Since so many of the numbers that appear above also appear in Series E, I will just note again how Series E starts. The first line is 1, 5, 6, 11, 17, 28, etc. The second line is 2, 10, 12, 22, 34, etc. So in effect this Series transforms horizontally via a leap from 1 to 5 and vertically by going from 1 to 2. It then proceeds from there. The golden number, 1.618, is named on Line 2, suggesting that it is necessary for 1 to divide into 2 before the golden number itself can be named. Here, it is Line 2, Column XIII. This Series E in fact gives square values that seem to be arrayed around a nucleus, 1000, of Line 2, Column XII. I will have to explain this interesting property later.

Given everything that has been said so far about this simple 2:1 rectangle, it is worth contemplating closely how it has squared the circle here. The first ingredient is the diagonal, AC. Then a circle is drawn around point A that cuts the diagonal at E. AE = AB which is Prime Unity or here, 5. This leaves 6.18 for the rest of the diagonal, EC. EC becomes the radius of a new circle with diameter 6.18 x 2 = 12.36068, centered on C. Now comes the most interesting part. This 6.18 radius, (the golden section and navel position of BC and the length of the Prime 5 x 10 rectangle), cuts the opposite side of the Prime 2:1, AD at G. So it has reached across the 2:1. This forms a new triangle: CGD

with 6.18 as its  hypotenuse and 5 (Unity) as its base. The short side is easy to calculate and turns out to be 3.6327, which is the square root of (6.18034)² – (5)² = 38.1966 – 25 = 13.1966, the square root of which is 3.6327. The angle formed by CGD, incidentally is that of the pentagonal triangle, 54°. So you have 54° + 90° + 36°. When this is doubled you have 54° + 54° + 72°.  This is also a way to construct a pentagon, by doubling GD. The remainder of AD is naturally 10 – 3.6327 = 6.36728. And it is this that becomes the radius of a circle with circumference equal to 40. The creative aspect of the 2:1 is also the way it generates dynamic equilibrium through complementary opposites that start with 1 becoming 2 and the diagonal that links opposite corners, A and C.

We have here 3 circles. The inner circle has a diameter of 10. The next circle is 12.36068. They relate as 1:1.236, the same length that determines the navel and the size of the Pelvic Vesica Piscis in the female. If the Thoracic Vesica is 1, the Pelvic Vesica in the female is 1.236, contained within the golden rectangle, EFGH. This golden rectangle holds more surprises ahead that I hope we will be able to get some handle on, as it involves gravity.

The third circle is 12.734. It relates to the inner circle as 12.734:10 or 4/π. And it relates to the middle circle as 12.734/12.36 = 1.03007. This latter ratio happens to be one category for a common male armspan in the horizontal position. The other most common is 1.045 which is probably from 1.0472. In my armspan, one is 1.03 and the other is 1.047. If you multiply them together you get the following:

1.03 x 1.0472 = 1.078689 = 1.618 ÷ 1.5 = 2/3 x 1.618

This 1.078689 holds the key to the male anatomy and how it differs from the female. Note the reciprocal of 1.078689 is .927. With the AU-Canon as a template for the meter, one can see that ‘927’ concerns the generation of the female Pelvic Vesica and hence the womb. Even the number for the Pelvic Circle is a ‘lunar’ number, 29,530,000 km, the Synodical Month being 29.53 days. Through laws of harmony – the way spacetime is broken down into measures, the day and the meter find a relationship through this number 29.53.

Note, too, that there are only 4 points where the inner ‘10’ circle coincides with the square. At all other points, it is contained by the square. This seems to establish the four cardinal points which may act as outlets. The inner circle, on the other hand, is wholly contained within the 12.734 circle.  But the square itself pushes beyond the outer circle at its diagonal points which are 45° to the cardinal points. In contemplating the King’s Chamber from this perspective, one can see that it has the same diameter as the inner ’10′ circle. So it is ‘The Circle-in-the-Earth-Square” and is wholly contained within the Earth Circle. Since the King’s Chamber is located deep within the Great Pyramid and when it was found by Al Mamun in 820 A.D. all the passageways were plugged with heavy stones that could not be moved. The feeling is that the King’s Chamber was designed to be completely hermetically sealed so that it could accumulate the very energy discussed above, by which the Earth’s magnetic field, electrostatic energy and gravitational ‘self-energy as it is called today, could reach a threshold capable of inducing the coveted metamorphosis symbolized by the scarab “Khepere” (The Transformations of Re). The harmonies locked into the paradigm of the King’s Chamber appear to be such that the essence of the Stone (Grail) that ‘burns the Phoenix to ash’, is capable of perpetuating the link between body and ka in the act of spiritual rebirth. At the same time this is what was known as the entrance to the Duat or Gate to the Netherworld through the portal of the 2:1 paradigm; the sba.

In the image above I have laid out the basis of the female anatomy with head, thorax and pelvic vesica. There is a great deal of information in this one image but it will be too complicated for right here. The Thoracic Vesica is 1/3rd of Unity (AB) and the Pelvic Vesica is 1/3rd of 1.236 or the golden section of the height, 2 and width of the golden rectangle, EFGH. I have laid 3 ‘grail tables’ over the navel position, but will discuss them at a later stage. The mouth of the Thoracic Vesica falls exactly on the mouth of the archetypal head marked with a dotted line. I put this image in to provide a sense of how the figure  gathers itself up within the basic geometry and even at this early stage seems to want to spring to life. The golden section within a 2:1 paradigm contributes enormously to this well-spring of life.

In the image below we see the same basic paradigm applied to the Astronomical Unit. Here the golden section is playing an even deeper role. The short length that is around 57,338,563 km is collapsed into the abdominal circle that will contain key organs including pancreas and kidneys; while the larger segment, 92,773,794 km collapses into a circle that is 29,530,815 km and becomes the womb. It is a striking coincidence that the lunar number associated in days with the Synodical Month (29.53066 days) should show up in meters or kilometers in the Pelvic Vesica and Pelvic circle. A human gestates according to the Synodical Month in exactly 266 days or 9 synodical months. One can say, provisionally, that the Synodical Month (29.53 days) has a formative influence on human gestation. It determines ovulation. And when the golden section is divided by 3 we here have 30,922,000 km which is another lunar number. In 25 years there are 309.2 lunations or synodical months. The width of the Parthenon is 30.922 meters. The internal width of the transepts at Chartres is twice that, or about 61.8 m, or some say a few centimeters more.  One second of arc at the Earth’s Equator is 30.922 meters. The Parthenon is also based upon ‘squared-circle’ geometry but I will have to show it elsewhere.

SQUARED-CIRCLE ARCHETYPE

Now take note of the difference between AG and AN. It is 1.3673 = 6.3673 – 5. And we can note an archetype here also worthy of contemplation. 6.3673 ÷ 1.3673 = 4.6578 = AG ÷ NG.

We have already seen this relationship in the Great Pyramid which I will now show again. The number that corresponds to 1.3673 is 60 cubits or 31.416 meters. This depends from the scale of 1: 43,200, proportioned to the size of the Earth. 432 is itself a member of an important number series I will call Series A, very familiar to the ancients. I will give some members right here:

SERIES A

27  54 108  216 432 864  1728 3456  6912  13,824  27,648  55,296  110,592

Best to note here that one prime interior measure at Chartres is from immediately inside the Royal Portal all the way to the interior Round Point of the Choir is 110.76 meters. Due to the Earth’s flattening, a degree of longitude lengthens by definition as one walks from the equator to the pole. At about 8° Lat the degree of longitude will be 110,591 meters (13th member of Series A). At 25° it will be about 110,764 meters. At Chartres itself a degree of Long is about 111,180 meters. This measure would be very easy to accommodate by simply starting from closer to the immediate door itself at the Royal Portal. Series A is amazing. In miles it gives the Moon’s diameter as 2160 miles or 11404800 feet. When this is divided by another member of the series, 172,800 you get:

11404800 ÷ 172800 = 66 feet

This is the height of the Great Sphinx and two obelisks that disappeared from Heliopolis near 1200 A.D. when the Templars were rebuilding Chartres. A shadow cast by that height at Noon on the Equinox would be ½ the Chartres Square: 11.5965 m.

