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Why Bother About Sacred Geometry? Mystery Brought to Light

Change your attitude, and you change your life. This is literally real. And that goes both directions, make no mistake about it, for negative, derogatory life, feelings or failures or for the positive. Change your perspective and change your eternity. This is literally real as well. But change your perspective? From where or what to where or what? What does this even mean? I will explore it a little in this paper.

What you see is what you get. I will believe it when I see it. We use sight as a basis for so very much in life, and quite properly so. It is quite true that our acuity in seeing is a matter of life and death. It is also precisely a matter for attaining to a higher, deeper, and broader level of enlightenment within ourselves.

From the Doxographers of Aetius, Plutarch and Theophrastus we learn that the Pythagoreans “all assume matter is subject to change, [they] assert that genesis and destruction in an absolute sense take place, for from change of the elements, modification and separation of them there takes place, juxtaposition and mixture, and intermingling and melting together.”[1] And in the Timaeus of Locri (ca. 100 C.E.) is taught “Of produced things, the substratum is Matter, while the reason of each shape is abstract form; of these two the offspring is Earth and Water, Air and Fire.”[2] Lucretius (ca. 100 -55 B.C.) described the universe as “raw material, or generative bodies, of ‘seeds’ of things. Or I may call them primary particles, because they come first and everything else is composed of them.”[3] Simplicius described the ancient atomists, Leucippus, Democritus, Epicurius ideas “said that the first principles were infinite in number, and thought they were indivisible atoms and impassible owing to their compactness and without any void in them.”[4]

In this very brief, incomplete survey from antiquity, we see the idea plainly before us that “stuff” is the reality. Plotinus did end up distinguishing the difference “between the individual bodies of the sensible world and the conditions underlying the very possibility of such things so individuated.”[5] Kurt Smith notes that what distinguishes matter from what we think is there is what for Kant “would call the empirical versus the transcendental. Inferior matter is, as Schafer contends, ‘what we

The Greeks saw the

But today we now know that “stuff” is not the only reality, even though we are all made of it, as is everything else we see and experience in life. Stuff is everywhere! Especially in our garages and basements which we overcrowd and make it difficult to get to. So, if it’s not real, then why is it all over the place getting in our way all the time? If all that crap is not real, then what is?!

Now-a-days it’s hip to talk not of particles as the

Yet, and yet… why stop at gravity? Einstein, without question has changed our perceptions, but why stop at such milestones? Is there yet more change in our perceptions that might help us see a deeper truth? Without question! “The world is made up of particles and fields, and nothing else; there is no need to add space as an extra ingredient. Newton’s space is the gravitational field… space is no longer different from matter. It is one of the material components of the world, akin to the electromagnetic field. It is a real entity which undulates, fluctuates, bends and contorts.”[12] And now we are led into geometry.

In biology we learned the DNA works in humans, not only because of how particles and the molecular components are the carriers of continuity in our lives (we change every cell in our bodies every 5-7 years), but the helix shape is responsible for its power. “The helix, which is a special type from the group of regular spirals, results from sets of fixed geometrical proportions… their proportions can be understood to exist a priori, without any material counterpart, as abstract, geometric relationships. The architecture of bodily existence is determined by an invisible, immaterial world of pure form and geometry.”[13]

Remarkably interesting as well is the complex twelve-fold symmetry pattern of photosynthesis of carbon, hydrogen, nitrogen and magnesium and chlorophyll in the molecular body of living organisms. “It appears that the same constituents in any other arrangement cannot transform the radiant energy of light into life substance. In mythological thought, twelve most often occurs as the number of the universal mother of life, and so this twelvefold symbol is precise even to the molecular level… this spatial awareness on a cellular level may be thought of as the innate geometry of life. All our sense organs function in response to the geometrical or proportional differences inherent in the stimuli they receive.”[14]

This twelveness is found throughout the ancient world symbology whether that of Israel’s tribes, or other nations aligning with the cosmos in China, Ireland, Iceland or as far south as Madagascar, or showing up in other symbols such as Britain’s King Arthur’s Round Table, all tied to the zodiac of constellations and alignments here on earth with stone works and temples, altars, pathways, etc.[15] It is also found in the mystical dimensions of the verbal description in the Jewish text

