24-METHOD FOR SQUARING-THE-CIRCLE with PRIME 2:1 (Post 11)
We have seen one method for squaring the circle already. We are now going to look at another method that starts with the same Prime 2:1 rectangle, ABCD. This time it will build the three grail tables on top of itself as if it were their mother. In fact the 2:1 acts like the mother and the father, not unlike Hexagrams #1 and #2 of the I Ching, which are complete inversions of one another. Hexagram #1 is all solid light lines (yang) and #2 consists of all broken, dark, Yin lines. They could be considered the “bellows” of Lao Tzu from which the myriad creatures enter the world. This is one way to look at the first two numbers, 1 and 2, of Column I of Series E, or the next two: 5 and 10. The golden number, 1.618, is named by Column XIII. We will see just ahead that tubulin protein dimer subunits assemble into 13 slightly skewed columns to form
microtubules. The 13 columns close up a hollow core interior consisting of ‘ordered water’ that is free of all ions. It also seems to have a 3 nanometer thick layer of ordered water surrounding the tubes as if insulating them. The total diameter may be 29 nm which suggests some sort of optimum resonance capacity that might harmonize with hydrogen bonds in DNA. Some think the ordered water may be the site of an event called ‘superadiance’ in which all the photons achieve a common resonance or quantum state. This might be a critical aspect of consciousness. Microtubules seem to have multiple functions that range from mitosis to synaptic repair, to memory, to information processing, to consciousness itself. But I wanted to emphasize that they form in 13 columns and the tubules themselves bundle in a circle of 9 MT’s per bundle. When you cut open a column and flatten it you see the dimers are staggered and when counted they have been grouped in a pttern that divides into 5 and 8 which give 13. The golden proportion may be the means by which superadiance ignites consciousness in waves.
We start by swinging the unity circle around point A and where it intersects the diagonal AC we mark E. We then golden section BC by swinging the compass from CE to CF.
This divides BC into BF and FC, where BF is 3.81966 and FC is 6.18034. By now we are very familiar with this pair of numbers. We can note that (6.18)² = 38.1966 so their golden section relationship remains harmonious in diverse ways. We now take 3.81966 as the radius of a circle that we create around the navel, marked here as “I”. Its diameter (NO) will be 7.64. 382 and 618 appear next to each other on Line 2 in Series E and 764 appears under 382 on Line 4 in Series E, immediately before 1236. All this suggests that these numbers are part of a process, a dynamic web that is regulated by the golden number. So in the image above we see the 3 grail tables centered over the navel. The circle has a diameter of 7.64. The square has 6 on a side. The 2:1 rectangle is 4 x 8. The trick is finding the diameter of the circle with circumference equal to the others, which is 24. The way to do it without computation is to golden section the Prime 5 x 10 rectangle. Series E rounds 3.81966 to 3.82. If we took the actual golden section as 3.81966 and doubled it we would get 7.639320119. If we multiplied this by pi from Series E we would get 7.639320119 x 3.1416 = 23.999688. This is close to 7.64 x 3.1415926 = 24.00176. If we split the difference we can use 7.64 and 24.
The more I work Series E the more convinced I am that it was one ‘Net of Numbers’ known to Moses and housed in the Ark of the Covenant. Under Column III on Line 4, we find 24, because 4 × 6 = 24. 7 steps away is 764 under Column X, a new Unity, often heralded by ‘7’ steps. 22/π gives 7. So 24/π gives 7.64. 7.64 is the diameter of a circle with a circumference of 24. This establishes a pattern between Columns III and X. Take Line 6, Column III. It is 36 and 36 ÷ π = 11.46. Line 8, Column III is 48. And 48 divided by π gives 15.28. Line 24, it is 144 for perimeter with a diameter of 144 ÷ π = 45.84 of Column X. So Column III gives the circumference and Column X gives the diameter.
I’ll just briefly mention another way that Series E can ‘square-the-circle’. As an example I’ll just take a number from Line 2. If we take, say 6.18 and make it the diameter of a circle we can find the square with equal perimeter by tripling Line 2 to Line 6 and counting over 2 Columns to Column XIII which is 4854. What we find is that 6.18 × 3.1416 = 4.854 × 4. If you take 1000 from Column XII, Line 2 and go down to Line 6, Column XIV you find 7854. By using the decimal system we can see that 10 × 3.1416 = 7.854 × 4. So the exact squared-circle ratio 4/pi (or 1.27324) is built into Series E. 6.18 ÷ 1.27324 = 4.854. Another example would be 7.64 of Line 4. Go to Line 12 (or 4 × 3) and count over 2 Columns to Column XII. There you find 6000. So 7.64 × 3.1416 = 6 × 4 and 7.64 ÷ 1.27324 = 6
Let’s return to 24 for perimeter.