Above I have marked the proportions where the ½-base is 1.618 = 220 cubits = 115.192 meters. The height is 280 cubits or about 146.6 meters plus a 1/2 – meter platform, totaling say 147.108 meters. The difference between the whole height and the square is 60 cubits or 31.416 meters. 280/60 = 4.666…

Since this is an important point I will also show this in a closer up version. It allows a better viewing of the King’s Chamber looking west. The 31.416 meters that can be seen at the top of the pyramid have been turned into the perimeter of the King’s Chamber. It is as if the whole intention was to take that 31.416 meters and turn it into the sba gate, the entrance to the Netherworld. But what the Pyramid Texts tell us is that the entrance into the Netherworld is  through and as the Eye of Horus. Therefore the King’s Chamber must be the place where the entire endeavor of the Egyptian Civilization reaches its apotheosis; namely the reconstitution of the Eye of Horus. The Egyptians saw this as an example of the principle of Maāt, or Cosmic Equilibrium. And they viewed the Squaring-of-the-Circle as a fundamental expression of Cosmic Equilibrium or Maāt.

Pi represents the horizon of a circle. Horachty meant Horus-of-the-Horizon. Sometimes Atum is referred to as “Him-of-the-Horizon”. The Berlin Papyrus suggests that the name is so mysterious that it cannot in fact be named or even uttered. It is experienced by the spirit of any candidate as too powerful and too mysterious or infinite to label. Encountering the horizon of Him in spirit form absorbs words, language, metaphors and was well-stated by the Hindu as just “that”. “That” is simply the essence of spirit that has shed the corporeal body and even, according to the Tibetans, the double or ka. Sages have long thought that pi, 3.1416… genuinely represents or points to that mysterious “Horizon”.

SQUARED-CIRCLE-MODULUS: 1.27324 ÷ .27324 = 4.659…

Around 1999 I stumbled upon a few clues that turned my head and helped me solve a number of outstanding problems with both female and male geometry. This occurred after I had a preliminary drawing based upon the AU for April 16^th, 1998 (~ 150,108,000 km). Two examples are particularly important. One is a macroscopic resolution which nevertheless has retained considerable mystery. It involves the proportional relationship between the Earth and the Moon. The other is a quantum-scale resolution that concerns the mean momentum energy associated with the nucleons (protons and neutrons) that compose the bulk of the Earth’s atomic mass.

Scientists do refer frequently now to “Mearth”, acknowledging that the behavior between the two bodies is not a typical planet/moon relationship but also possesses aspects of a binary planetary system. For example the Moon is always falling toward the Sun in a concave orbit and even when approaching the waxing square in its orbit around Earth, never goes convex to the Sun the way other moons do, for example around Jupiter and Saturn. Earth and Moon fall together toward the Sun a greater distance than ever the Moon climbs against the Earth in any given time frame. Now, let’s take the  mean diameter of the Earth as about 12,732 km and the diameter of the Moon as 3476 km. Together they add up to 16,208 km which is close to the number of (4/π) ² or 1.2732 x 1.2732 = 1.621. I have shown this above in the drawing of the Great Pyramid. 1.621 is just a little bigger than the golden number, 1.618, but geometers often consider ways that these two great factors work together or dovetail. If you add the Earth and Moon together and divide by the Moon you get: 16,208 ÷ 3476 = 4.66. The Moon is behaving exactly the way the 31.416 meters are behaving at the top of the Great Pyramid. It is important to emphasize that the Egyptians appeared to follow the soli-lunar cycle religiously – literally. They were not only aware of its physical influences, but were equally aware of its spiritual influences and so kept 29 altars, one for each day of the synodical month which is also the  female infradian biological clock.

In the image above I used an Earth radius of 6367 to be consistent with the Squared-Circle illustration shown earlier. I could also have used a cut through the Earth according to its tilt which would be about 6370.5. But for now I will stay closer to the mean for convenience.

The difference between the Moon and our Squared-Circle archetype is that the Moon and Earth have become separated by about 384,401 km between their respective centers of mass. THEY ALSO SHARE A COMMON CENTER OF MASS THAT IS LOCATED ~ 1700 KM BELOW THE EARTH’S SURFACE. This I am calling in the human canon, the SCOM circle or ‘Shared-center-of-Mass’ circle that we have already seen without comment. Its relative size within the final canon is remarkable because it coincides with the golden section! We have to discuss the Astronomical Unit first and how the separation between the Sun and Earth/Moon influenced details now incorporated into the geometry of the male in particular. Then we can appreciate the proportional size of the SCOM circle in the final canon. But that it coincides with the width of the golden rectangle is a remarkable example of the harmony among Earth, Moon and human anatomy.

The Moon is 27.3% of the Earth’s diameter or 3476/12,732 = .273123. It is instructive that the Sidereal Month is 27.3217 days or 27.32% of 100 days. After the mean period of human gestation which is 9 Synodical Months (265.77 days), there are almost 100 days remaining in the year (99.48). Incredibly, this remaining period is where the tradition of the progressed lunation cycle derives from because it shows that 1° approximately equals 1 day which approximately becomes 1 year. Perhaps this suggests that most of us ought to be able to reach 100 years of age if circumstances permit. It’s like an internal implicate clock that simply requires reasonable explicate expression – maintaining natural rhythms – to make it to 100.

Meanwhile the Moon during our time, has taken up a position in the sky where it has about the same angular diameter as the Sun when viewed from Earth. Most scientists of the last few hundred years have regarded this coincidence as meaningless. But as science begins to think more in terms of systems, webs of circumstances, patterns and phases within patterns, it is possible that we will come to see this coincidence as meaningful. My guess is that we will see a type of gravitational resolution in the order of a form of gravitational self-focusing. This is what an eye does. The eye does it largely in response to electromagnetic radiation or light. We will look at the geometry of the eye in a future post, once we have examined another method for squaring-the-circle. But at the level of planetary bodies, gravity plays a much bigger role and in fact the Earth and Moon are bound together in a very shared gravitational field. Just a planetary body on its own is an eye-like resolution. It is gathering spacetime around itself in a shared field. The Earth and Moon are completely immersed in the Sun’s field but the Moon’s tidal influence on the Earth is greater than the Sun’s. The Sun’s inverse square gravity is greater on the Moon than the Earth’s is even though the Earth is so much closer. The Moon’s tidal gravity is about 2.18 times greater than the Sun’s tidal gravity, while the Sun’s inverse square gravity is 2.2 times greater than the Earth’s is on the Moon. There is a form of decussation going on between tidal gravity and inverse square gravity. Tidal gravity is approximately an inverse cube force that is stronger across shorter distances. It is my conviction that this sets up a  tidal gravitational variation cycle around the Earth that helps set both Circadian and Infradian biological clocks located in the suprachiasmatic nucleus just above the optic chiasma in the hypothalamus. Shifting locations suddenly induces ‘jet-lag’. Probably the cause is upsetting the entire metabolic clock that has adjusted to the tidal gravitational variation cycle that stimulates very subtle levels of cell activity such as cytoskeletal microtubules which are involved with consciousness itself. So jet-lag involves even how we think and how our muscles work. It would take up too much space to theorize here how it came to be that the Moon and Earth share these crucial proportional relations. But it suggests to me a very ancient history when both bodies could be shaped together by the same forces operating in the same field. I don’t think the Moon came wandering in from outside. My guess is that the Moon and Earth were once so close together that they were able to be separated in a proportional manner. Now, not only is the Moon proportioned as .2146 or 1/4.659 of the combined diameters, but it has also taken up a position where it appears the same size as the Sun only from the surface of Earth. This suggests to me a form of symmetrical gravitational self-focusing in a shared field. This also increased the harmonious properties of the symmetry. But I am suggesting that this ’4.66′ factor is not merely harmonious; it is fundamental. The builders of the Great Pyramid were very cognizant of it since they based the dimensions of the King’s Chamber very precisely upon this factor: 280 ÷ 4.66 = 60, or 146.4 ÷ 4.66 = 31.416 meters. And this we have seen is based upon a scale model of the Earth set at 1: 43,200. At Chartres and the Great Sphinx the scale is set at 1:432,000 instead. From an engineering point of view, Chartres could manage a 36 meter vault (118.65 feet) but not a 360 meter vault!!

