid brain damage. But you are placing your hand exactly where the 2 axes of the Vesica Piscis cross one another. I repeat: this is at 3/4th the height or 1.5 out of 2. The two axis of life here happen to have a specific ratio: 1:1.7320508… which is the square root of 3. Moreover, it lies on the upper branch of a Prime 6-pointed star of a descending equilateral triangle.
Since the Thoracic Vesica Piscis has turned out to be so prominent at this point in the path, it makes sense to discuss the male thorax that took me much longer to appraise. As much contemplation was required as creative activity. What needs to be stressed, however, is that I derived the male thorax and pelvic vesicas from proportions established first in the female. The female program was first or primary. Moreover, I found there to be a reciprocal relationship between thorax and pelvis. The prime difference between male and female biologically is that the female pelvis must accommodate the gestation of a new human being from fertilized egg to a 9-month baby that has out-grown the womb. To establish the interdependency or conservation law between thorax and pelvic Vesica, I will just return to the female for a moment.
Female Pelvis Vesica Piscis and the Golden Rectangle.
The second biggest breakthrough was situating the Pelvic Vesica. It did not actually take long and was inspired by an error passed along by one of my favorite writers that I will come to ahead. He had suggested that a particular book, sculpted at Chartres, was a golden rectangle and that it was related to teaching. I was inspired by this only to discover later that the proportions appear to be 1:1.68 which is the Major 6th in music. Relative to Middle C as the fundamental, it is A-440. But I was so convinced by the golden rectangle after Doczi’s celebration of its properties, I had already convinced myself that it had to be the proportional container for the womb that all of us issue from! So I tried it out with minimum variations. In fact what I found at a small scale with a dullish pencil was that the slightest deviation from the ratio of the golden rectangle and the woman did not merely look ugly, she also looked wrong, dysfunctional. An overweight woman will usually not look ‘wrong’; only overweight. I learned quickly that there was a significant difference between the two. For health, efficiency, balance and flexibility, it seemed that the basic relationship between the two Prime Vesicas had to be that which inheres between the Prime 2:1 and its Golden Rectangle. What they share is the height, namely “2”. So, it was simple. The Pelvic Vesica is inscribed within the width of the Golden Rectangle (EFGH) which you can see below situated between V and U, which passes through the pubic bone or ½ the height.
Half is the primal basis of sexuality. Within the female, the prime regulating function that determines the vital relationship between thorax – organs of the woman’s life and her ability to nourish all parts of her system and the pelvis within which all of us will be gestated and from which we will be born – is the ratio 1.236 or 2 ÷ 1.618. So:
RS ÷ TU = 1.236 = EF ÷ AB.
The Golden rectangle is characterized by EH ÷ EF = 1.618 = 2 ÷ 1.236. 1.236 also situates the mean navel position which separates assimilation from elimination. The newborn human is unique in that it assimilates for 9 months and is meant to be ‘eliminated’ as soon after that as possible. 9 months is about 3/4th of a year. So the harmonies must be exceptional. The Golden Rectangle circumscribes just such a combination of harmonies. The male construction is a little different as we will see. The initial drawings were rudimentary and looked like this one below where emphasis has been placed on the “Unity” square and ‘1.236’ square, HGIJ .
As more data was added, these drawings started to transform. Here I was trying to emphasize the bare essentials only: the 2:1, ABCD; the Golden Rectangle, the 1.236 x 1.236 square, HGIJ, and their interaction at the point of golden section across HG. But even this is packed with information. It takes time to introduce the contents of the Pelvic Vesica because it was inspired by a study of the Astronomical Unit which gradually became clearer with time. One note of harmony is that the square shown within the Pelvic Vesica turned out to be exactly 1/16th the area of the ‘golden square’, HGIJ. This square has the same perimeter as the circle contained by the Pelvic Vesica with circumference equal to ‘1.236’ or GI.