This underlying geometry found in life and the cosmos is also found in arithmetic, that is, number. “We obtain wholes by linearly adding together discrete units, such as 1+1+1. In systemizing number in this way we forget that number continuity really only progresses through a wave-like alternation (odd-even), that there are no discontinuous magnitudes nor discrete parts, that a unit can never exist outside of a contiguous form/flow. That is why the Greek word

It is to the one Pythagorean aspect of the Sacred geometry that becomes so very intriguing however, that of the relation of the diagonal to the sides of a cube that is most stimulating, which I get to just shortly. “Within the human consciousness is the unique ability to perceive the transparency between absolute, permanent relationships, contained in the insubstantial forms of geometric order, and the transitory, changing forms of our actual world. The content of our experience results from an immaterial, abstract, geometric architecture which is composed of harmonic waves of energy, nodes of rationality, melodic forms springing forth from the eternal realm of geometric proportion.”[21] In the fragments of Philolaus he mentions “the world’s nature is a harmonious compound of limited and unlimited elements; similar is the totality of the world in itself, and of all it contains.”[22]

It is exactly this, in fact, with which the idealized relationship exists between the sides of a square to its own diagonal splitting it into two perfect triangles from corner to corner of the square, between both the rational elements of number and the irrational aspects. In Psalms 104:2 we read where God wears a garment of light and stretches out the heavens like a curtain. Now the idea of wearing a garment of light is the limit theme to fit within a delimitation, and yet the stretching out of the heavens reveals not the elements with substance, rather the unlimited statement of process.[23] The process of “drawing” the cosmos with His energy, his power, the ability to produce “stuff” out of the background geometrical proportions. We imitate that creative process when we engage (as the Zohar constantly encourages) in doing the geometry for ourselves in imitation of the Creator.

John Michell has demonstrated throughout numerous delightful books that “All over the world the traditional units of length, area, weight, and capacity are related to each other and derive from one original canon of cosmology… the cosmological origins of the English and megalithic miles… occur naturally in the measurement of all classes of phenomena from the human to the astronomical scale. Their sanctity is not therefore a matter of arbitrary convention or superstition but inherent.”[24]

“Physicists have discovered that the deeper they push into matter, it looks like the cosmos of the Pythagoreans and Platonists. Each atom is a Pythagorean universe, the sight of eternity in a gram of sand, consisting of an arithmetic number of particles, geometrically distributed in space, dancing and vibrating like a miniature solar system to the music of the spheres...matter and energy are but different aspects of one, underlying continuum. Advancing to the subatomic level, quantity becomes quality, energy becomes information...the science of physics, proceeding from matter to energy, from energy to intelligence (i.e., pattern,

While it’s true that Walter Burkert demonstrated the concocted story of the Pythagoreans putting men to death for revealing the ancient secret mathematics they discovered, to be so much pixie dust, it’s also interesting that he missed the significant meaning of the diagonal to the square in the esoteric traditions. The so-called

But to the esoteric traditions it has enormous significance. With the square being divided at the corners diagonally, we discover that it is exactly this process, and this process alone, which allows for the square to have the potential for growth. The diagonal becomes the new base through which it can multiply its size. And here is the significance of this. “It was the incommensurability of the diagonal to the side that was the great secret the Pythagorean initiate was vowed to keep, not, as generally taught, because this fact destroys the rationality of Pythagorean geometry, but precisely because it is a revelation of the vital necessity of the suprarational, infinite element in all finite structures, that the square root contains a power whose nature is, in truth, unutterable in terms of finite measure and that points to the deepest of cosmic mysteries.”[27]

“The two types of numbers, rational and irrational, represented two completely different types of being. The whole numbers were related to manifestation and were the terms to be used in calculation. Every aspect of the phenomenal world was seen to be a fixed instantaneous moment caused by the interaction of complementary components, a moment trapped between light and dark, life and death, day and night, between formation, disintegration and reformation. An arrested formation was epitomized in ancient geometry by the

“When we see a revolving sphere it presents us with the notion of an axis. We think of this axis as an ideal or imaginary line through the sphere. It has no objective existence, yet we cannot help but be convinced of its reality; and to determine anything about the sphere, such as its inclination or its speed of rotation we must refer to this imaginary axis. Number in the enumerative sense corresponds to the measures and movements at the outer surface of the sphere, while the universal aspect of Number is analogous to the immobile, unmanifest, functional principle of its axis.