Terrestrial Nucleon Speed

The next example of the involvement of squared-circle relations came as a real surprise. I had read Fritjof Capra’s “Tao of Physics” in 1981 and had come upon two pieces of information that were memorable but which I assumed were extremely ‘ballpark’ in intention. However, in 1999, when I came upon this proportional relationship between Earth and Moon, I was also checking physics because of the fact that gravity was presumed to be operating in the vacuum at the speed of light just like electromagnetic radiation. So gravitons were supposedly a bit like photons: massless, spin-2, etc. And it was thought they might couple at some point and operate in the vacuum together. At the nanometric scale of microtubules, I thought this might be interesting especially if quantum gravity was a real factor and was capable of ‘tripping’ the quantum wavefunction as it is called.

One day I just intuitively checked Capra’s ‘nucleon speed’ against the speed of light and gravity in the vacuum, believing that this had to be proportioned like any other significant factor that has contributed to the ability for our bodies to build up at the Earth’s surface into what we are today. Capra had given a strangely rounded figure for the ‘Heisenbergian momentum’ associated with the mean energy of all terrestrial nucleons (protons and neutrons) which make up the bulk of the mass of the Earth. We are to picture entities that are ‘boiling’ away in their extreme confinement, far removed from the First Bohr Radius where the electrons begin to be distributed in wave patterns around the nucleus. The nucleons can’t get too close together and can’t get too far apart. Either way there is a powerful force of repulsion or attraction keeping them locked in place in a state of extreme agitation that can be expressed in terms of ‘speed’ or ‘momentum’ relative to position as calculated by Heisenberg’s complementary Uncertainty Relation which makes use of Planck’s tiny little quantum constant as its frame of reference. Capra gave about 40,000 miles per second worth of momentum or speed. He did not make his calculation on the basis of color theory with quarks and gluons as factors in how that momentum energy would be distributed or shared. But I wrote him in 1999 and asked him if I could take the figure of  64,374 km/s seriously? I basically told him what I wanted to do with it. I also asked him about his electron speed that was close to 1,000 km/s compared to 2188 km/s for an electron in the First Bohr radius. Capra did not feel it was necessary to modify or place his figure in any sort of parenthesis. He clearly had confidence in basic rules that had allowed him to calculate such a figure for all the nucleons on Earth. The Earth is not very massive so there is nothing going on here within the atomic nucleus such as we see inside our Sun or larger stars if fusion-burning is the correct explanation for the way our Sun is powering its light.

Here is what is so interesting about a figure of 64,374 km/s. Apparently nothing can move through the vacuum faster than light which is about 299,792.458 km/s. A massless photon cannot exceed “c”. If its energy is increased it will increase its frequency, not its speed through the vacuum. It will increase its agitation while moving. The frequency is important because the visible light spectrum is only a tiny segment of the great spectrum of electromagnetic energy. And we humans are tuned to the visible spectrum to a high degree. So frequency matters to humans – among a host of things – such as temperature, atmosphere, oxygen supply, air pressure, gravity acceleration, etc. Our Earth is quite the ally, friend, nurse and sanctuary!

If gravity also moves at this rate (299,792 km/s), then it might be important to know the proportional relationship between the confined momentum of our mass-carriers – the same nuclei that support the atoms found in carbon, calcium, iron, potassium, chlorine, oxygen, hydrogen and so forth. These atoms then heap up together into real gravitational effects which become significant at the macroscopic scale. In our bodies each atom does not possess much gravity. Electromagnetic forces are relatively much, much stronger. One way to represent that strength is through speed. So that at the First Bohr Radius we can represent the energy and the strength of the electromagnetic interaction by dividing the speed of light by the very important Fine Structure Constant also known as ‘alpha’. Alpha has an effective value of 1/137 so that the speed of the electron in hydrogen can be calculated as c x 1/137 = 299,792.458 ÷ 137 = 2188.266 km/s. The relationship between the electron in the First Bohr Radius and all other electrons on Earth is approximately 2.188 or ~ 2.2. We had just said that the Sun exerts about 2.2 times more inverse square gravity on the Moon than Earth does and the Moon exerts about 2.18 times more tidal gravitational influence on the Earth than the Sun does. The Moon is moving 2.2 times faster in its orbit around the Earth than the Earth is moving at its Equator: 1.023/.4651 = 2.2. The Earth rotates faster in a day because it has far less distance to go. At the Equator it has to go about 40,075,035 meters. It does this in about a day. But the Moon has to go much further even though it is traveling 2.2 times faster. It has to make about 2,415,262.7 km to get around the Earth in its 27.3217 day Sidereal Orbit. This is about 60.26 times greater in distance. So, naturally, 60.26 ÷ 2.2 = 27.3, which is the days in the sidereal month, the Moon also being .273 or 27.32% of the Earth’s diameter. The Moon is locked into a synchronized roll as it orbits the Earth. It rotates the same rate that it revolves as if the friction (and gravitational resolutions) were acting like a fixed axle and it were a cart on wheels constrained to a gravitational path. But this imparts cyclic regularity to the Earth/Moon/Sun dynamic. For the ancients, the fixed regularity of these cycles of motion of Moon, Earth and our relationship to the Sun constituted the most basic laws governing all aspects of life on Earth. Everything was subsumed under these cycles of motion. The constellations like Sirius, Orion, the Great Bear, Alpha Draconis, Regulus and so forth mixed their influences in with this basic Triadic Paradigm. Man himself was sculpted on the Potter’ s Wheel by these cycles or arcs of motion.

These examinations encouraged me to treat Capra’s ‘nucleon second’ in a similar fashion. I was imagining an inversion where a balance was struck between macroscopic resolution (and regularity) and microscopic or quantum balance where, through confinement, the one energy slows down in one respect since it is changing from massless energy to mass-energy through the curvature and self-enclosure of spacetime, but by containing that energy forces it to speed up according to specific rules or at least proportions. I strongly believed that proportion would have to play a role because the vacuum was being involved in the curvature or self-gravity; or self-attraction process. The spacetime vacuum was involved and proportion was one indicator that described the involvement. It was curving itself and reacting to its own confinement; its own self-enclosure. I could easily picture the squared-circle effect as a self-reaction: a geometric tendency toward dynamic equilibrium akin to hydrostatic equilibrium. Well, one day I simply related Capra’s figure to the speed of light:

299,792.458 ÷ 64,374 = 4.657

I definitely gulped when I saw this. It invoked what we had just seen above with the way the 5 x 10 rectangle squared the circle:

6.3673 ÷ 1.3673 = 4.6578 = AG ÷ NG

With time I came to call this 4.6578 factor, the “Squared-Circle Modulus” (SCM) because I found a path from the Fine Structure Constant all the way through Capra’s nucleon speed to the speed of light with some interesting stops along the way – all through the number 4.6578 or there abouts. I will show this path later. As a ratio it was behaving similarly to the golden number. But what is this factor? And why do we find it also associated with the Earth/Moon proportions where by a sort of inversion, the Moon is acting analogously to the internal agitation or activity of the highly confined nucleons – the carriers of what we call mass-energy: energy that is densely confined and so starts to behave like matter? To repeat: in the one instance the SCM is referring to speed or a rate of propagation of an energy relative to the vacuum. In the case of nucleon momentum, it is not to be pictured like a speeding bullet moving from A to B. Rather it is a self-contained distribution of motion-energy, highly confined and relative to its position (or confinement). In the case of the Moon it primarily refers to diameter or the amount of space occupied by a celestial body in a gravitational field. In the case of the nucleons, the speed (64,374 km/s) is certainly offering resistance to gravitational collapse. It also seems to be imparting stability to the identity and lifespan of protons in particular. In the case of the Moon, speed is certainly playing a role in its position but the SCM is describing the volume of space that it occupies as if Earth and Moon had been sculpted proportionally out of a single mass. So what is so interesting here is the proportion itself, namely ~ .2146, or 1/4.66, that links speed to volume.