Line I above 24 is 6 and 24 is on Line 4. 4 x 6 gives 24. The circle has just been squared. The 2:1 with the same perimeter is easy: it is the inverse: 24/6 = 4. The 2:1 will be 4 x 8. We should just dwell on Line 4 for a moment. We are looking to square the circle and what better line to look to than ‘4’. ‘4’ means squareness. Now in the K.C. the square with equal perimeter to the 5.236 x 10.472 perimeter is 3/4th its length which is 7.854 of Line 6, Column XIV. 24 is the corner of the first 3 x 4 rectangle: Line 4, Column 3. So 4 x 6 gives 24. Then Columns X and XI give 764 and 1236 which we have seen are the golden section of a 1 x 2 rectangle .764 + 1.236 = 2. But 7.64 is now seen to square the circle when it is the diameter of a circle and 6 is the side of a square. A rectangle that is 4 x 8 then completes the 3rd Grail Table:
4 x 6 = 24 = 8 x 3.
Line 4 goes on to give 5236 again, the royal cubit value and then 13708 in Column XVI which is the square of 4 and a member of the power of 2 Series. 13708 is a valuable archetype because in physics it invokes the dimensionless coupling constant, 1/137.036, (Fine Structure Constant) that determines the strength for the electromagnetic interaction at about the first Bohr Radius. This strength is relevant in biology because it determines the coupling strengths among electrons of biochemicals critical for life-structures and life processes. For example, the creation of ATP from oxygen, glucose and phosphate bonds, is a high speed operation. The brain and muscles need trillions of these phosphate bonds to transfer information while thinking, moving and so forth. So the speed of the bonding and transfer process is critical for optimum function. The electron speed has to be just right if it is to function properly and is determined by the Fine Structure Constant as c/137 = 2188 km/s. We have already seen the importance of 2.618 in the calculation of π as 6/5 x 2.618 = 3.1416. This appears in Series E under Column XIV, the royal cubit column, on Line 24. Once again ‘24’ gives a circumference. 24 hours is a circumference of time and space. But 5.236 is Unity in the King’s Chamber, or 52.36 cm. 52.36 x 2.618 = 137.08.
137 is also a reasonable measure for the overall length of Chartres excluding the Saint Piat Chapel. It is simply 5.236 x 26.18 which is the diameter of a circle with the same surface as the Chartres/Sphinx square with 23.193 m on a side.
Note that 84.72, 68.54 and 137 all appear in Series E. In cubits they appear again as 161.8, 130.9 and 261.8 respectively. My guess is that the builders of Chartres possessed this Series E and understood it in a way that I have only a glimmering of.
If you look at Column XVI you see 6854 in Line 2. This is the overall width for the 2:1 rectangle that measures the whole cathedral. 84.72 is the width of its cousin golden rectangle with length 137 meters. 8472 is next to 5236 on Line 4. 137 is under Column XVI. To find the square with the same perimeter as this all-embracing 2:1, it is just
68.54 x 6 ÷ 4 = 411.24 ÷ 4 = 102.81.
Now, 411.24 appears as 41124 under Column XVI on Line 12. The square with equal perimeter is 10281 under Column XVI on Line 3. It is well worth mentioning that the South Tower at Chartres has been measured from its base to the point of the steeple as 102.64 meters. The difference here is 17 cm. This is a small enough discrepancy over ~ 337 feet, to consider seriously the possibility that the South Tower is also the length of the side of a square with the same perimeter as the commanding 137.08 x 68.54 m rectangle. For centuries the South Tower has been regarded with special reverence. There is a simplicity and harmony that satisfies immediately. It seems very likely that the height was taken from the measure of a square with the same perimeter as the overall 2:1 that frames the complete cathedral. And as we know from the human body, this length is ¾ the height or 1.5 x its width of 68.54 meters. In our anatomy this is the place of the heart, the center of the Thoracic Vesica Piscis. I think many visitors to Chartres would agree that the South Tower goes straight to the heart.
Here is another fine example of the Three Grail Tables. The diameter of a circle with the same perimeter as the 102.81 square and 137 x 68.54 rectangle is found on Line 1, Column XIV as 130.9 meters. So you take 41124 of Column XVI, Line 12 and divide it by 31416 of Line 24, Column XIV to get Line 1, Column XIV. This is still a play upon ‘24’ because 24 Lines separates 1309 from 31416. Meanwhile if you take 130.9 as a diameter of a circle Series E will find for you the circumference and the square and 2:1 rectangle with the same perimeter. That shared perimeter is Line 12, Column XVI: 411.24 Given the importance of the 100 x 61.8 meter golden rectangle for measuring interior measures including the transepts, it is worth noting the harmonious difference between 100 and 130.9. It is 30.9 which is ½ the transepts. And 30.9 is very close to 1 second of arc at the Earth’s equator and is also ½ of 61.8, the exact internal width across the transepts. A quick way to know the 2:1 and circle with equal perimeter is through the ratio of the common arm-span and the royal cubit: 137.08 ÷ 1.0472 = 130.9 so that 137.08/130.9 = 1.0472. The difference between 137.08 and 130.9 is exactly 6.18 meters or a tenth part of the interior measure across the transepts. Chartres is an endlessly self-interacting crucible of harmony that seems to be feeding itself with the inner wisdom revealed by Series E.