In this next image we see the abdominal and pelvic circles more clearly. These two circles that touch at the golden section (navel) that I came to label ‘abdominal’ and ‘pelvic’, were derived when I divided the
Astronomical Unit by the golden section and then collapsed each length into a circle which touch each other at the navel on line OP above. The widths of the two Vesicas are .3333 and .412. Their relationship is of course: .412 ÷ .3333 = 1.236. If ever there was a key to life offered by the cornucopia we call the 2:1 rectangle, it is 1.236068 = 2.236068 – 1 = 2 ÷ 1.618034…..
The Astronomical Unit
The inspiration came from a hunch and then stuck. I was interested in the hydrostatic pressure that surrounded the Sun out into the planetary field. I was particularly interested in the vital aspect of conditions on Earth at 1 AU from the Sun. When using the metric system, the diameter of the Pelvic Circle was the same number as a circle with perimeter equal to the perimeter of the Great Pyramid and a square cast by the shadow of the Great Sphinx at noon of the Equinox and also the square table identified at Chartres by an invaluable researcher (Louis Charpentier) coming up in the pages ahead. It also had the same value in the metric system as days in the synodical month. The date that worked very well as a module was around April 16 for the year I was doing this (1998). The Earth is about 150,108,000 km from the Sun at that time (~ 1.0034 AU). The golden section divides it as 92,772,000 km and 57,336,898 km. The perimeter around the socket stones of the Great Pyramid is about 927.7 meters. The Sphinx square is about 92.78. At Chartres the square is 92.7724 meters for perimeter. A circle with the same perimeter is 29.53037 meters. The value of the synodical month is 29.53066 days. What is particularly important about the 29.53 meter circle is that its radius, 14.76518 m, appears to have been used in determining the elevations of Chartres through triangulation. So, it is this 29.53 circle that seems to help set the scale of Chartres. But in any case, that is how I came to place those two circles. It was by collapsing the lengths of the golden division into two adjacent circles which touch at the navel. The larger one is the enclosure within which we are all gestated. We will see this same principle at the Great Pyramid in the most remarkable way when we get to the ‘squaring-of-the-circle’. The size of the King’s Chamber is not a meaningless scale. Its dimensions are set by the size of the whole Pyramid in a unique way.
Male Thoracic Vesica
I noticed that men had proportionally longer arms than women. A very standard armspan became the fathom over time: say 1.854 m for a height of nearly 1.8 m. This also corresponds to a minute of terrestrial arc (Earth Circumference ÷ 21,600). Women had absorbed more energy by a conservation law into the pelvis. Men have more energy available for the thorax and arms. So the male was going to exceed unity for the thorax and be less than 1.236 for the pelvis. I decided to take the thoracic ratio from the function. What happened was I found the arms at the same time by creating a circle over the heart instead of the navel, as Da Vinci had done, and set the angle at the ideal angle of botany which divides the circle by the golden section. A ratio emerged that dovetailed
more than one function. When I saw that the arms were an expression of the heart and that the upper limit of the heart was the Aortic Arch, I could see that that limit was being set by the golden number itself or 1.618 from the soles of the feet. 2 ÷ 1.236 = 1.618. But the center of the heart at 1.5 was the center of the heart circle and was the way to restart a stopped heart. So I divided 1.618 by 1.5, giving: 1.618 ÷ 1.5 = 1.07868 = 2/3 x 1.618 = 1.33333 ÷ 1.236. I felt this was a tension and supplied a pressure, a force, that set in motion the arms and hands from the thorax. Moreover, 1.3333 marks the solar plexus and this divides by 1.236 into the same ‘force’. So I decided to try it as the width of the space within which the male thoracic Vesica should be inscribed; i.e. ‘wz’,
1.07868 does a lot of work here. It is the width within which the overlapping circles of the Thoracic Vesica is contained. It is the radius of the ‘heart circle’. It measures from the soles of the feet to the tops of the femur heads where they plug into the pelvic structure. It measures from the crown down to the base of the male sex organ. And it is the ratio between the solar plexus and the navel. The solar plexus is a power point. It is between the kidneys. It is the level of the pancreas. It is the center of blood traffic from spleen to liver; kidneys to prime blood vessels; and the portal vein that takes all that has been absorbed by the small intestine to the liver for refinement into glucose – our main source of energy. The navel, represented by 1.236 is the point that separates assimilation – controlled by the solar plexus – from elimination, whether waste products or new life. And 1.3333 ÷ 1.236 = 1.078689. The limit of the sex organ from which new life will emerge is set by 1.078689 as measured from the crown. Moreover, when we compare the relative sizes of male and female Thoracic Vesicas, the ratio is 1:1.078689. And when we compare the relative sizes of male and female Pelvic Vesicas, it is 1.236 ÷ 1.145898 = 1.078689.