The Moon appeared to me like a point of macroscopic inversion in a gravitational field intimately shared with the Earth and dominated by the Sun which determines with its mass and distance the speed the Earth and Moon can travel around it (29.786 km/s as the mean).

10,000 ÷ (4.6578/1.619) =  3476 km

Ten thousand kilometers is the Circle-in-the-Earth square. 3476 is the Moon’s diameter. 1.619 is intermediate between the golden number and the square of 4/π.  4/π is 1.2732395… The square root of the golden number is  1.2720196. Their ratio is 1.000959, so they are very close. One could say that the square root of phi is involved in determining the height of the Great Pyramid and helps to set the difference between square and circle. Probably one of the most fundamental coincidences in the whole of geometry is that between 4/π and the square root of phi. When 4/π is used the corner angles of the Great Pyramid are about 51.854 Degrees.

Here is another observation that may add another section to Ariadne’s thread that helps us to keep connected inside the labyrinth of knowledge. We saw that to obtain the SCM- factor in the Great Pyramid - so that it was 31.416 meters - it meant dividing the Earth by 43,200. This gives a height of about 147.1469 including a 1/2-meter platform for the Earth’s Polar Radius. Minus the platform gives a height of 280 cubits. With a 220 cubit square 60 cubits remain. 60 x .5236 = 31.416 meters. We also saw that when we used an Astronomical Unit for the day the Earth is 150,108,000 km from the Sun very interesting numbers are spawned. We can also take the same AU and divide it by 43,200 since it is such an interesting proportion in its own right:

150,108,000 ÷ 43,200 = 3474.72 ~ Moon’s Diameter

Curiously, back in the 1980’s a team from MIT measured the Moon’s diameter and got 3474.8 km, not 3476 that we see most commonly in use today. I checked with a member of that team back in 2000 and he still insisted that the Moon’s diameter was 3474.8 km. But in any case it is close enough to make the point. Where there is life, harmony tends to be operating on multiple self-interacting levels and with effort humans can glean some of those levels and the nature of some of the harmonies themselves. This becomes quite intuitive, like composing music, and it is true that ultimately it is not prone to analysis because by its very nature it holds together as a web of existence, life, reality.  The life of a ‘thing’ is in the web that it shares with other ‘things’. In this instance you have two proportions dovetailing on an AU of about 150,108,000 km. When I used this date the numbers broke down harmonically in such a way that I started to fine tune the canon of human proportions by using those numbers! And when the Moon is used as a measuring rod, it measures the same distance 43,200 times. Moreover, the Sun measures to nearly the same distance 108 times, another member of Series A.

So what might be going on here? What might this tell us about squared-circle relations? And why did the Egyptians take the same difference that concerns us here – recalling that in a sense, the Capra speed is acting like a difference between mass-energy and massless energy – and turn it into an ‘empty’ enclosure as if it were an inversion of mass-energy, the mysterious Cavern of Sokar or Broad Hall of Osiris? Were they not attempting through the enclosure to bring about an acceleration, a quickening, a metamorphosis that they symbolized by the scarab beetle? Moreover, they often replaced the scarab with various forms of ‘bnbn’ which is none other than the Benben Stone brought by the Phoenix which in turn burns itself to ash in the House of the Phoenix only to rise again probably as Horus, symbolized by the falcon or the Eye of Horus. It seems these inner caverns were designed to accumulate energy, spirit as a food for the ka to facilitate the very ‘leap’, the very spiritual ‘mutation’ that many of us seem to crave. If the Capra nucleon speed refers to the matter pole (Egyptian “Seth”), perhaps the empty caverns were referring to the ‘spirit-pole’. The same proportion was being applied to opposite ends of the spectrum. This is a good point to mention Eschenbach’s definition of the Grail in Parzival.

The Grail and the Phoenix

So, this enclosure determined by the SCM, brings about the proportional size of the King’s Chamber. Not only has the difference of the Squared-Circle Modulus (SCM) become the perimeter 31.416 meters or 60 cubits, where 60 is a common number for measuring time, but 31.416 meters is also exactly 1236.84 Br. inches. In 100 years there are 1236.8 synodical months. Half the perimeter is 618.42 inches symbolic of 50 years and 618.42 synodical months. 1/4 of the perimeter is 309.2 inches symbolic of 25 years and 309.2 synodical months. All these numbers are found in Column XI of Series E. There is the feeling that the King’s Chamber was also a clock and that the clock was tuned to both small and great cycles of time. It acted like Solomon’s Holy of Holies which was closed up all year except once, admitting the High Priest alone to enter its pristine state like an oracle.  Here is what Eschenbach says about the ‘stone’, as if the stone were acting like a ‘pyr’ or an accumulation of electromagnetic energy capable of undergoing a metamorphosis in the thick darkness pregnant with suspense and mystery:

“By virtue of this Stone the Phoenix is burned to ashes, in which he is reborn. – Thus does the Phoenix molt its feathers! Which done, it shines bright and lovely as before! … This Stone is also called “The Grail”.  (Parzival, p. 239)

We all know the Phoenix burns itself to ash in a singular act of self-immolation upon an altar of its own making. The self-enclosure is also the ‘altar’. Since there was nothing in the King’s Chamber except the coffer, the coffer was likely, then, one and the same as the altar upon which ‘the Phoenix burned itself to ash’. The Grail, then, is the essence of the Phoenix itself, the mystery of the eternal spirit within and its relationship with the surrounding universe, the Cosmic Web. It molts; it does not die! The coffer within the King’s Chamber has been called a ‘fount of spirit’. For the Egyptians and Gothic Builders at least, the Squared-Circle seemed to act as an eternal gate when Circle and Square have been reunited; when their difference has been transformed and possibly transcended. The King’s Chamber must have been seen as the Cavern of Osiris/Sokar within which that union and transformation is realized. The diagonals of rectangles have long been treated as transcendental functions. In the King’s Chamber you have 15 cubits in the end wall, 22.36 cubits in the floor and 25 cubits in the Great Diagonal which is also 13.09 meters – so prominent at Chartres. Recall that at Chartres the 13.09 circle seems to determine the internal round point of the choir. We can see 1309 below at the head of Column XIV, the same column that also contains 2618, 5236, 7854, 10472, 15708, 31416. The width of the King’s Chamber is 5.236 meters; its length is 10.472 meters and its perimeter is 31.416 meters.

So much more can be said about this famous chamber – especially when one examines the Pyramid Texts – but I will have to do this at a later point – it is too detailed. What should be said, however, is that when ‘circle’ and ‘square’ are reunited, all the opposites are reunited, probably in part due to the presence of the square root of phi (1.2720196…) in the radius of the circle (Height of the Great Pyramid). In Egypt, Horus was the Eye or Circle; Seth was the Square, the one who dismembers, opposes, obstructs and imprisons the coveted luminary: the principle by which light (and sound) become intelligible. Seth is the matter pole that must be transformed from passion into wisdom. When the Two Combatants, as they were called, are reconciled by Thoth-Hermes through the agency of Isis - undoubtedly by way of the altar supported by the 2:1 Table - the “Eye” is restored; the Phoenix is reborn; the scarab transforms from a worm into a winged insect of startling beauty. It is my belief that the ancient Egyptians believed in this metamorphosis sufficiently enough to build the Great Pyramid.