Again we have another pattern. Line 12 gives the perimeter while Line 1 gives the diameter but 2 columns back. Let’s take 809 of Column XIII we will find the perimeter on Line 12 at Column XV as 25416. 500 of Line 1, Column 12 will have a perimeter of 15708 on Line 12 at Column XIV. And this goes on until it goes out of phase. Series E is continuously ‘squaring-the-circle’ undoubtedly because of the presence of the golden number. So we can see a sort of inversion occurring here. When we are on the right in the vertical Column 3 we have the perimeter. When we go to the larger numbers in Column X we find the diameter. But when we are in the horizontal Line 1 we have the diameter and when we head down to the larger numbers we find the perimeters on Line 12.
137.036 is an interesting number. If you take the SCM as a constant ratio, say 4.6558 it creates a curious progression.
29.433 x 4.6558 = 137.036
137.036 x 4.6558 = 638.0122
638.0122 x 4.6558 = 2970.4572
2970.4572 x 4.6558 = 13,829.85463
13,829.85463 x 4.6558 = 64,389
64,389 x 4.6558 = 299,792
6 steps gets from a speed that is close to the Earth’s circular velocity to the speed of light. This is ~ 4.6558 raised to the power of 6 or 10,185
Curiously, Sirius B used to be about the Sun’s size and has now collapsed into a White Dwarf with a diameter that is 80% that of Earth which is about 10,185 km. If we divided the diameter of Sirius B by 4.66 we would get 2188 which is the speed of electrons in the First Bohr Radius or 299,792/137. A White Dwarf is thought to be held up from complete gravitational collapse by a ‘gas’ of degenerate electrons. What bet their speed averages to about 2188 km/s and that determines its size. In other words the Fine Structure Constant is affecting the collapse ratio of a White Dwarf. And the SCM is involved in establishing a progression.
The more one works these numbers the more one thinks of the phrase under the sculpture of the Ark at Chartres: “Here things take their course. You are to work through the Ark.” “Archa Cederis”. According to what we are seeing, “archa” must refer to the relationship between the circle and everything else. When things divide square-ness sets in. Square-ness is part and parcel of inverse square law forces: gravity and electromagnetism. Gravity and electromagnetism hold together the solar system, the planets and biological structures. The nucleus of the atom with their nucleons hold together the matter. And according to Capra the speed of that matter in its confined state is around 64,374 km/s. This number occupies a position in the little chart just shown above. Its relationship to the constant c is the SCM: ~ 4.6558, which is derived from the ideal SCM, 4.666 or 1.27324/.27324.
According to this number, this relationship, 4.66, the speed of light in the vacuum is being altered when energy (E) is bent or curved (Archa) into some type of enclosure or self-enclosure. The E reacts to its own enclosure and eventually will feel the reaction. It will become conscious of the reaction when a threshold has been reached. The square suggests an acceleration which is radius dependent within a field. The circle is in part a reaction to the square and vice versa. It seems that when they are in balance – when they have the same perimeters – something focuses. What focuses is an energy. Something opens up. The Eye opens and sees. When our vision is blurry we feel ill. So health and resolution are equal. And resolution seems to depend in part from squared-circle resolution. The underlying geometry is the basis of health. The etymology of the English “health” goes back to heath and the heather that is found there. This in turn came from a more ancient root denoting open, unploughed country. So health was originally associated with the most natural and unrestricted state of life. This would be the place of the Wild Mountain. The geometry that gives form would have to remain in communication with what is wild to preserve health. The golden number and squared circle geometry are probably doing just that and in fact this open relationship between what is wild and what is contained is likely a good way to characterize their function.
Let’s review this 24-method for squaring the circle.
The way to create this image is to simply golden-section the Prime 2:1, ABCD (5 x 10), into 3.82 (BF) and 6.18 (FC). You then center a circle on the navel, point I. You take 3.82 (BF) as its radius. You inscribe a circle with diameter 7.64. Its’ perimeter will be:
7.64 x 3.1415927 = 24
The relationship between 10 and 7.64 is 1.309, so the reciprocal of 1.309 is .764. Recall that 1.618 ÷ 1.236 = 1.309. And this ratio framed the Ra-Horachty stele from the earlier post. I cannot emphasize the importance of this ratio enough. It should be emblazoned on a flag and be seen fluttering in the wind outside churches and courts of law. It is easy to use whole numbers to create a square and 2:1 rectangle that are 24 on the perimeter. The square is 6 x 6 and the 2:1 is 4 x 8. 10/8 = 1.25 (which is the square of 1.118), the Major 3rd in music; and 10/6 is 1.66666, very close to A-natural (440) (440/264 = 1.66666). This paradigm is packed with geometrical information that gives it remarkable self-focusing properties. For the moment I should just mention again that I am rounding up 3.82 and 7.64 as in Series E. In fact they are bothirrationals and appear in this paradigm as 10/2.618034 = 3.81966 and 10/1.309016 = 7.63932.