Lets shift our analogy to the two dimensional plane. If we take a circle and a square and give the value 1 to the diameter of the circle and also to the side of the square, then the diagonal of the square will always be (and this is an invariable law) an ‘incommensurable’, irrational number. It is said that such a number can be carried out to an infinite number of decimal places without ever arriving at a resolution. In the case of the diagonal of a square, this decimal is 1.14142… and is called the square root of 2. With the circle, if we give the diameter the value 1, the circumference will also always be of the incommensurable type, 3.14159… which we call pi.

If we invert the principle remains the same. If we give the fixed, rational value of 1 to the diagonal of the square instead, and to the circumference of the circle, then the side of the square and the radius of the circle will become the incommensurable irrational type: 1/sq.root 2 and 1/pi.

This is a relationship. A formal relationship and these are absolutely knowable, self-evident realities. They demonstrate graphically a level of experience which is universal and invariable.

The irrational functions are a key opening a door to the higher reality of Number. They demonstrate that Number is above all a relationship...within the concept of Number there is a definite, finite, particularizing power and also a universal synthesizing power, the external aspect of Number (exoteric), and the internal (esoteric) aspect of Number.”[29]

All things have this aspect, since all things are geometric patterns with number proportions, ratios, and relationship. We are finite, yet have the infinite built right within us, as does everything. By doing the geometry, seeing, creating, and thinking about what we are doing, we come to recognize these very interesting facts about ourselves and our world.

Endnotes

1. “The Pythagorean Sourcebook and Library,” compiled and translated by Kenneth Sylvan Gutherie, Edited and Introduced by David Fideler, Phanes Press, 1988: 309.

2. “Pythagorean Sourcebook,” p. 291.

3. Lucretius, “On the Nature of the Universe,” translated by R. E. Latham, Penguin Books, 1951: 28.

4. In G. S. Kirk, J.E. Raven, “The Presocratic Philosophers,” Cambridge University Press, reprint, 1981: 407-408.

5. Kurt Smith, “Matter Matters, Metaphysics and Methodology in the Early Modern Period,” Oxford University Press, 2010: 55.

6. Smith, “Matter Matters,” p. 55.

7. Rudy Rucker, “Infinity and the Mind,” Birkhauser, 1982: 2-3.

8. Paul Davies, “The Cosmic Blueprint,” Simon & Schuster, 1988: 12.

9. Paul Davies, “Other Worlds, Space, Superspace and the Quantum Universe,” Touchstone Books, Simon & Schuster, 1982: 12-13.

10. Paul Davies, “The Cosmic Blueprint,” p. 154.

11. Paul Davies, “The Edge of Infinity,” Simon & Schuster, 1982: 15.

12. Carlo Rovelli, “Reality is not What it Seems, The Road to Quantum Gravity,” Random House, 2016: 65.

13. Robert Lawlor, “Sacred Geometry, Philosophy and Practice,” Thames & Hudson, 1982: 4.

14. Robert Lawlor, “Sacred Geometry,” p. 5.

15. John Michell and Christine Rhone, “Twelve-Tribe Nations, Sacred Number and the Golden Age,” Inner Traditions, 2008.

16. Leonora Leet, “The Secret Doctrine of the Kabbalah,” Chapter 7; Daniel Matt, “Zohar,” Pritzker edition, 12 vols., Stanford University Press, “Hayyei Sarah,” Vol. 2, 2004: 224 (1:129a)’

17. Matt, “Zohar,” “Hayyei Sarah,” vol. 2, p. 225. (1:129a).

18. Matt, “Zohar,” Pritzker edition, Vol. 4, 2007, “Shemot,” p. 2 (2:2a). See on the Pythagorean cosmos, S. K. Heninger, Jr., “Touches of Sweet Harmony, Pythagorean Cosmology and Renaissance Poetics,” Huntington Library, 1974: 156-194.