SQUARED CIRCLE MODULUS (SCM) AND HUMAN GEOMETRY: THE MOUTH

Before demonstrating the second method for squaring the circle that also starts with the 5 x 10 Prime 2:1 rectangle and also uses the golden section division of 3.82 and 6.18 (also from Series E, Line 2, Columns X & XI), I should just note the first prominent application of the SCM in a human being. It has already been alluded to, but now that a bit more has been discussed, it is worth looking at in better detail. We are interested in that difference between circle and square where a sort of inversion takes place at a limit. At the macroscopic limit we have the Moon that has assumed the same angular diameter in the sky as the Sun. The Moon has harmonized with the Earth to set the monthly rhythm that governs female ovulation that alternates month-by-month, ovary by ovary. It is a clock-like oscillation. In the male there are no ovaries but there is a longing in the depths of the psyche and soul for another form of birth that has likely contributed fundamental elements to initiation rites of all races and all creeds – in pursuit of the mystical marriage of the  inner man with the inner woman in such fashion that a gate opens for the spirit to find its passage into open territory. It is as if the ‘squared-circle’ were a fundamental analogy for the mystical marriage of male and female. When the marriage is successful a candidate experiences a state of exaltation that stands outside sexuality, but yet also overflows with a higher level of energy.

At the other pole we find another limit set by the mean rate or energy of agitation, movement, speed, in a state of severe confinement far removed from the electrons that mediate all biological activity that supports conscious life on Earth. This energy is deep in the atomic nucleus, where the mass-energy has reacted proportionally to its confinement. More important than the highly suggestive ‘speed’ offered by Capra, who made the calculation himself with the aid of Heisenberg’s Uncertainty Principle along with a few other factors, is its proportional relationship with the massless constant we call ‘c’ that also may determine the speed of gravity in the vacuum, if indeed gravity has an ‘exchange particle or mediator’, similar to photons. And that relation is 299,792.458 ÷ 64,374 =  4.657. But this is the squared-circle modulus (SCM). And it has shown up with the Moon - that is also partly a planet - and has determined several biological and possibly spiritual rhythms. So, (Earth + Moon) ÷ Moon = 4.66. At one pole the SCM applies to comparative  speeds and at the other pole it applies to comparative  diameters. Through an inversion the SCM relates speed to diameter. Earlier we saw how the ratio between Solar pulses and Earth shouts related as 9/69, just as the head relates to the whole body height as 9/69 or ~ 13/100. And since the Sun and Moon have the  same angular diameter when viewed from Earth, one wonders if the head is not a  marriage of both Sun and Moon and the respective ‘motions’ each imparts to the  head’s formation.

The SCM has also has shown up in a most fundamental way at the Great Pyramid – determining the dimensions of the King’s Chamber (perimeter = 60 cubits = 31.416 meters) – within which most likely occurred the apotheosis of the Pyramid Age. We see it below as the  difference between the radius of the circle TR (280 cubits) and the 1/2 square SR (220 cubits).

Now let’s look at the human female where the square PQTU replaces JKLM seen in the Pyramid image above. Let’s take the Prime Circle with length “2” (or 150,108,000 km) and let’s square that circle by equal perimeter and see where the SCM falls.

The circle is OV and when it is squared the square becomes PQTU. The difference between the 2:1 PQNM and the radius of the circle is clear. I did not put a letter in, in order to avoid cluttering the space. However we will take the measures arithmetically to satisfy all curiosity. But the first point is obvious.  The difference falls on the mouth of the adult. PQ passes right through the opening of the mouth. In other words the upper side of the square PQTU passes through the mouth of the standing adult. One could argue that the geometry of the squared-circle is the underlying cause of the position of the opening of the mouth of the standing adult. Incredibly, as some may know already, the King’s Chamber was associated with a recondite ceremony or ritual known as the “Opening of the Mouth”. It was also called “Splitting open the Mouth” and “Splitting open the Eye”. In human anatomy, however, the functions are so complex, that the mouth position would have to be caused by more than one factor alone. However, the coincidence with the squaring of the circle is a remarkable one.

The distance from the mouth to the vertex (O) is acting like the 31.416 meters at the top of the Great Pyramid that becomes the perimeter of the King’s Chamber. It is acting, proportionally, like the ‘nucleon second’ compared to the speed of light. By becoming denser the speed has been slowed and has also become absorbed into the properties of the nucleons in their condensed confinement. It is also acting, proportionally, like the Moon relative to the Earth + Moon, which stimulates and regulates biological processes probably in more than one way.

Now if the diameter of the circle is 2, its radius is 1. Its circumference will be 2π or 6.2831853. The square with equal perimeter will be 1.5708 on a side. Half that will be .7854. These numbers appear in Series E on Line 6 and Line 12:

If we add 1 to this .7854 we get the height from the ground: 1.7854. Another route to get to the mouth is to divide the height simply by 1.118 (1/2 the square root of 5) which goes in the opposite direction from the diagonal. The diagonal of a double square is obtained by multiplying the length by 1.118 giving 2.236. To obtain the mouth from the diagonal it is 2.236 ÷ (1.118) ² = 1.7888 = 2.236068 ÷ 5/4. The difference between this and 1.7854 is negligible. It is for all intents and purposes exactly coincident. So: 2 ÷ 1.118 = 1.7888 = 1 + .7854. Note that in the King’s Chamber the diagonal of the end wall is 7.854 meters. With this as the side of a square you have a perimeter of 7.854 x 4 = 31.416 which is the same as the King’s Chamber itself and also the same as a circle with diameter 10.

31.416 appears on Line 24, Column XIV above. The 3 Tables that bore the Grail are instantly evident in the King’s Chamber (Circle has diameter of 10; Square has side of 7.854; Rectangle is 5.236 × 10.472). How interesting that the same factors are literally opening the mouth of the standing adult! The ‘place of coincidence’ and the ‘line of difference’ are the very factors that open the mouth. In the Pyramid Texts the Egyptians refer to both ‘Opening the Mouth’ and ‘Splitting open the Mouth’. They also interchange this with ‘Splitting open the Eye’ by which they mean the Eye of Horus. But in an important reference to Sokar they also mention ‘splitting open the egg’.

I should mention that just as male blood pressure tends to be around 120/80 which is 3/2 or 1.5, the center of the thorax from the ground, the female blood pressure tends to be 110/70 which is 1.5708. The relationship between 1.618 and 1.5708 is 1.030048 – probably the most common female armspan. The male can be looked at as 1.618/1.5 = 1.078689, which I will make as the  overall armspan for the male as it is 1.03 × 1.0472 when in the angled position we have been seeing. The relationship between male and female blood pressure standards is none other than the ratio of the Royal Cubit: 1.5708 ÷ 1.5 = 1.0472 = π/3. The woman has a larger range and her systolic pressure is lower than the male. By a conservation law, these nuances show up in different functions.

Let’s check the ‘mouth’ of the female Thoracic Vesica. It divides unity by 3, so the horizontal cross is .333333 or WX here below. Half that will give the ½ base (WN) of an equilateral triangle or .16666666. The base of this triangle is shown below as WX. The apex is unmarked but falls on the mouth. 1/2 WX falls in the center of the heart, marked N.

The height of the triangle is easy to determine as it is the root of 3. So you have .1666666 × 1.73205 = .288675.  Since the middle of the Thoracic Vesica sits on the 1.5 latitude marked above, we add 1.5 to .288675 = 1.788675. So we now have the following: 1.7888, 1.7854, 1.788675. In the image above I have marked 1.7854 to label the upper side of the square with 1.5708 on a side. Once again they cannot be distinguished from one another. This is extraordinary harmony without any question. It is well worth enumerating.