7.63932 x 3.1416 = 23.999 ~ 24
The way to ‘square-the-circle’ with the prime 5 x 10 2:1 is through the golden proportion. An ‘irrational’ diameter times the irrational pi produces a whole number; in this case 24. This is an inversion. Normally we think of pi as irrational and operating at the limitless circumference of a circle. We think of the diameter as being finite. But here the diameter becomes ‘irrational’ and the circumference becomes ’24’. 24 is a nice number. We have used it to divide the ‘circle’ of the day for a long time. The hour has relevance in human rhythms. 24 hours sets the Circadian rhythms in the brain which according to the paradigm we have just seen fall in the King’s Chamber: Note how the floor of the KC divides the Unity Cube into .382 and .618! This is also the level of the Great Step.
The floor of the Queen’s Chamber corresponds precisely to the Foramen Magnum which is where the spine plugs into the brain proper.
In any case, it is the golden division within the 5 x 10 rectangle that sets the scale of the 3 Tables. But it is as if the 5 x 10 is the underlying support of the ‘3 Tables that bear the grail’. This will have consequences when we look at the nanometric scale of the tubulin protein dimers (4 x 4 x 8 nanometers in volume) which may be information-processing automata in the cell, but also the scarab beetle that undergoes a metamorphosis from worm to winged insect of extraordinary beauty. This does not just happen out of thin air. There must be geometrical forms that facilitate this remarkable transformation of form. Something is ordering it in its cocoon. Three tables bore the Grail. The Grail is the essence (in the Stone) by which the Phoenix is burned to ash and from which it rises again as lovely as before!
1.309 is acting as a key of transformation and this must have something to do with the relationship between 1.618 and 1.236: respectively within a 2:1 rectangle, its golden number and its golden section. 1.236 is also the width of its cousin golden rectangle within which the female Pelvic Vesica is constructed. So 1.236 in the female is the relationship between heart and lungs above and womb below. When it is related to the golden number we get 1.309, at the headwater of Column XIV of Series E. So here it is the relationship between parent and child; Prime 2:1 and grail circle: 10 ÷ 7.64 = 1.309. We can also see in Series E that these numbers are lined up through a diagonal from 1236 to 1500 to 1618 to 1309. And we can note further that 1.618 ÷ 1.5 = 1.078689, the width that contains the male Thoracic Vesica and is also the armspan set at the Idea Angle of botany which divides a circle by the golden section (137.5° + 222.5°).
If I were asked to be as specific as possible as to what I think “archa” refers to in the Latin inscription at Chartres, I think I would say the arcs that swing out the golden section and golden number within a 2:1 rectangle that in turn generate the three grail tables. The Ark that contains the root of transformation from an angular to circular table is the 2:1 rectangle. Through it things take their course. Concern for equality of perimeter seems to reflect a concern for the conservation of energy or spirit throughout the transformations within the 3 tables; the 3 phases.
This image is remarkable. This beautiful insect, carefully chosen by the Egyptians as one of the symbols of the entire opus of man’s metamorphosis, has arisen from a worm through this combination of three enclosures set over the ‘navel’ of the prime 2:1. This 5 x 10 rectangle that generated the circle with circumference 24, remains active in shaping the scarab. It measures its width if the middle claws were tucked in slightly to touch the body. But the whole length is framed precisely by the 4 x 8 rectangle which extends a little above the 5 x 10 by the ratio 1.0472. The “1”, or 7.64 – circle, exactly limits the antennae. The 4 x 8 (EFGH) determines the main width of the body minus the middle claws. The fore claws are limited by the 4 x 8 for extent, but are shaped by the diagonal of the prime 5 x 10 for angle of attachment to the body! They are in a precise complementary relationship with the opposite hind claw also determined by the diagonal of the Prime 5 x 10. The Square, IJKL, cuts through the eyes and marks details of the rear legs/claws. The precision of these interactive forms can only be expected in view of the precision of the scarab’s natural beauty.
But the key to this entire image is the relationship between the Circle with a circumference of 24 and the Prime 5 x 10 that created it. 10/7.64 = 1.309 = 1.618/1.236. This then allows the other 2 tables with perimeters of 24 to be generated. They are all centered upon the navel of the 5 x 10, or 6.18 from the base DC. There is the distinct feeling that the golden section is at work in the process of metamorphosis that transforms the worm into a winged insect. It is the feeling that the golden proportion is the “Stone” itself, the essence most pure, born in the midst of the activity of the 2:1 rectangle. The entire 5 x 10 seems to focus itself at the navel as if it were a new umbilical link to the 3 new Grail Tables. The volatile urge within the worm awakens and flows through the portal of that umbilical link as if magically. The 3 Tables that bear the grail limit themselves as they distribute the extrusion of energy among the three shared formats: square, circle, 2:1 rectangle. The 2:1 is the key to the limits and function of the main body of the insect. But the circle with equal perimeter carefully limits the antenna and the rear of the body.