It made sense then, to discover the ‘pressures’ that are built up by a complex conservation law in the male pelvis through these factors. What made sense was to return to the female program: namely 1.236, the width of the golden rectangle, and divide it by the male Thoracic Vesica Complex, 1.078689. 1.236068 ÷ 1.078689 = 1.14583854 = 1.5 ÷ 1.309
This ended up serving perfectly as the width within which to inscribe the male Pelvic Vesica. We have already seen the prime importance of 1.309. There is much more to come that shows the amazing function of 1.309 which is why it speaks volumes in the Ra-Horachty image. It acts as a ‘bridge’ from a prime 2:1 rectangle, to a marvelous way to ‘square-the-circle’.
Vesica Piscis at Chartres Cathedral
The Vesica Piscis as we now know is intimately connected with the Equilateral triangle and an angle of 60°. Below we see the importance of the root of 3 and 60° angle when the ‘opening’ of the triangle is 14.765 m. It corresponds with the height of the top capitals from which the entire Ogive springs at Chartres. 14.765 is half of 29.53.
29.53 x 432,000 = 12,756,960 which is an excellent approximation of the Earth’s equatorial diameter. This means that the base of the triangle with height square root of 3, represents the equatorial radius of the Earth at a scale of 1: 432,000. At Gizeh the scale of the Great Pyramid is 1:43,200 and the shadow cast by the Sphinx – also 60° – is the same as Chartres: 1:432,000. Note that the first elevation at Chartres is determined by the Minor 3rd, 6/5 for the hypotenuse. This gives a height of 9.794 meters which is the fall rate of gravity at the Earth’s equatorial surface in one second of time. The average for the planet is about 9.81 m/s². There is an easy way to find this figure. Simply divide the Earth’s polar axis by the number of geographical seconds in the circumference: 12,713,500 ÷ 1,296,000 = 9.81 meters. The fall rate at the latitude of Chartres is about 9.81
The Vesica Piscis is so important in the way it bridges opposite or complementary worlds that it should come as no surprise that it would make a fundamental appearance at Chartres, since the builders of Gothic Cathedrals viewed Christ as that state of being in which the adversarial conflict between the opposites was overcome and redeemed. Christ was seen as the royal principle of grand unification. He was seen as the testament on Earth to ‘Truth’ or what the Egyptians called “Maāt” who also embodied ‘Cosmic Harmony’ and ‘Cosmic Equilibrium’. The Vesica Piscis is one of the finest geometrical forms that expresses that end. So I will include a prominent example of it right here.
Below is the Royal Portal at the West Front of Chartres, dating to around 1150 A.D. You can see Christ in the tympanum of the central door. Every pilgrim in medieval times who reached Chartres entered the Cathedral with bare feet through this central door. This whole first elevation with its three doors is proportioned the same as the first elevation inside the cathedral. With Unity as the base, the hypotenuse is 1.2 and the height, .6633.