19. Leonora Leet, “The Secret Doctrine of the Kabbalah,” p. 83.

20. “A Greek-English Lexicon,” Compiled by Henry George Liddell and Robert Scott, Oxford at the Clarendon Press, With a Supplement 1968, Reprinted 1983: p. 240.

21. Robert Lawlor, “Sacred Geometry,” p. 5.

22. Fragments of Philolaus in “The Pythagorean Sourcebook,” p. 168.

23. Leonora Leet, “The Universal Kabbalah,” Inner Traditions, 2004: 24, 25.

24. John Michell, “City of Revelation,” David McKay Co., 1972: 108, 109. See also his fantasmagorical delicious book “How the World Is Made, The Story of Creation According to Sacred Geometry,” Inner Traditions, 2009. Also his mind boggling book “The Dimensions of Paradise, Sacred Geometry, Ancient Science, and the Heavenly Order on Earth,” Inner Traditions, reprint, 2008. Fantastically interesting is how his gematria on so much of this coincides and correlates very well with Thomas Simcox Lea and Frederick Bligh Bond, “Materials for the Study of the Apostolic Gnosis,” (1920), reprint from Kessinger Books. See as well, David Fideler’s “Jesus Christ Sun of God, Ancient Cosmology and Early Christian Symbolism,” Quest Books, 1993. Profitable also is Mark Pawson, “Gematria, the Numbers of Infinity,” Green Magic, 2004.

25. David Fideler, “Introduction,” in “The Pythagorean Sourcebook,” p. 45.

26. Walter Burkert, “Lore and Science in Ancient Pythagoreanism,” Harvard University Press, 1972: 463.

27. Leonora Leet, “The Universal Kabbalah,” p. 5.

28. Robert Lawlor, “Sacred Geometry,” p. 38.

29. Robert Lawlor, “Sacred Geometry,” p. 11, 12.

Change your attitude, and you change your life. This is literally real. And that goes both directions, make no mistake about it, for negative, derogatory life, feelings or failures or for the positive. Change your perspective and change your eternity. This is literally real as well. But change your perspective? From where or what to where or what? What does this even mean? I will explore it a little in this paper.

What you see is what you get. I will believe it when I see it. We use sight as a basis for so very much in life, and quite properly so. It is quite true that our acuity in seeing is a matter of life and death. It is also precisely a matter for attaining to a higher, deeper, and broader level of enlightenment within ourselves.

From the Doxographers of Aetius, Plutarch and Theophrastus we learn that the Pythagoreans “all assume matter is subject to change, [they] assert that genesis and destruction in an absolute sense take place, for from change of the elements, modification and separation of them there takes place, juxtaposition and mixture, and intermingling and melting together.”[1] And in the Timaeus of Locri (ca. 100 C.E.) is taught “Of produced things, the substratum is Matter, while the reason of each shape is abstract form; of these two the offspring is Earth and Water, Air and Fire.”[2] Lucretius (ca. 100 -55 B.C.) described the universe as “raw material, or generative bodies, of ‘seeds’ of things. Or I may call them primary particles, because they come first and everything else is composed of them.”[3] Simplicius described the ancient atomists, Leucippus, Democritus, Epicurius ideas “said that the first principles were infinite in number, and thought they were indivisible atoms and impassible owing to their compactness and without any void in them.”[4]

In this very brief, incomplete survey from antiquity, we see the idea plainly before us that “stuff” is the reality. Plotinus did end up distinguishing the difference “between the individual bodies of the sensible world and the conditions underlying the very possibility of such things so individuated.”[5] Kurt Smith notes that what distinguishes matter from what we think is there is what for Kant “would call the empirical versus the transcendental. Inferior matter is, as Schafer contends, ‘what we