The squaring-of-the-circle is the geometrical means by which this whole distribution of vital energy is facilitated and the golden section is the way the circle-is-squared. Through this geometry, what was in the worm is conserved. Through the 3 tables the geometry can blossom, sharing the characteristics of all three tables as well as the underlying Prime 2:1. Looked at this way, there is a striking parallel between the Egyptian symbolism of the scarab “Khepera” and the Grail. Automatically, then, we have to conclude that there is an even older parallel between the Eye of Horus and the Grail. The Eye of Horus at one point in the Pyramid Texts was said to ‘go up from the head of Seth’. If it takes 3 tables to bear the ‘grail’, one must empathize with the Egyptian tendency to give one phenomenon many ‘names’. Each name refers to different aspects of its nature. The Egyptians were confident of unity and appeared not to be reductionists. I think we have projected our own western reductionism on a civilization that we did not understand. Their spiritual preoccupation was with the final metamorphosis into pure spirit. This occurs when the plurality of names returns to the unity of spirit. The golden proportion and the squaring of the circle seem to be the means by which this metamorphosis takes place.
This paradigm also lends itself to a reinterpretation of the ‘brazen serpent’. There is the feeling that the 3 grail tables, generated through the golden section of a prime 2:1 rectangle, is the means by which the serpent is raised in the wilderness. The wilderness does not simply mean ‘sin’. It refers to the Wild Mountain of the Grail romances; untamed forces; the chaos of the alchemist that provides the extra energy for spiritual transformation once it has caught itself, like the uroboros, in its own ‘grail vessel’.
It is worth looking at other examples of where these 3 grail tables supported by the navel of a prime 2:1 rectangle, seem to be functioning. One overriding feature that seems to be common to all examples is the capacity to self-focus the diverse energies supported by the four tables in response to the surrounding environment. Obviously this self-focusing capacity must start small: at the microscopic scale. So I’ll show the tubulin protein ‘dimer’ first, based on drawings now familiar to many. According to Stuart Hameroff, who initiated the interest in the role of microtubules in the phenomenon of consciousness, these drawings are really quite schematic, ‘cartoons’ as he now calls them. But I will apply them to the ideal geometry anyway in order to illustrate the principle.
The world of tubulin protein is a science unto itself because they are at the border between quantum and relativistic scales. They straddle and ‘partake’ of both behaviors – probably by alternation like most phenomena we know of. To the human mind this alternation can appear as contradictory. Being at the border between quantum and classical behaviors, they in a sense, appear to embody the ‘gap’ in our knowledge of how the mathematical descriptions of the two behaviors unite. Because Stuart Hameroff and Roger Penrose have persisted in their determination to explain the role played by microtubules in the physiology and physics of consciousness – in an effort to unify mind and brain matter – it is worth at least glancing at an image of one subunit known to be a double sphere occupying a tiny space that is 4 x 8 x 4 nanometers. Because they assemble into tubes consisting of 13 columns, they appear to either side as 2:1 constructions. They contain a hydrophobic pocket that houses a single, relatively free electron that moves back and forth between the alpha and beta spheres – a bit like a built-in switch! Hameroff now tells us that these hydrophobic pockets may be located in various parts of the double sphere. This schematic diagram here, then, is a sort of geometrical ‘what if?’, where the geometry can be seen as a bridge, a continuity that links the basic dimer subunit of cytoskeletal microtubules all the way to the newborn human infant. The geometry might play a role in the internal self-organization of all early one-celled creatures which are capable of learning and reacting in accordance with that learned state to the changing environment that engulfs them. According to this model, the golden section gets the ball rolling and may well be the means by which the learning and adapting takes place, making of a living thing something much more than just a crystal.
Note that ordered water, that behaves like crystal, is surrounding the tubes and is contained within the tubes’ cores, but the hydrophobic pocket has no water in it at all. Its electrostatic properties have expelled water entirely from the pocket where one single electron is housed. The change in dipole state of the dimer results in a change in position of the electron. This is accompanied by a change in shape or conformation of the dimer so that it appears to open and close or nod at its ‘mouth’. In the image above I have shown it in transition between the two positions and am offering a geometrical solution as to how it distributes its self-organizing mass during the conformational shifting. In its open state it is 4 x 8. In its closed state it may go down to a shared distribution between hexagon and square with equal perimeter – meaning 24 nm. I am also suggesting that the hydrophobic pocket may be 1.888 nm which is the overlap of both golden sections taken up and down, so that 8 ÷ 4.236 = 1.888. 4.236 is the golden number cubed, which invokes a volume and enclosure of a high order of harmony. This might play a role in expelling the water from the pocket. This is a zone of optimum stability and harmony that keeps the slightly larger (56,000 Dalton) Alpha sphere in contact with the slightly smaller (54,500 Dalton) Beta sphere. I should also note again that each tube consists of 13 columns of dimers (slightly skewed or staggered) but microtubules assemble in groups of 9, yet another example of an Ennead. Since MT’s are involved in separating self-replicated DNA during mitosis, this Ennead is quite fundamental. It also appears in the tail piece of the sperm and so is involved with a sort of whip-like and rotational locomotion that guides the sperm in its journey toward fertilization. 9 can handle a fundamental acceleration/deceleration cycle generated by 3 x 3. This is highly creative and interactive.