The Vesica forms the frame, the place where Christ, from the Book of Revelation, sits on his throne as the hero, the redeemer, the protector of knowledge, of the mysteries and of the mystical marriage of all the opposites known and unknown. All pilgrims passed through this Royal Portal from a south-west to north-east direction – sometimes lead by the Bishop himself – to the labyrinth designed to offer them the possibility of a second birth. The path of the labyrinth at Chartres is exactly 500 royal cubits, the same cubit used in the Great Pyramid. This amounts to 261.8 meters which can be expressed as 100 x the Square of the Golden Number or 2.618 x 100. This Royal Portal dates to around 1150 A.D., or about 16 years after the fire of 1134. Such a comforting order opening the way for any pilgrim who has braved every type of danger to get to Chartres – usually on foot! Such a glorious radiance emanating from men and women sculpted in what was once obdurate stone. Such venerable age transformed into the well-spring of ageless youth. Such majesty comporting itself in perfect humility! I myself could not wait to get to Chartres to photograph this royal entrance. Just as with the visit to the Great Sphinx, I owe the journey to the Navigator which was in competition at the Cannes Film Festival in 1988. Kely and I drove from Cannes and reached Chartres at the end of May. I got up at dawn and walked the hill as millions had done before me. The outside was so venerable, so old with grace that I was not prepared for what was inside this glorious crucible: an ageless grandeur penetrated with an equally ageless beatitude. I did not enter the Cathedral until Kely had arrived after a much needed sleep. When we both went in we could not talk to each other for at least two hours. Our words were carried off by the spirit of the world that has been with Chartres before ever there was a cathedral or crypt. At Chartres words are shed the way shoes are shed when entering a mosque. What freedom, when the heart gets a few days to swim in silence! I have never liked the fence that impinges upon this portal’s natural state, but I understand it must be protected. People have been carting away pieces of the Great Pyramid for centuries. It would be a disaster to lose even a single granule of this medieval masterpiece.
Below is an enlargement of Christ sitting above the central door in the tympanum. In red we can see that this heavenly ruler is framed by a 2:1 rectangle. The vesica itself is further framed by a root-3 rectangle which determines the base of the tympanum. The carving of the vesica itself is precise. Moreover, a 2:3 rectangle frames the fused circles (omitted here), echoing the whole Royal Portal itself which is a close variation of these proportions.
The ratio 2:3, the Major Fifth, is the only interval that all other notes in the scale will relate to harmoniously. We see the 5th in the second elevation at Chartres, recapitulating the end wall of the King’s Chamber: Unity x 1.118. At Chartres it has been given as 16.5 meters. 14.765 x 1.118 = 16.507. One must marvel at the workmanship exhibited in this carving; the precision of the vesica itself, carved out by the two perfect circles which have fused in consummate equilibrium. What better symbol could be utilized to demonstrate one’s faith in the potential of the human heart! What greater way to celebrate and teach the true goal of both birth and rebirth! The Vesica Piscis is not ambiguous. It is rigorous without being severe; perfect without being confining.
We also see Christ framed by four figures in which the 3:4 principle is observed again: in a group of four, 3 behave similarly while the 4th behaves differently. This principle frequently shows up in dreams but even with the Four Forces of Nature, we can unify three, while the fourth (gravity) holds court with infinity. So it cannot join with the other three (electromagnetic, electroweak, strong nuclear). The four figures surrounding Christ, who is the 5th or quintessence, are representative of the Four Evangelists. Three are depicted as animals, while the fourth is human. But there is also likely an allusion to astrology – a science mentioned with considerable fanfare in Wolfram von Eschenbach’s Parzival, being written as the Cathedral was being rebuilt following the 1194 fire. The wise and scathing sorceress, Cundrie, not only knew geometry and three languages, she also knew the language of the stars. The three animals, lion, ox and eagle, likely refer to the Fixed signs Leo, Taurus and Scorpio while the man, said to be Saint Mathew, represents the last fixed sign, Aquarius, usually depicted as a human waterbearer. Note that by reducing four 2:1 rectangles to three, it is as if the one that would not go along with the others has been assimilated into the Vesica where it becomes ready for transformation. So the Vesica, in a sense, is an image of a crucible of transformation that results when 4 reduces to 3 and the extra unity has gathered itself up into an overlap or a self-amplification that can result in a qualitative exaltation of being. Christ is that exaltation! Symbolically it is as if the very gravity captured in a second of time in that first elevation – symbolizing the fall of Adam, of Lucifer into a troubled Earth – has been captured and with faith, knowledge, courage and honesty the cathedral comes to represent the means by which spirit rises again to escape the very gravity that had initially claimed it.
I will end on the reminder that the choice of a Vesica to frame Christ at the Royal Portal is likely timeless and based on many nuances of one knowledge. When you see this woman below you ask yourself: is this cathedral not descriptive of us all? We are all reflected in it.
In the same way I learned to check the reciprocals of almost all numbers to discover the particular number’s unique relationship with unity, I also find that when I have put so many years into a labor such as this image – so much geometrical function – it is instructive to contemplate its inverse state. The computer can quickly offer a version that would be impossible for me to work out by hand. It resembles an x-ray and is an opportunity to see through the forms into the vacuum substrate from which – like the spirit of the world – we draw that invisible breath of something extra.
I think what gets to me the most in these inverted images, is a feeling of such simplicity. Our first stance when we are born from this geometry – a geometry that builds up from the vacuum in discrete bundles of energy – is joy, delight, pride of presence in the right way. An eternal “I am!” “I am the fountain!” I am the gate!” “I am the way!” What maintains this through the cycle is the 2:1 rectangle (ABCD) and its close cousin, the golden rectangle (EFGH).
New Zealand: An Island ‘Ark’
I find it so fitting that the very place where all this geometry started to coalesce – so quickly, it nearly made my head spin several times a day – was a place that is exactly bound by a 2:1 rectangle and then divided into two islands by the golden section!!! That place is New Zealand which strikingly resembles the foot-knee complex of a human being. Note in the image just shown how
precise the golden section is as the whole landmass attempts to handle the collision of two massive tectonic plates: the Pacific from the east and the Australian from the west. They invert in the middle of the South Island. In the far south the Pacific moves over the lighter Australian. In the north the heavier Pacific dives under the lighter Australian. The golden section handles this inversion of forces! It is as if the northern ‘feet’ of the South Island are literally tip-toeing along the golden section arc! It literally seems to have separated the two islands while also keeping them together in a dynamic way. The action of the golden section within a double square is occurring on the surface of the Earth sphere. So it is handling shared forces that converge and diverge all over the terrestrial surface.
New Zealand was the perfect place for me to start an authentic pilgrimage. At a level that seemed to be measured by the depth of the trenches that tumbled away from this Island Arc system, surrounded by more of the Earth’s water than any other piece of land, I feel a gratitude that has never been shaken from my system. There’s an old saying: the earth makes the man. I know what that means having spent 11 months in New Zealand.
A seed was sown there when the 2:1 rectangle entered my mind. It sprouted quickly and started to grow. No matter what, I could not stop that growth. It has been a complex imposition – a bit like an earthquake – on others, most especially my wife, Kely. At times the imposition took us to a breaking point. The whole enterprise would seem to collapse like an old tree. But somehow, a seed or two of the whole would break off and fall back into the fallow Earth, only to sprout again even stronger and better informed than before.
In an image like this, one can see that man is as busy as a machine. One can understand and feel the human obsession with the machine. Without any doubt there is a machine-like dimension to man. However, the operation of the golden number and its remarkable family ensures that man will never be solely a machine. It ensures a forever-renewable depth to our humanity. It does not merely transcend the machine, the constellation of beautifully ordered crystals. It infuses the machine with a vitality and a spiritual breath that is a portal to other dimensions. The soul participates in that wider frame of reference, whether consciously or not.