A. Height divided by  half of the square root of 5 = 2 ÷ 1.118 = 1.7888

B. Squared-Circle falls at exactly 1 + .7854 = 1.7854

C. The mouth of the female Thoracic Vesica Piscis falls at 1.788675.

The fact that these prime geometrical factors converge on exactly the same point, indicates that the mouth is capable of genuine refinement as well as being a cornucopia of flow, an exchange between inside and outside, diastole and systole, inhalation and exhalation, inspiration and expiration. The mouth articulates and shapes what has been formulated in the vocal cords (throat) by the motor activity that comes from the brainstem via the recurrent laryngeal nerve that wraps around the Aortic Arch of the heart after having left the Vagus Nerve (N.X). The Aortic Arch is at the latitude of the golden number (1.618) from the ground. The Aortic Arch signals the medulla and pons as to the state of the blood: oxygen and carbon dioxide levels. It strikes me that the recurrent laryngeal nerve needs to maintain that somewhat cumbersome route to reach the larynx because it is related to  respiration and the emotions associated with respiration and heart rate. The lungs dominate the upper half of the Thoracic Vesica, demonstrating that they are formed within the unity circle. The breath supports the vocal apparatus. The heart is also very much involved with speech as is the brain and brainstem. Circuits that drive the larynx also pass through the heart and are undoubtedly related in a prime way to Nerve X, the Vagus Nerve.

The motor control of the larynx and mouth can be manipulated by yoga to conserve the almost automatic flow that engenders ideas, emotion and speech. When a peak is reached – especially when the kundalini is added to the equation – it is possible to experience some sort of mystical initiation. Here words are experienced differently. They have great power and authority and can bring about transformations. They induce dramatic changes. But it is as if your own will is not the source of the vocal energy. When the Tibetan kuden has donned his 30 pound helmet and has entered his oracular trance, he is capable of extremely powerful and authoritative displays of words. But later when he has recovered from his mediumistic trance he has no recollection of the counsel he gave. So the source of words has a mysterious origin and there is both consciously acquired knowledge and deeper levels of knowledge that can use the same vocal apparatus. To see these fundamental geometrical functions predetermining the position of the mouth indicates that the source is complex and in part transcendental. No wonder Saint John begins his gospel with “In the beginning is the word…” In the Pyramid Texts it is said at one point: “I am the Great Word”. And this Great Word had liberating effects!

It seems the best way to assimilate unusual information is thematically. The memory likes themes, probably because cognition itself holds together thematically; in patterns. So I should mention while we are here that there is yet another convergence on the mouth of note. This one is the tip of a GP-pyramid, the base of which is 1.236 or 2 × .618, the same span and same latitude within which the Pelvic Vesica Piscis is constructed, namely the  width of the Golden Rectangle, EFGH. In the accompanying illustration it is base RS set at the level of the pubic bone or ½ the height. The apex of the pyramid falls on the mouth. The apothem of the pyramid, moreover, is Unity.

This is another striking example of harmony, so proof is in order. If we put the compass point alternately on R and S, open it to Unity at E and F  and then swing it into two interpenetrating circles they will intersect on the mouth with the apothem of the pyramid as 1 and the ½ base as .618034. It is easy to calculate the height of this archetypal pyramid from the pubic bone. We just take the square root of (1 – .381966) = .6180434. So the square root of .618034 is .78615. We add unity to this to get the distance from the ground: 1.78615. Once again we can see how close these four convergences are:

A. Height divided by half of the square root of 5 = 2 ÷ 1.118 = 1.7888

B. Squared-Circle falls at exactly 1 + .7854 = 1.7854

C. The mouth of the female Thoracic Vesica Piscis is 1.788675.

D. The apex of the 1.236 pyramid is 1.78615

We can also go the 4/π route as well, giving .618034 × 1.27324 = .7869. When added to unity you have 1.7869.

These convergences are not inconsequential. They are not trivial geometrical components. Rather they are the most significant and fundamental geometrical components that relate directly and harmoniously to the Prime 2:1 that spawns and supports them.

THE SIGNATURE OF THE GOLDEN NUMBER: .618

It is quite interesting to see how versatile and harmonious .618 in fact is. And .618 is the signature of the golden number. It is the difference, the proportion that gives the golden number its meaning and operating capacity. The  reciprocal of .618 is 1.618, the golden number itself. The square of .618 is its own mate in the golden section: .381966 which will play the pivotal role in the second method coming up for squaring the circle. Recall that .618 ÷ .382 = 1.618. And .618, the ½-base of this golden rectangle pyramid, can be used two ways to obtain the height of the pyramid that reaches the mouth from the pubic bone. The square root of .618 gives a height of .7861 and so 1.7861 from the ground. And it also can be used as .618034 × 4/π = .7869 = 1.2732395 × .618034. This architecture is so strong and so flexible and so harmonious in the way inside relates to outside that growth, cell-division and multiple activities can co-function in a dynamic environment such as the Earth’s surface. The mouth’s versatility is astonishing.

In the meantime we can see that we are dealing here with several points of constructive harmony – especially the constructive use of .618. Not only is it ½ the base RS, but it is also SJ and RI. This links the pelvic bone (1/2-point) to the golden number (1.618) that marks the horizontal armspan and the Aortic Arch of the heart. So IJSR is an important 2:1 rectangle that works harmoniously with PQNM, another important 2:1 rectangle that links the Pelvic bone to the mouth very precisely. As archetypes they relate as follows:

1.5708/1.236068 = 1.2708. This number appears in Series E on Line 6, Column XV next to 7854 and .7854 is the height of this ‘1.236’ pyramid. .7854 × 1.618 = 1.2708.

I will confess to the reader that there are times when I wish I could assign a tone to each of these geometrical ratios. I then wish I could ‘play’ the orchestra of sounds that have come together in the human body. That way the experience could be direct and non-analytical. Here, the reader must supply much of their own imagination to experience the interactive significance of these fundamental ratios. What I have found is that contemplation has proven to be a little more important than analysis. But contemplation takes  time. In any case, there is no question that in the female, the golden rectangle, with width 1.236, plays a marvelous, even beautiful role, in the way it works with the Prime 1:2 rectangle. The mouth emerges as the fount of life and wisdom as the tradition insists! The geometry helps us to understand how it can be the source of wisdom.

The Human Face

It is now important to examine the geometry of the face to see where the mouth falls relative to its immediate environment and then see how that harmonizes with the 4 methods already cited which apply to the body as a whole. I will start with a brief synopsis of how I became convinced of one statistic that demonstrated that the head tended to be 13/100 of the whole height. I found this particularly interesting at Chartres because the 100 meters was in evidence as an important internal measure. Its diagonal seemed to be giving the height of the tallest tower (North). And the inside measure across the transepts suggested a golden rectangle was also determining important interior measures, namely 61.8 meters. The fact that a 13 meter circle was setting everything from the round point of the choir to the entrance to the Royal Portal and the diameter of the Labyrinth, I decided to take the 13/100 seriously. But then I came upon a curious work called “The Opening Eye” by Frank McGillion (1980) that referred to a New York Times article (mentioned earlier) from January, 1976, written by Walter Sullivan called “Doubts are raised on why the Sun shines.”  It discussed different scientific teams who had discovered a pulse in the Sun that was like a gigantic heartbeat – every 160 minutes - or 9 beats daily. Another impressive example of a Great Ennead. But this little book also mentioned something called an Earth ‘shout’ that is an electrostatic retort from the Earth’s ionosphere as if in response to the Sun’s pulses – or in any case a build up of electrostatic energy as if in a large resonance cavity. The Earth shout is 69 times a day (see L. Watson, The Gift of Unknown Things, 1977). The relationship between the Solar pulse and the Earth ionospheric ‘shout’ is 13/100, or more exactly .13043478… So I made this relationship between the Solar ‘heartbeat’ and the Earth ionospheric ‘shout’ the proportion of the human head.

Now I will introduce the head of a young woman without the geometry and will then put the geometry in to show the position of the mouth.

In the case of this young woman I have divided the height of the head in half at the rims of the upper eyelids. I have divided other faces in half at different points of the eye with equal success. But the first operation is to divide the particular head in half. That becomes the length of a 2:1 rectangle. The 2:1 is expanded to a golden rectangle (WXYZ) and it is also doubled to a Prime Square (EFGH). The Prime 2:1 (ABCD) is then able to be broken down into smaller 2:1 rectangles which can be further broken down by the golden section and so forth.

When the Prime square, EFGH, is divided in half at IJ through the upper eye lids, the mouth immediately becomes a very harmonious place geometrically-speaking. The diagonals IG and JH immediately determine the  angle of the upper lips. It is exact. I never associated the diagonals of a double square on its side with beauty. But here the angle formed by the descending diagonals is obviously creating a bewitching beauty. The angle HGI is 26.56°. The lower lip is further defined by another diagonal of a smaller 2:1 rectangle, KLCD, that also descends at 26.56°. KLCD is ¼ of the Prime 2:1 rectangles and 1/8^th of the whole Prime square. By extending CK and DL, the base of the ear lobes are delimited, so in fact a larger pair of complementary 2:1 rectangles are involved linking V to D and U to C. These are 3/4^th the Prime 2:1 rectangles and they also overlap, reaching across the Prime 2:1, ABCD.

The opening of the mouth can be looked at two ways: within the whole square and within a ¼ 2:1 rectangle, “abcd”, that exactly measures across the width of the mouth from corner to corner. So the width across the crease of the mouth is 1/4^th of the Prime square and ½ of the Prime 2:1 rectangle. Its height is ½ the height of the whole head. The mouth is opened very precisely by the golden section of this 2:1 rectangle, ‘abcd’. ‘xd’ becomes .382 and ‘ax’ becomes .618. This means that from the vertex or crown of the head to the crease of the mouth is 1.618 and from the base of the head to the crease is .382. And 1.618 ÷ .382 = 4.236 = (1.618)³. Also notice that just like the Vesica Piscis the nose is exactly 1/3^rd the width of the Prime 2:1 and 1/6^th the width of the Prime Square. Its triangular shape is precisely determined by the diagonals of the Prime 2:1, ABCD. Meanwhile the ears meet the face exactly at the boundary of the golden rectangle. What is most striking for me about this geometry is the way the opened eyes from upper eyelids to xy, or the opening of the mouth, works with the golden section to produce the overall feeling of concentrated alertness and sense of purpose that this face exudes. The next remarkable observation is the coincidence of the intrafacial measure from mouth opening to vertex (1.618) and the overall measure within the body taken as a whole (.2146). Within the head  taken as 2 the SCM length is 1.618. Within the whole body height taken as 2, the same span is .2146. Two significant harmonies act together to open the mouth. According to the infrafacial geometry, the opened eyes play a significant role in determining te opening of the mouth in this particular person.

We can now readily compute the intrafacial position of the mouth and see how it relates to the other overall methods that integrate the mouth into the whole body. As it turns out with a total height of “2” for the body when we add the percentage of the whole represented by the base of the jaw to mouth portion, namely .0249085, to the height of the body as far as the base of the jaw or .8695652 we get .894473709 of the whole, so:

2 x .894473709 = 1.7889474

We can now add this to our list of geometrical factors responsible for opening the mouth.

A. Height divided by half of the square root of 5 = 2 ÷ 1.118 = 1.7888543

B. Squared-Circle falls at exactly 1 + .7854 = 1.7854

C. The mouth of the female Thoracic Vesica Piscis is 1.788675.

D. The apex of the 1.236 pyramid is 1.78615

We can also go the 4/π route as well, giving .618034 × 1.27324 = .7869. When added to unity you have 1.7869.

E. (.618 x 4/π) + 1 = 1.7869

F. Intrafacial location (.049821653) added to 1.73913 = 1.788947

First of all, note again that I did not use exactly 13/100 for the head size. I used the ratio between 9 and 69 sensing that this relationship between the Sun’s potential pulses and the Earth’s ionospheric response (which is also tidal by the way), is probably fundamental and helping to support electrical and magnetic brainwaves – rhythms that support patterns, themes and musical-like waves that support cognitive processes and memory patterns.

9/69 = .130434782

This leaves a body height to the base of the head of .869565217. The jaw-to-mouth % of the head is .190983 and this is .024910829 of the whole body. This is added to everything up to the jaw:

.024910829 + .869565217 = .894476029

This is 89.447% of the whole height or the height divided by 1.118 as we have seen. So, .894476 is the reciprocal of 1.118. Remember that 1.118 is the ½ diagonal of a 2:1 rectangle and is the ratio between length and diagonal. The height of the KC is 1.118 × Unity or 11.18 cubits or 5.854 meters.

With a height of “2” with this configuration we have from the ground:

.894476 × 2 = 1.788952

This next illustration shows how Nature has fine-tuned the sculpting of this face. The same ¼ 2:1, here IJKL, when golden sectioned the other way across ‘ut’ determines the opening of the nostrils. So in one rectangle, golden sectioned in both directions – down and up – what is determined is both points where air is taken in and breathed out. uM or tN have a value of 1/(1.618)³ or 1/4.236.  It is a particularly stable region and we will see with cytoskeletal microtubules that this region houses a relatively free electron in a hydrophobic pocket.  In the image above also note now the even smaller 2:1, here ‘abzy’. When it is golden-sectioned from below it forms the bottom limit of the lower lip.  The V of the upper lip that completes its beauty is formed by the golden section arcs of IJKL, the same arcs that determine the crease or opening of the mouth. When I make the jaw-to-mouth distance “1”, the mouth width is 1.309 and 1.309 is 1.618/1.236 = 1.309. We are back to the proportions of the Egyptian Ra-Horachty stele. The rectangle, MNKL is part of what makes this mouth beautiful, edited, functional. This individual happens to write very well, and tries very hard to articulate her truest thoughts and impressions. She is as gifted with imagination as with the desire for ‘truth’.

Splitting Open the Mouth = Splitting open the Eye.

It was mentioned earlier that the Egyptians used different phrases to say the same thing. One event or phenomenon had several aspects to it. So “Splitting open the Mouth” also was referred to as ‘Splitting open the Eye.” The Eye of Horus was depicted in various ways, one of the most important being the Uraeus on the brow. This was the raised serpent or kundalini. In an image from Karnak it is striking to see that the measure of the jaw-to-mouth is identical to the measure of the Uraeus itself that is seated on the brow. Note that the Uraeus (Maāt, Eye of Horus) sits on 3.236 from the base. Above it is 2 to the vertex. 3.236/2 = 1.618. So Splitting open the Mouth or ‘Splitting open the Eye’ also refers directly to the golden number and golden section of the whole head. In the Pyramid Texts it is stated that the Eye of Horus is recovered from the forehead of Seth. I will discuss this in more detail in future posts.

This image below combines these same geometrical factors with squared-circle relations and angles. I did this in 1990 to demonstrate a principle, not attempt to draw an actual head.

In the next post I will demonstrate another method for squaring the circle and how it seems to be used by Nature to work a few wonders. We will also return to the Astronomical Unit to get some clues about the male geometry and the origin of the heart. The human being is an unapologetically profound creature. Our depth cannot be separated from the geometry that supports us.

El Matenavegante: La Gran Pirámide de Keops: pi por la raíz de fi es ...

La Gran Pirámide de Keops: pi por la raíz de fi es casi cuatro

Cuaderno de bitácora: publicamos hoy otro de los artículos que en su día aparecieron en doDK. Este artículo fue escrito hace más de seis años. En él explico un descubrimiento que hice por mí mismo, una extraordinaria coincidencia que se da en las proporciones de la Pirámide de Keops y que implica, necesariamente, una no menos extraordinaria coincidencia entre dos de los números más conocidos de las matemáticas.