To help picture the scale of operation here, note that 4 nm is 1/100 the wavelength of violet light and 8 nm is about 1/100 the wavelength of infrared. Each dimer contains about 900 amino acids; 450 per monosphere, already displaying the fundamental properties of 2:1. The conformational shifting of the dimers when assembled into tubes is important because it seems to correspond to different states of electrical polarization. In this way they may well be capable of processing electrical information and conducting that information within the cell and possibly to the synapses. But the tubes are also capable of becoming insulated by ordered water that may allow their hollow cores to act as wave guidance systems or super-radiance devices whereby quantum coherence becomes a possibility. With 9 also involved with the 13 columns we have a rather complete picture of self-regulating, self-organization and information processing. Recall that the golden number names itself in Series E at Column 13, Line 2. I won’t go any further into the mathematics of the quantum wavefunction here (see Roger Penrose for clear explanations in “Shadows of the Mind”, 1994)) and how MT’s may be involved with all their golden proportion relations, but it is worth mentioning that for decades (after 1971?) Stuart Hameroff believed that anesthetics were working by seeping the chemical into the hydrophobic pockets of MT dimers, binding up the free electron and shutting off consciousness. And this is how the idea got going. There is so much more to say about the behavior of microtubules in the cytoplasm of the cell. There is the cell’s sol/gel cycle involving calcium. There is a metabolic input. There is the notion of feedback loops that would set up an amplification wave needed to generate the quantum wavefunction but also to contribute to the collapse of the wavefunction that causes matter to ‘choose’ a particular state over an array of possible states described by probability distributions. What I wanted to do here was to show the possible role of geometry even at this tiny scale, in guiding the morphology within which quantum coherence might be able to thrive thereby allowing mind and brain to cyclically act as ‘one’ while yet retaining the principle of difference through alternation, dipole behavior, and of course the underlying exclusion principle of Pauli. I sometimes feel that the golden number retains the secret key that allows a stream of consciousness to be maintained because it divides things unequally. This staggered asymmetry – yet harmony – allows what is separated to be joined in a mobius like action that feeds back into itself. There is no learning like self-learning. But self-learning is also the suspense of self-transformation – or at least the potential for it. Just plain proliferation does not mean relatedness. The golden proportion and its family allows continuous relationships among parts whereby proportion becomes the secret ‘glue’ of the conscious present. As I have said repeatedly, the secret asymmetrical, but harmonious, signature of the golden proportion is .618.
Now let’s look at a similar shape but at a larger scale: same geometry. This is the very familiar human fetus. Though much larger in size it still conforms to the same shape as
the dimers and is also prone to ‘nodding’. Its height is wholly contained within the ‘8’ rectangle (in blue here), but its width including large forehead and lower dorsal region, fill out the original width of the Prime ‘5 x 10’ rectangle: and once again, with amazing precision! The arms, ears, eyes and heart are contained precisely in the area corresponding to the hydrophobic pocket of the dimer as if this slot which is 8/4.236 or simply 1 over the golden number cubed, were of particular importance. Here the heart is regulated perhaps. All the needs of all growing cells will have to be supplied by the heart which distributes everything through the blood. In this pocket a pressure differential is established and regulated which is one reason why electron ‘speed’ in the First Bohr Radius is so crucial. It is part of the efficiency with which oxygen and carbon dioxide are managed by hemoglobin and the iron atom that is at the heart of hemoglobin.
Next we will look at the same set of tables as they apply to the newborn infant after 9 months gestation. First of all note that crown to knees fits in the 8×4. But the feet take it ‘back’ to the 10 × 5 (ABCD). The baby’s width, excluding the arms is exactly ½ the width of the 4 × 8, so ‘2’. But the height of the head is exactly ½ the width or ¼ the length of the 5 × 10! So the head is 1 × 1.25 or qr/ru. The throat of the baby, who does not have much of a neck, corresponds to the center of the thorax in the adult.
Now note how the heart occupies the upper half of the ‘hydrophobic pocket’. In the dimer the upper half would be in the Alpha sphere which is slightly larger than the Beta sphere and carries an excess of negative charge. The head is what develops most prominently first in the fetus and is proportionally large. So presumably the head is carrying a higher charge than the caudal pole. It would appear to contain the seed from which all other growth extrudes. Here the heart is lodged between the golden section of the 5 × 10,
the center of the entire complex of smaller grail tables and the golden section of the 4 x 8, indicated by mn. The infant’s navel is the center of the 5 × 10 where the diagonals cross and the genitalia are at the lower point of golden section of the 5 ×10. To summarize, here we see a real co-operation between the 4 x 8 rectangle and 5 × 10 rectangle. Particularly striking is the head that has its height determined by the 5 × 10 and its width determined by the 4 × 8. The process of dimidiation of the 2:1 rectangles seems to be in evidence all the way to the adult. Note that the heart seems to be angled by the diagonal EG, which is upper right to lower left of the 4 × 8 rectangle. The two hands appear to be ‘plucking’ the invisible diagonals of the 4 × 8 as if they were musical strings – the strings that make the baby want to sing.