*ex post*identify as the (‘in itself’) structureless ‘fabric’, which underlies material-matter as we know it of the bodily universe. As Plotinus puts it, ‘What is called matter is said to be some sort of ‘substrate’ (*hypokeimenon*) and ‘receptacle’ (*hypodoxein*) of forms (*eidos*)...Both Armstrong and McKenna in their translations of the Enneads render this ‘deeper’, underlying matter the unlimited and the Undelimited (*to apeiron*), respectively.”[6]The Greeks saw the

*apeiron*(infinity) entirely in the negative light since it was “indefinite, or undefined… the original chaos out of which the world was formed...*apeiron*need not only mean infinitely large, but can also mean totally disordered, infinitely complex, subject to no finite determination. In Aristotle’s words,... being infinite is a privation, not a perfection but the absence of a limit…”[7]But today we now know that “stuff” is not the only reality, even though we are all made of it, as is everything else we see and experience in life. Stuff is everywhere! Especially in our garages and basements which we overcrowd and make it difficult to get to. So, if it’s not real, then why is it all over the place getting in our way all the time? If all that crap is not real, then what is?!

Now-a-days it’s hip to talk not of particles as the

*Urstoff*(original stuff), but fields.[8] “Reality, inasmuch as it has any meaning at all, is not a property of the external world on its own but is intimately bound up with our perception of the world - our presence as conscious observers.”[9] On top of this, the universe is now understood to be a “huge self-regulating self-sustaining mechanism [organism is even more accurate] with a capacity to self-organize ad infinitum.”[10] But further now than this, most interesting for this paper, is the idea that we should no longer talk about gravity as a force, but as geometry. “Gravity is geometry.”[11]Yet, and yet… why stop at gravity? Einstein, without question has changed our perceptions, but why stop at such milestones? Is there yet more change in our perceptions that might help us see a deeper truth? Without question! “The world is made up of particles and fields, and nothing else; there is no need to add space as an extra ingredient. Newton’s space is the gravitational field… space is no longer different from matter. It is one of the material components of the world, akin to the electromagnetic field. It is a real entity which undulates, fluctuates, bends and contorts.”[12] And now we are led into geometry.

In biology we learned the DNA works in humans, not only because of how particles and the molecular components are the carriers of continuity in our lives (we change every cell in our bodies every 5-7 years), but the helix shape is responsible for its power. “The helix, which is a special type from the group of regular spirals, results from sets of fixed geometrical proportions… their proportions can be understood to exist a priori, without any material counterpart, as abstract, geometric relationships. The architecture of bodily existence is determined by an invisible, immaterial world of pure form and geometry.”[13]

Remarkably interesting as well is the complex twelve-fold symmetry pattern of photosynthesis of carbon, hydrogen, nitrogen and magnesium and chlorophyll in the molecular body of living organisms. “It appears that the same constituents in any other arrangement cannot transform the radiant energy of light into life substance. In mythological thought, twelve most often occurs as the number of the universal mother of life, and so this twelvefold symbol is precise even to the molecular level… this spatial awareness on a cellular level may be thought of as the innate geometry of life. All our sense organs function in response to the geometrical or proportional differences inherent in the stimuli they receive.”[14]

This twelveness is found throughout the ancient world symbology whether that of Israel’s tribes, or other nations aligning with the cosmos in China, Ireland, Iceland or as far south as Madagascar, or showing up in other symbols such as Britain’s King Arthur’s Round Table, all tied to the zodiac of constellations and alignments here on earth with stone works and temples, altars, pathways, etc.[15] It is also found in the mystical dimensions of the verbal description in the Jewish text

*The Sefer Yetzirah*, correlating to the twelve diagonals of the Cube of Infinite Space as the most sacred geometric diagram which heavenly pattern was imitated on earth in the ancient city of Jerusalem at the center of the world, including that of the temple in the center of Jerusalem, the nave in the center of the temple, the ark of the covenant in the center of the nave, and in front of the ark the Rock of Foundation, from which the world was founded, all cubes, nestled within cubes, in signifying association linking with the cosmological reality of the infinite Cube of Space fittingly.[16] “The blessed Holy One has formed everything so that this world corresponds to the pattern above.”[17] The twelve tribes, according to the “Shemot,” are originally from the Cosmological Tree of Life, which extends out into infinite space.[18]This underlying geometry found in life and the cosmos is also found in arithmetic, that is, number. “We obtain wholes by linearly adding together discrete units, such as 1+1+1. In systemizing number in this way we forget that number continuity really only progresses through a wave-like alternation (odd-even), that there are no discontinuous magnitudes nor discrete parts, that a unit can never exist outside of a contiguous form/flow. That is why the Greek word

*arithmos*denotes the definite and discreet, but also means rhythm and unbroken interrelatedness… Plato’s monadic ontology implies that every number presupposes a definite and discreet unit taken from a limitless, homogenous field. Contemplation of it thus provides access to the contemplation, not only of a limit, but also of the limitless. These extremes are the fundamental tension in Pythagorean thought.”[19] The Liddell-Scott Greek Lexicon clearly shows this interesting aspect of the “discreet” harmoniously included within the “contiguous flow” of numbers in the ancient Greeks usage of the term in its various contexts.[20]It is to the one Pythagorean aspect of the Sacred geometry that becomes so very intriguing however, that of the relation of the diagonal to the sides of a cube that is most stimulating, which I get to just shortly. “Within the human consciousness is the unique ability to perceive the transparency between absolute, permanent relationships, contained in the insubstantial forms of geometric order, and the transitory, changing forms of our actual world. The content of our experience results from an immaterial, abstract, geometric architecture which is composed of harmonic waves of energy, nodes of rationality, melodic forms springing forth from the eternal realm of geometric proportion.”[21] In the fragments of Philolaus he mentions “the world’s nature is a harmonious compound of limited and unlimited elements; similar is the totality of the world in itself, and of all it contains.”[22]

It is exactly this, in fact, with which the idealized relationship exists between the sides of a square to its own diagonal splitting it into two perfect triangles from corner to corner of the square, between both the rational elements of number and the irrational aspects. In Psalms 104:2 we read where God wears a garment of light and stretches out the heavens like a curtain. Now the idea of wearing a garment of light is the limit theme to fit within a delimitation, and yet the stretching out of the heavens reveals not the elements with substance, rather the unlimited statement of process.[23] The process of “drawing” the cosmos with His energy, his power, the ability to produce “stuff” out of the background geometrical proportions. We imitate that creative process when we engage (as the Zohar constantly encourages) in doing the geometry for ourselves in imitation of the Creator.

John Michell has demonstrated throughout numerous delightful books that “All over the world the traditional units of length, area, weight, and capacity are related to each other and derive from one original canon of cosmology… the cosmological origins of the English and megalithic miles… occur naturally in the measurement of all classes of phenomena from the human to the astronomical scale. Their sanctity is not therefore a matter of arbitrary convention or superstition but inherent.”[24]

“Physicists have discovered that the deeper they push into matter, it looks like the cosmos of the Pythagoreans and Platonists. Each atom is a Pythagorean universe, the sight of eternity in a gram of sand, consisting of an arithmetic number of particles, geometrically distributed in space, dancing and vibrating like a miniature solar system to the music of the spheres...matter and energy are but different aspects of one, underlying continuum. Advancing to the subatomic level, quantity becomes quality, energy becomes information...the science of physics, proceeding from matter to energy, from energy to intelligence (i.e., pattern,

*logos*) and from intelligence to Nous...the phenomenal universe is a mixture, a synthesis of limited and unlimited elements.”[25]While it’s true that Walter Burkert demonstrated the concocted story of the Pythagoreans putting men to death for revealing the ancient secret mathematics they discovered, to be so much pixie dust, it’s also interesting that he missed the significant meaning of the diagonal to the square in the esoteric traditions. The so-called

*Grundlagenkrisis*- the dilemma or crossroads - of the Greeks discovering the irrational square root 2 diagonal of a square, while being made sinister by many ancient authors as Burkert amply demonstrates, Burkert says “For the Pythagoreans who were concerned with the number theory described by Aristotle, and for the cosmology summarized in the phrase ‘everything is number,’ the irrational has obviously no importance.”[26]But to the esoteric traditions it has enormous significance. With the square being divided at the corners diagonally, we discover that it is exactly this process, and this process alone, which allows for the square to have the potential for growth. The diagonal becomes the new base through which it can multiply its size. And here is the significance of this. “It was the incommensurability of the diagonal to the side that was the great secret the Pythagorean initiate was vowed to keep, not, as generally taught, because this fact destroys the rationality of Pythagorean geometry, but precisely because it is a revelation of the vital necessity of the suprarational, infinite element in all finite structures, that the square root contains a power whose nature is, in truth, unutterable in terms of finite measure and that points to the deepest of cosmic mysteries.”[27]

“The two types of numbers, rational and irrational, represented two completely different types of being. The whole numbers were related to manifestation and were the terms to be used in calculation. Every aspect of the phenomenal world was seen to be a fixed instantaneous moment caused by the interaction of complementary components, a moment trapped between light and dark, life and death, day and night, between formation, disintegration and reformation. An arrested formation was epitomized in ancient geometry by the

*diaphantine*triangle, which is a right angled triangle with all three sides equal to whole numbers, such as the 3,4,5 triangle. This latter is traditionally called the Sacred Triangle, ‘sacred’ meaning fixed or permanent… on the other hand, the irrational roots symbolize the constant creative process of acting and reacting energy. The immeasurable gestating force emanates from the incomprehensible Unity. That which is comprehensible is no other than a momentary limitation of this One, indefinable Being into a definable moment ‘Necessarily, then, all that is definable arises out of an indefinable All.’”[28]“When we see a revolving sphere it presents us with the notion of an axis. We think of this axis as an ideal or imaginary line through the sphere. It has no objective existence, yet we cannot help but be convinced of its reality; and to determine anything about the sphere, such as its inclination or its speed of rotation we must refer to this imaginary axis. Number in the enumerative sense corresponds to the measures and movements at the outer surface of the sphere, while the universal aspect of Number is analogous to the immobile, unmanifest, functional principle of its axis.

Lets shift our analogy to the two dimensional plane. If we take a circle and a square and give the value 1 to the diameter of the circle and also to the side of the square, then the diagonal of the square will always be (and this is an invariable law) an ‘incommensurable’, irrational number. It is said that such a number can be carried out to an infinite number of decimal places without ever arriving at a resolution. In the case of the diagonal of a square, this decimal is 1.14142… and is called the square root of 2. With the circle, if we give the diameter the value 1, the circumference will also always be of the incommensurable type, 3.14159… which we call pi.

If we invert the principle remains the same. If we give the fixed, rational value of 1 to the diagonal of the square instead, and to the circumference of the circle, then the side of the square and the radius of the circle will become the incommensurable irrational type: 1/sq.root 2 and 1/pi.

This is a relationship. A formal relationship and these are absolutely knowable, self-evident realities. They demonstrate graphically a level of experience which is universal and invariable.

The irrational functions are a key opening a door to the higher reality of Number. They demonstrate that Number is above all a relationship...within the concept of Number there is a definite, finite, particularizing power and also a universal synthesizing power, the external aspect of Number (exoteric), and the internal (esoteric) aspect of Number.”[29]

All things have this aspect, since all things are geometric patterns with number proportions, ratios, and relationship. We are finite, yet have the infinite built right within us, as does everything. By doing the geometry, seeing, creating, and thinking about what we are doing, we come to recognize these very interesting facts about ourselves and our world.

Endnotes

1. “The Pythagorean Sourcebook and Library,” compiled and translated by Kenneth Sylvan Gutherie, Edited and Introduced by David Fideler, Phanes Press, 1988: 309.

2. “Pythagorean Sourcebook,” p. 291.

3. Lucretius, “On the Nature of the Universe,” translated by R. E. Latham, Penguin Books, 1951: 28.

4. In G. S. Kirk, J.E. Raven, “The Presocratic Philosophers,” Cambridge University Press, reprint, 1981: 407-408.

5. Kurt Smith, “Matter Matters, Metaphysics and Methodology in the Early Modern Period,” Oxford University Press, 2010: 55.

6. Smith, “Matter Matters,” p. 55.

7. Rudy Rucker, “Infinity and the Mind,” Birkhauser, 1982: 2-3.

8. Paul Davies, “The Cosmic Blueprint,” Simon & Schuster, 1988: 12.

9. Paul Davies, “Other Worlds, Space, Superspace and the Quantum Universe,” Touchstone Books, Simon & Schuster, 1982: 12-13.

10. Paul Davies, “The Cosmic Blueprint,” p. 154.

11. Paul Davies, “The Edge of Infinity,” Simon & Schuster, 1982: 15.

12. Carlo Rovelli, “Reality is not What it Seems, The Road to Quantum Gravity,” Random House, 2016: 65.

13. Robert Lawlor, “Sacred Geometry, Philosophy and Practice,” Thames & Hudson, 1982: 4.

14. Robert Lawlor, “Sacred Geometry,” p. 5.

15. John Michell and Christine Rhone, “Twelve-Tribe Nations, Sacred Number and the Golden Age,” Inner Traditions, 2008.

16. Leonora Leet, “The Secret Doctrine of the Kabbalah,” Chapter 7; Daniel Matt, “Zohar,” Pritzker edition, 12 vols., Stanford University Press, “Hayyei Sarah,” Vol. 2, 2004: 224 (1:129a)’

17. Matt, “Zohar,” “Hayyei Sarah,” vol. 2, p. 225. (1:129a).

18. Matt, “Zohar,” Pritzker edition, Vol. 4, 2007, “Shemot,” p. 2 (2:2a). See on the Pythagorean cosmos, S. K. Heninger, Jr., “Touches of Sweet Harmony, Pythagorean Cosmology and Renaissance Poetics,” Huntington Library, 1974: 156-194.

19. Leonora Leet, “The Secret Doctrine of the Kabbalah,” p. 83.

20. “A Greek-English Lexicon,” Compiled by Henry George Liddell and Robert Scott, Oxford at the Clarendon Press, With a Supplement 1968, Reprinted 1983: p. 240.

21. Robert Lawlor, “Sacred Geometry,” p. 5.

22. Fragments of Philolaus in “The Pythagorean Sourcebook,” p. 168.

23. Leonora Leet, “The Universal Kabbalah,” Inner Traditions, 2004: 24, 25.

24. John Michell, “City of Revelation,” David McKay Co., 1972: 108, 109. See also his fantasmagorical delicious book “How the World Is Made, The Story of Creation According to Sacred Geometry,” Inner Traditions, 2009. Also his mind boggling book “The Dimensions of Paradise, Sacred Geometry, Ancient Science, and the Heavenly Order on Earth,” Inner Traditions, reprint, 2008. Fantastically interesting is how his gematria on so much of this coincides and correlates very well with Thomas Simcox Lea and Frederick Bligh Bond, “Materials for the Study of the Apostolic Gnosis,” (1920), reprint from Kessinger Books. See as well, David Fideler’s “Jesus Christ Sun of God, Ancient Cosmology and Early Christian Symbolism,” Quest Books, 1993. Profitable also is Mark Pawson, “Gematria, the Numbers of Infinity,” Green Magic, 2004.

25. David Fideler, “Introduction,” in “The Pythagorean Sourcebook,” p. 45.

26. Walter Burkert, “Lore and Science in Ancient Pythagoreanism,” Harvard University Press, 1972: 463.

27. Leonora Leet, “The Universal Kabbalah,” p. 5.

28. Robert Lawlor, “Sacred Geometry,” p. 38.

29. Robert Lawlor, “Sacred Geometry,” p. 11, 12.

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