 

[Vista de las tres grandes pirámides de la planicie de Giza o Gizeh. No hay que confundirse: la pirámide de Keops, la Gran Pirámide, es la que está más a la derecha, más atrás en la foto. La del medio es la de Kefrén, la segunda en altura, aunque en la foto parece más alta por estar más cerca, y la tercera la de más a la izquierda, la de Micerinos. La pirámide de Kefrén es muy fácil de reconocer porque conserva en su parte superior algo del revestimiento original. Es muy frecuente que se hable de la Gran Pirámide de Keops y sin embargo en las imágenes, erróneamente, aparezca la pirámide de Kefrén, la más fotogénica de las tres]

 

La gran pirámide de la planicie de Gizeh, la conocida como pirámide de Keops, siempre ha sido una fuente de misterios, y la mayoría están aún por resolver. Sus medidas han sido estudiadas exhaustivamente por todos los inquietos de los enigmas antiguos, y con los datos obtenidos podemos afirmar que los Egipcios no construyeron la pirámide dándole unas medidas al azar, sino que sus proporciones mantienen unas relaciones matemáticas muy interesantes entre sí.

La gran pirámide medía originalmente 147 metros de altura, y el lado de la base tenía una longitud de 230 metros, aproximadamente. Hoy en día la pirámide es un poco más baja, porque a lo largo de los siglos y sobre todo en la Edad Media ha sido utilizada de cantera artificial. Las piedras de las que estaba compuesta se han ido partiendo y tallando en ladrillos más pequeños para servir de material a algunos monumentos levantados en el pasado en la ciudad de El Cairo. Así la pirámide, que en su origen tenía una superficie pulida y blanca y estaba rematada por una punta de oro, se puede contemplar hoy como cuando contemplamos una casa vieja y a punto de derrumbarse, en la que se ven los ladrillos porque la capa de yeso que recubría la pared se ha caído con el tiempo.

Hace ya muchos años descubrí en cierto libro que las proporciones de la pirámide guardaban una importante relación: cuatro veces el lado de la base dividido por dos veces la altura daba el número pi. Esto es lo mismo que decir que si tomamos la altura de la pirámide como radio de una circunferencia, la longitud de la circunferencia coincide con el perímetro de la base.

Si tomamos como datos los que hemos mencionado anteriormente, h = 147 metros, y b = 230 metros. Haciendo la cuenta, 4·b = 920, 2·h = 294, y dividiendo ambas cantidades obtenemos 3'1292517..., es decir, aproximadamente 3'13. Teniendo en cuenta que tanto la altura de la pirámide como el lado de la base se han tomado de forma aproximada, es normal esperar que el resultado no coincida exactamente con el número pi.

Si tomamos en cuenta unas medidas más exactas, como las que aparecen en el libro De las mentiras de la Egiptología a las Verdades de la Gran Pirámide, de Luis García Gallo, la altura sería de 146'7 metros y el lado de la base de 230'4 metros (aproximadamente). Volviendo a hacer los cálculos con estas dos nuevas aproximaciones tenemos que 4·b/(2·h) = 3'14110429... y aquí ya nos vamos aproximando más al número pi. De hecho el error es del orden de 0’016%.

El error es mínimo y totalmente admisible ya que en arquitectura, lo mismo que en todas las demás ciencias aplicadas, las medidas tienen un límite de precisión. De hecho, los cuatro lados de la base de la pirámide no miden exactamente lo mismo, sino que se diferencian en algunos centímetros. De la misma forma las desaparecidas Torres Gemelas no eran exactamente igual de altas, sino que una era un poco más alta (creo que como medio metro) que la otra. A todo esto hay que añadir los estragos del tiempo sobre los monumentos. Las medidas obtenidas son aproximadas sobre una estimación de lo que la pirámide medía cuando la construyeron, hace casi cinco mil años, porque ahora las medidas son muy distintas...

La relación entre b y h se puede expresar así:

Es decir, la proporción entre b y h es como la de pi a 2.

Consultando la página de matemáticas Epsilones descubrí algo nuevo para mí. Según el historiador Heródoto, los Egipcios construyeron la gran pirámide de tal forma que el área de cada una de las caras triangulares laterales coincidiera con el área de un cuadrado de lado igual a la altura.

Teniendo en cuenta lo que acabamos de decir, nos encontramos con las siguientes fórmulas:

Vamos a buscar la proporción entre a, b y h:

Dividimos por b cuadrado y consideramos a/b como una incógnita:

Hemos suprimido la solución negativa porque tanto a como b son números positivos (estamos tratando con longitudes de la pirámide).

De repente nos ha aparecido el número áureo, fi), un número no tan conocido como pi, pero muy importante en la historia de las matemáticas:

De aquí tenemos la relación entre a y b, y por ende entre b y h:

Con esto tenemos que la proporción entre a y b es como la de fi a 2, y la proporción entre b y h es como la de 2 a la raíz cuadrada de fi.

Resumiendo, si los Egipcios construyeron la pirámide con las proporciones mencionadas por el historiador Heródoto, entonces la pirámide de Gizeh es proporcional a una que tenga como altura de una de las caras laterales a fi y como lado de la base a 2:

Entonces surge la cuestión de si ambas propiedades de la pirámide son consistentes, la de pi y la de fi. ¿Cuál de las dos propiedades es la que guió a los constructores de la pirámide? ¿O los constructores quisieron incluir adrede ambas características en su diseño?

Supongamos que somos los constructores, y el faraón nos ordena que levantemos una pirámide en la que el perímetro de la base dividido entre dos veces la altura dé el número pi. Como ya conocemos el número pi, sólo tenemos que preguntarle al faraón la altura que quiere que tenga, y tras unos cálculos sencillos, obtenemos todas las dimensiones, el lado de la base, la longitud de las aristas, etc. Pero el faraón nos dice poco después que además quiere que el área de una de las caras laterales sea igual al área de un cuadrado de lado igual a la altura.

¿Pueden ser posibles ambas cosas? Nosotros ya hemos hecho los cálculos de todas las dimensiones y ya casi nos hemos puesto manos a la obra... Sólo podemos esperar que la suerte nos acompañe y que efectivamente y casi por casualidad se cumpla la segunda condición que nos pide nuestro rey.

¡Y la suerte está de nuestro lado!

Para que se cumpla la condición de pi, b y h tienen que estar en proporción de pi a 2. Para que se cumpla la condición de fi, b y h tienen que estar en proporción de 2 a raíz de fi. Si queremos que se cumplan las dos condiciones, ambas proporciones han de ser iguales:

Bueno, esto no es cierto exactamente, pero sí aproximadamente:

De hecho el error que se comete es menor al 0'1%. Eso quiere decir que con un error del 0'1% podemos construir una pirámide que cumpla las dos condiciones, guardando dentro de sus proporciones al número pi y al número fi. Y la pirámide de Keops es un ejemplo de ello.

Maravilloso, ¿verdad? Y todo porque pi por la raíz de fi es casi cuatro.

 

Notas: no fui el primero en descubrir esta coincidencia entre los números pi y fi. En el libro de Martin Gardner, Los Mágicos Números del Doctor Matrix, en el capítulo de las pirámides, se habla sobre la curiosa relación entre el número pi y el número fi que posibilita que la Gran Pirámide de Keops cumpla dos propiedades matemáticas diferentes. Sin embargo, honestamente, no leí ese contenido del libro hasta este mismo año pasado, 2009, seis años después de escribir este artículo.

Por otro lado, han quedado plasmados mis esfuerzos para expresar la notación matemática en un artículo de la web. No soy muy experto todavía en esto, y la solución que encontré en su momento fue la de usar el editor matemático del Microsoft Word para escribir la expresión que quería, y luego guardar dicha expresión como archivo de imagen, para incluirlo en el artículo. Los gráficos de las pirámides los realicé con el sencillo programa Paint que viene incluido en Windows.

Para algunas otras curiosidades matemáticas de la pirámide de Keops, entre las muchas que tiene, recomiendo leer mi artículo en el blog Vientos de eternidad.

http://elmatenavegante.blogspot.com.ar/2010/01/la-gran-piramide-de-keops-pi-por-la.html