Next I have shown the morphological similarity between a global magnetohydrodynamic simulation of the Earth’s Magnetosphere and the human eye. I have placed the same tables in the upper half of a 2:1/hexagonal image that embraces the whole eye right to the cornea. The ‘hydrophobic’ pocket falls on the anterior half of the lens. The lens complex with cornea and so forth distributes its functions right through the grail complex as if the self-focusing properties of the tables were their most important feature.
What is important about this morphological comparison is the way the Earth is acting similarly to the lens of the human eye. When the geometry is added it seems we are looking at the means by which energy has been gathered up, or gathers itself up, and self-focuses through the ‘grail tables’. Comparable shapes result within differences of density and medium. But basic ingredients remain the same. Clearly for the medium to self-focus, the grail tables must build up an inner pressure: some of which is electrostatic. This then resolves the material into an ordered state which in the case of the eye permits perception.
The Magnetosphere looks different when examined from different points of view and in different ways. One rather extraordinary cross-section strongly resembles a lizard. The Earth seems to be acting like an electromagnet in the position of the nostrils above the mouth. Snakes and presumably lizards are 95% olfactory in character, which is to say their cerebral cortical development is 95% olfactory derived. A human is 80% visual. An eagle is 95% visual! So it is fascinating to see a common geometry at both ends of the sensory spectrum.
There is the sense of electromagnetic self-organization that involves decussation of electric current that weaves together these forms including volume. Once a vessel has formed around the energy that has crossed over and curved back upon itself, it is then able to build up. As it builds up consciousness becomes increasingly dominant as if it were the reaction to both internal and external stimuli. In this image the Earth acts like a seed around which lines of force gather. The creature always seems to have been formed around this little seed as if consciousness itself was vastly greater than the seed through which it is threaded!
In the image below one can see that the decussation of optic and olfactory is stacked on top of each other at the same site. It is marked by EF which is the golden section division of the Prime 2:1, ABCD. I have put in the ‘7.64’ circle to point out that it seems to create the boundary for the visual re-association area of the cerebral cortex. The ratio that dominates this boundary is once again 1.309 or 1.618/1.236.
Note that the angle separating the eyes is the same angle found in the Synodical Month, namely 29.1°. This angle runs to the back of the head. So in 29.53 days from New Moon to New Moon, the Earth and Moon together travel through 29.1°. And from the Sidereal Month (27.3217 days) to the Synodical Month (29.53066 days), the Moon travels 29.1° in 2.2 days.
Here in a slice through the brain at the level of the optic chiasm, we see the 29.1° recapitulated again.
With a feeling for the morphological shape of the magnetosphere, I would like to show the ‘teardrop’ shape that is familiar to many as an analogy to a rather intriguing image of Hour Ten of what is in the Duat of the Egyptians. We see an unmistakable portrayal of a well-known breathing technique practiced at advanced stages of ‘spiritual yoga’ where the intention has become the reconstitution of the Eye of Horus. The technique involves raising the kundalini by alternating breathing through the nostrils. The finger is used to pinch one nostril while one cycle is performed by the other. What happens is that the electromagnetic pathways stimulated by the very primordial olfactory decussation mechanism at the White Anterior Commissure, eventually with respiration and heart rate registered in the pons and medulla oblongata, stimulate the spine and provoke the
kundalini in the sacrum. This raises the ‘serpent in the wilderness’ – suggesting there can be a constellation of chaotic psycho-spiritual forces which makes available a large amount of energy. This then reaches an apotheosis in the realm of the ka whereby the Eye of Horus self-organizes around the mystical marriage of Horus and Isis eventually resulting in spirit-state or spirit-form. What is striking here is that the two figures are not seated on anything concrete. There is no question that the figure on the right is using the right hand and arm to ‘pinch’ one nostril while the figure on the left is using the left arm and hand to pinch the opposite nostril. There are two serpents for the two halves of the spine. I placed the teardrop-shaped image of the magnetosphere below it to suggest how this great form has been recapitulated in a human endeavor on earth with a certain instinctual self-knowledge. With this breathing technique, it is not smell that matters but rather the meeting of various forces that converge electromagnetically to stimulate the kundalini. This undoubtedly involves heart rate and chemoreceptors on the Aortic Arch which communicate information back to the Medulla Oblongata in the lower brainstem. This in turn will influence the spinal column and probably the state of the cerebrospinal fluid that bathes the brain and runs down to the sacrum creating a cranio-sacral pulse. CSF is formed in choroid plexi that receive blood and then remove large percentages of material from the blood, dripping a clear, colorless fluid with a lower specific gravity into the four ventricles of the brain. CSF is renewed around the brain about 4 times daily, which is about the same as the number of tides in a day.
The alternating breathing balances, amplifies and stimulates the complete cranio-sacral complex. It is as if it acts upon a hydraulic pump that primes the well of the sacrum, drawing that sacred fire-water up the spine in a wave that will reach the King’s Chamber in the head and reconstitute the eye.
Possibly one of the most beautiful, startling, pure, compelling, inspiring and direct expressions of this intention of the Egyptian way can be found in Utterance 524 of the Pyramid Texts (Faulkner translation is just fine for this one) from PT 1233 to PT 1243. Note also that the endeavor appears to be reuniting the Red and White Crowns which in various texts are referred to directly as the Eye of Horus. They mean of course its function and its purpose within a human life; not its ornamentation. Utterance 524 is called “a text concerning the Eye of Horus”. It is a great one. It is complete. It is startling and it helps to understand the meaning of the image above. It strikes me that the sphere you see materializing atop the two serpents is probably the same sphere the image maker sculpted into the Ark at Chartres. It is the same endeavor, the same purpose, the same transpersonal inspiration:
“I am he who prevents the gods from becoming weary in seeking the Eye of Horus; I searched for it in Pe, I found it in On [Heliopolis], I took it from the head of Seth in that place where they fought [meaning Horus and Seth]. O Horus, stretch out your arm to me; O Horus, take your Eye; may it go forth to you, may it go forth to you when I come to you. May the Eye of Horus come to you with me, upon me forever.” (PT 1242-43). The analogy between this and the story of David and Goliath is striking. David takes Goliath’s head after slaying him by hitting him with a single stone in the middle of the forehead. Goliath is very close to the story of Seth. Seth is the material force that takes the Eye and imprisons it. The candidate for spiritual metamorphosis must retrieve the Eye. In Egypt it was found on Seth’s head and must be returned to Atum – the Lord of the Horizon. The King’s work is not merely for himself; it is for the totality, gods included. We can recall that π is the ‘horizon’ of a circle the diameter of which is ‘1’. The King’s Chamber in meters has a perimeter of π x 10 or 3.1416 x 10 = 31.416 m. One imagines that they intended it to be an inner gate or portal (sba) that opens the road or way to Him of the Horizon, Atum, the original, eternal.
Entrance to the Netherworld and the Golden Section
With this in mind take a look at the famous ceiling of Seti I of the XIXth Dynasty in the Valley of the Kings. This ceiling image portrays the entrance to the Netherworld. It is very obviously designed. All I have done is highlight it.
ABCD is an exact 2:1 rectangle. It is the same as the ceiling of the King’s Chamber. This 2:1 captures a rotation of the heavens around a celestial pole that shows the summit of the circumpolar North Stars such as the Big Dipper, here shown as the Bull of Heaven chained or tethered like Seth to the ‘Thigh Bone of the Horus Eye’, guarded by the pregnant Hippo, Isis, with Draco? (Alpha Draconis?) (or Sobk perhaps?) surmounting her to join the rotation. The standing Osiris King with up-raised arm connecting sky to ground, is pointing to the exact golden section of the 2:1 rectangle – the portal to heaven. The Bull is exactly ½ the width (AD or BC) of the 2:1 ABCD, implying that it has a value here of 1.309 compared to the Thigh of the Horus Eye held by the Hippo with value ‘1’. The height of the standing King with up-raised arm is also 1.309. In this context the full length of the opening is 5.236 and its width is 2.618. In Utterance 524 just mentioned at PT 1238-39 it is stated:
“Atum summons me to the sky, and I take the Eye of Horus to him. I am the son of Khnum [Ram-headed Divine Potter], and there is no evil which I have done. Long may this word be in your sight, O Re; hear it, O Bull of the Ennead, open up my road, make my seat spacious at the head of the gods, that I may take the Eye of Horus to him and that I may cause to reknit for him that which went forth from his head.” (PT 1238-38, p196-7 Faulkner).
Note, also that when the thigh-bone or thigh of the Horus Eye is taken as ‘1’ which it is when enumerated in the Wadjet Eye, it divides Unity also by the golden section so that the length overall equals 5.236 which is the same as the archetypal Pyramid Profile of the Great Pyramid. The top of the mnj.t or Thigh Bone is marked by the hands of Isis and the outermost curve of her pregnant belly. The total Eye is 64. It is then dismembered by Seth. When it is reconstituted it adds up to 63, not 64.
Unity remains the thigh bone of the Eye of Horus also known as ‘qd’ and “mnj.t” with a value of “1”. In the Seti I ceiling Horus mounts to the sky on the “thigh bone” itself, with the Bull supported by his harpoon and tethered by the mnj.t – the seed or essence of the Eye of Horus. In some instances the “thigh bone” is referred to as a Mooring Post. In the Pyramid Texts the Mooring Post refers explicitly to Isis or Nephthys. Isis was associated in turn with Sothis or the bright star Sirius. The suggestion is that the Egyptians were treating Sirius as some sort of celestial Mooring Post, in turn associated with the axis of rotation of the heavens.
It is this essence that has everything to do with the Grail – the essence of the stone by which the Phoenix purifies and renews itself and cannot die. This is the inspiration that caught the minds and souls of the Templars and formerly, the entire Egyptian Civilization. When the essence is known; when the keys to the kingdom have been recovered then ‘things take their course”. You work through the Ark which carries the Eye of Horus. It appears that the King’s Chamber was designed to act as just such a portal: