“Geometry existed before the creation." -Plato SACRED GEOMETRY
Sacred Geometry means different things to different people, depending on their interest in and entree to the subject. Broadly, it includes everything from the beauty of Platonic solids and the Golden Mean spiral and Fibonacci sequences to Pythagorean and Vedic mathematics, alphanumerics, kabbalistic glyphs, Merkabah Mysticism, Bucky Fuller's tetrahedral Synergetics, to artistic subject matter, such as Fractals, Cymatics, Yantras, Mandalas, and magic squares and circles. Even crop circles showcase sacred geometry, regardless of their origin.
To me geometry is "sacred" when it reflects Nature, and our own nature, as experienced externally and internally. There is evidence that geometry and hieroglyphics developed through cymatics, the acoustic resonance between sound and form. Simple sand on a drumhead will produce many classical geometric forms and embryonic letters or numbers. Our DNA forms our bodies in the same way with sound and coherent light (see my Bioholography paper). It is the song of life.
Sacred Geometry is also the archetypal realm of topological dynamics, hyperdimensionality, and metaphysical geometrical templates. "God is always geometrizing", Plato said.
Curiously, the primordial geometrical matrix of the kabbalistic Tree of Life directly mirrors the dynamic geometrical transforms of atomic structure. They found it by looking within themselves. The Tree of Life elegantly models the entire universe in minimal graphic elements - lines, spheres, and colors. This resonance is the basis of modern meditations on the Cube of Space or Throne Chariot of God (Merkabah).
In the early 80's, I had my own unique revelation about this nested geometric figure, based on my previous work in Synergetics, Hermetic qabalah and physics. I have purposely not studied other systems to keep my vision clear and original. Others may have carried their insights and articulations further, but some of mine are unique. I have also not commercialized them. After all, ultimately we must each remain true to our own vision.
Comprehending the physics of the plenum by wrapping our minds around it is one thing, a concept; living from that place experientially is another. Take a long, hard look inside and see what you find. You may find the Universe smiling back at you from that Abyss.
The geometrical paintings and forms presented here are a small portion of my artistic body of work. See my other sites for more psychological, symbolic, and mythopoetic artwork.
The History of SACRED GEOMETRY by Iona Miller
"Whoever reflects on four things, it were better he had never been born: that which is above, that which is below, that which is before, and that which is after". --Talmud, Hagigah 2.1
ABSTRACT: Quantum cosmology attempts to describe how the universe emanated from the void. Kant said space and time are the necessary categories of thought. Einstein taught us to think in terms of an interwoven spacetime. We comprehend our experience in terms of space-time geometry. A dynamical geometry, the psychophysical twelve-dimensional matrix, informs our very being. Consciousness-researcher David Chalmers suggests that perhaps the universe exists in terms of psychophysical laws, and that consciousness may involve both an information state and experiential state.
The maximum number of dimensions allowed to a symmetrically-divided sphere is twelve. The 12-dimensional model appears to offer a psychophysical solution to consciousness, quantum gravity, the origins of life, and the birth of the universe--the traditional subject matter of Qabala. We demonstrate that Kether is in Malkuth--information has both a physical and experiential aspect. The Right and Left Pillars of the Tree correspond to Right (spatial/behavioral) and Left Hemispheric (temporal/psychophysical) functioning.
Rabbi Rahumai said: What is the meaning of the verse (Proverbs 6:23), "And the way of life is the rebuke of admonition"?
This teaches us that when a person accustoms himself to the study of the Mystery of Creation and the Mystery of the Chariot, it is impossible that he not stumble. It is therefore written (Isaiah 3:6), "Let this stumbling be under your hand." This refers to things that a person cannot understand unless they cause him to stumble.
The Torah calls it "the rebuke of admonition," but actually it makes one worthy of "the way of life." One who wishes to be worthy of "the way of life" must therefore endure "the rebuke of admonition." --Kaplan, The Bahir
ORIGINS OF THE GEOMETRICAL MODEL:
An examination of the prehistory of religious ideas begins with the conception of surrounding space. When humans began to walk erect they transcended the typical condition of primates.
It is because of our vertical posture that we organize space into four horizontal directions radiating from an"up/down" central axis. We automatically organize the space around our bodies as extending forward, backward, to right, left, upward and downward.
Thus we orient ourselves to the apparently limitless, unknown and threatening extension. We need a method of orientation because it is impossible to survive long in the vertigo brought on by disorientation. Our experience of space is primarily oriented around a "center" and explains the importance of paradigmatic divisions of our experiential field.
This model of experiential space is projected into mythical, celestial space as the cosmogony. The very theory of celestial models continues and develops the universally disseminated archaic conception that man's acts are only the repetition (imitation) of acts revealed by divine beings. The divine modality is defined by the powers and "transcendence" of space. Pre-cosmogonic chaos can be conceptually ordered, and this is an archetypally divine act. The numinous character of divinity increases by becoming brighter. Light is considered the particular attribute of divinity, Initial Perfection.
In Paleolithic times, familiarity with different modalities of matter gave rise to imaginative activity. The first signs of ancient religious sense came from burial rites. Early inventions, such as primitive tools and domestic skills gave rise to imaginative analogies. Through activites such as sewing, shaping statuettes, and making hunting tools objects came to be laden with symbolism. The imaginary world was created and enriched by intimacy with matter.
This imaginary realm was inadequately grasped in figurative and geometric creations of various prehistoric cultures. This imaginative experience is still accessible to us. There is a continuity to this plane of imaginative activity which permeates throughout human history and spiritual notions. The imaginal activity of the ancients had a mythological dimension. Many of the supernatural figures and mythological events which appear in later religious traditions, were probably discoveries of the Stone Age. For millennia Mother Earth gave birth by herself, through parthenogenesis. Born from the Earth, man returned there when he died.
The development of agrarian cultures ushered in notions of circular time and cosmic cycles. The confrontation between two cosmogonic principles, time and space, meant a new orientation to both inner and outer life. A settled existence organizes the "world" differently from a nomadic life. The seed "dies" and is then reborn in order to multiply. Thus death ensures a new birth. Agriculture demands a different relationship to the seasons and weather--to earth and sky, and this had a deep impact on religious values. The theme is one of periodic renewal.
"For religious creativity was stimulated, not by the empirical phenomenon of agriculture, but by the mystery of birth, death, and rebirth identified in the rhythm of vegetation. In order to be understood, accepted, and mastered, the crises that threaten the harvest (floods, draughts, etc.) will be translated into mythological dramas. These mythologies and the ritual scenarios that depend on them will dominate the religions of the Near East for millennia. The mythical theme of gods who die and return to life is among the most important."
". . .The agrarian cultures develop what may be called a cosmic religion, since religious activity is concentrated around the central mystery: the periodical renewal of the world. Like human existence, the cosmic rhythms are expressed in terms drawn from vegetable life. The mystery of cosmic sacrality is symbolized in the World Tree. The universe is conceived as an organism that must be renewed periodically--in other words, each year. 'Absolute reality,' rejuvination, immortality are accessible to certain privileged persons through the power residing in a certain fruit or in a spring near a tree. The Cosmic Tree is held to be at the center of the world, and it unites the three cosmic regions, for it sends its roots down into the underworld, and its top touches the sky."
"...The Cosmic Tree is the most widespread expression of the axis mundi; but the symbolism of the cosmic axis probably precedes--or is independent of--the agricultural civilizations, since it is found in certain arctic cultures." (Eliade, 1978).
The cosmic axis defines and reiterates the divine energy flow between Sky (Kether) and Earth (Malkuth). It reiterates our ancestral vertical posture on the cosmic level, drawing a polarized line between the celestial and terrestrial. When we are in sacred space, we become that cosmic axis, incarnate. It is a cross-cultural, universal model.
In the Hebrew adaptation of this cosmic Tree model, there are differences and similarities to the older cults of western Asia. Archaic ideas about the creation of the world were taken up and reiterated. Mesopotamian legends formed much of the raw material. However, the main distinction from the agricultural fertility cults was that the Hebrews did not worship the Earth or forces of nature. This represented a break from conventional religious forms and was the new basis for the clan's spiritual life and ethos.
But, they were unavoidably pre-conditioned by the dominant Mesopotamian culture. Living on the outskirts of this society they incorporated notions, such as a Law or code. The very idea of a code is Mesopotamian, and cannot be found in ancient Egypt.
The primary difference in orientation is shown by the fact that the Hebrew Tree of Life reverses the neolithic notion of an earth-rooted sacred Tree. The qabalistic Tree is rooted in Heaven, with its branches extending downward toward earthly manifestation. The emanation is from the formless limitless light into corporeality by means of geometric unfoldment, from pure energy converted into matter. But, the Mesopotamian influence is seen here as well.
Sumeria revered a triad of great gods, analogous to the Supernal Triad (1-2-3) of the Tree of Life; followed by a triad of planetary gods (4-5-6), followed by lesser gods. This cosmic rhythm is sustained in the qabalistic Tree. Like the fertility cults, the cosmic axis is conceived of as a relationship between a primordial couple (Elohim; God and Shekinah, or Malkuth, the Bride; the Right (masc.) and Left (fem.) Pillars of the Tree). Their union is a hieros gamos, or sacred marriage which results in the manifestation of all things.
Qabala also incorporates the themes of circular and cosmic time, by valorizing the cycle of birth, death, and rebirth (initiation). This is the archaic idea of the periodic renewal of the world which resurfaces as ritual, magical exercises and disciplines. It is primarily for the agriculturalist that the "true world" or space in which he lives is the "center of the world." It is consecrated by rituals and prayers, and in that sacred space communication with divinity is effected.
Habitation of a sacred place led to the cosmological symbolism of sacred architecture. The Sumerians were the first to erect monumental temples, and to record myths of the quest for immortality. They invented notions about a 12-fold Zodiac, planetary astrology, magic and divination, spirits and demons, and later (from Babylon) angels. Akkadian religious thought also contributed the importance accorded to the personal element in religious experience. These were incorporated into Qabala in such ideas as the path of return, planetary spheres, spiritual hierarchy and demonology.
All this percolated down from the earliest neolithic cultures with their primal root-metaphors:
Cults of the dead and of fertility, indicated by statuettes of goddesses, and of the storm god. Beliefs and rituals connected with the "mystery" of vegetation.
Assimilation of the identity of woman/cultivated soil/plant, implying the homology birth/rebirth (initiation).
Very probably the hope of a postexistence. A cosmology including the symbolism of a "center of the world" and inhabited space as an imago mundi.
This represents a cosmic cycle of chthonian fertility and life/death/postexistence. These root metaphors are powerful, and have persisted into modern times in our religions. Many of the primal notions were incorporated into the classic literature of written Kabbalah in the Middle Ages, particularly the Sephir Yetzirah, the Bahir, and Zohar, and the codification of the Tree of Life and Cube of Space or Throne-Chariot.
These qabalistic texts exhibit considerable variation. We can see that these spiritual ideas about orientation in space/time have evolved through the centuries. We have every reason to believe that our view will continue to evolve to a new understanding of the meaning of the Universe or cosmic existence (where we are), our existence (who we are), and post-mortum continuation (where we are going).
Throughout most of the history of Kabbalah, the Tree of Life wasn't standardized or very geometrical. The emanations were contemplated in a variety of forms. The geometrically regular triplet array came much later. Path positions and attributions differed markedly, and there are several arrays of the emanations devised for various purposes. Even prior to formalization of the Tree of Life, the visionary experience of the Throne Chariot was pursued as "the Work of the Chariot."
This work, or Merkavah mysticism is the meditative branch of Qabala. But when it came to this central image, or template of divinity, there was much discussion in the evolving esoteric tradition over the pattern of the Chariot of Light. Unlike the Sephiroth which are not spatial, but qualities of Nothingness, the Chariot is a template or spiritual projection--a form and state which arises during mystical meditation. Both its rewards and the extreme dangers of this practice by the impure are covered in the Bahir, The Book of Illumination, published in 1175. This is one of the oldest Kabbalistic texts, and contains the earliest discussion of the Sefirot and reincarnation.
This work emphasizes meditative techniques to allow seers to develop profound astral visions of "God's Throne" by themselves becoming "chariots" or vehicles to the divine. They used the Hebrew equivalent of mantras and mandalas to facilitate their practice. Since the Jews were in the Babylonian exile and their earthly Temple had been destroyed, perhaps it was an attempt to internalize sacred space--to worship in an inner Temple.
The origin of this meditative practice goes back to Ezekiel, and according to Kabbalah, his vision of the Chariot was in the Universe of Yetzirah, the astral realm of Formation. This is the level where it is said God "fills all worlds." We look down into our own soul to see Him. Once again, the opposition to fertility cults is emphasized:
"Specifically characteristic of agriculturalists, cosmic religiosity continued the most elementary dialectic of the sacred, especially the belief that the divine is incarnated, or manifest itself, in cosmic objects and rhythms. Now such a belief was denounced by the adherents of Yahweh as the worst possible idolatry, and this ever since the the Israelites' entrance into Palestine. But never was cosmic religiosity so savagely attacked. The prophets finally succeeded in emptying nature of any divine presence. Whole sectors of the natural world--the "high places," stones, springs, trees, certain crops, certain flowers--will be denounced as unclean because they were polluted by the cult of the Canaanite divinities of fertility. The preeminently clean and holy region is the desert alone, for it is there that Israel remained faithful to its God. The sacred dimension of vegetation and, in general, of the exuberant epiphanies of nature will be rediscovered only late, in medieval Judaism." (Eliade, 1978).
This total and violent rejection of cosmic religiosity and nature symbolism was apparently sublimated in the work of the Chariot, because as we now know, its geometry actually reflects the deepest secrets of nature and the cosmos in terms of the formation of all possible things, from the macrocosmic to microcosmic.
Religiosity graduated from the physical to the metaphysical realm, and became a "way of knowing." Metaphysical knowledge presumably gave man control of himself, instincts and actions, and permitted living a fully worthy life. Events no longer reflect the eternal rhythm of the cosmic cycle or depend on the stars, they develop in accordance with God's plan.
The advent of syncretism in the Hellenistic world brought exposure to Greek ideas. The Greeks tried to impose their way of life on the Jews. Both Arab and Jewish philosophers engaged in scholastic metaphysics which proposed that the universe must have had a beginning in time, hence a Creator God, which implies the unity of creation, and shows that the soul is of God. Truth is one. The acquiring of truth is a religious duty. Reason and revelation are complementary. Still, philosophy alone is not a religious enterprise. The goal of philosophy is right action.
A theology which equates God's law with order and stability in nature still mistakes the creation for the dynamic transcendent Creator, worshiping creation rather than Creator. Philosophy suggested that immortality was an achievement of the soul which has activated its full potential for knowing. This is perhaps the subtle mistake or stumbling of the Daath-level, over-intellectualizing, rather than obediently following the Lord's Will as the Bible dictates. True knowledge of things divine comes through love of God. Only love admits one to God's supernal mysteries.
One of the most original religious creations of the Hellenistic period is the personification of wisdom as the Shekinah. In Greek form, Sophia as a divine and personfied entity appears comparatively late in the Hermetic writings. This feminine counterpart of God plays a major role in the metaphors of polar dynamics in the Tree of Life.
In Proverbs, Shekinah declares that Yahweh created her before the oldest of his works, that she was firmly set from the beginning, before earth came into being. This echoes the notion of a virtual state prior to and underlying physical manifestation, a virtual matrix of formation.. Wisdom emerged from the Lord's mouth. Among the realm of Jewish-inspired "intermediate beings" between man and God, Shekinah was elevated to supreme authority, the mediatrix. She is divine immanence.
Intuitive perception of the subtleties of metaphysical reality has evolved as philosophies have shifted. The more we know about the empirical nature of phenomenal reality, the more our intuitive concepts come into harmonization with the nature of Reality. At the risk of error, "stumbling" and admonishment, we can conceive of a postmodern view of the Tree and the Throne-Chariot, with analogies to current physics theories. But this theory may be more than analogy or metaphor--it is the way things are, and therefore phenomenological. It unites psyche and matter in the alchemical Unus Mundus, or One World.
An evolving esoteric tradition allows us to course-correct symbolic, intuitive notions about the nature of reality. In this pursuit, we are not trying to improve or defile Qabala, but employ our spiritual sensitivity to engage in true speculation, or seeing in the sense of the seer. A medieval Kabbalist said, "the philosophers can only surmise what exists in the metaphysical realm, while the Kabbalists can actually see it."
Three events influenced the evolution of Kabbalah in the Middle Ages: the printing press; discovery of the New World; and the Spanish Inquisition. As Kabbalah began to be written down, there was a need for outlining its organizing principles, systematizing it in a philosophical structure. We can only deduce what is correct from the considerable variations of Qabala by the logical derivation of formulas and true vision, not allegory, or even metaphor. Truth is one.
Our modern model meets religious, psychological, and physical criteria for depicting this ancient core image with even more clarity, making it crystal clear. Theological flexibility is a qabalistic tradition. Theosophical speculation is primarily based on insight rather than systematics. Through it we weave reason and revelation into a seamless unity.
------------------------ Re-Visioning Sacred Geometry [Illustrations can be found at this article's original home in Synergetic Qabala]
A RE-VISIONING OF THE COSMIC TREE
"537.131 Six Vectors for Every Point: The behavioral interpatterning frame of reference of the six degrees of freedom in respect to omnidirectionality is of course the vector equilibrium, which embraces the three-dimensionality of the cube and the six-dimensionality of the vector equilibrium. Experience is inherently omnidirectional; ergo, there is always a minimum of 12 "others" in respect to the nuclear observing self. The 24-positive and 24-negative vectored vector equilibrium demonstrates an initially frequenced, tertahedrally quantized unity of 20; ergo, the Universe as an aggregate of all humanity's apprehended and comprehended experiences, is at minimum a plurality of 24 vectors." (R. Buckminster Fuller, 1979).
All events can be described mathematically in space time. According to Fuller there are six vectorial moves for every event. Each of the vectorial moves is reversible, hence 12. Therefore, all positional differentials in the Universe derive only from the sixness of the 12 degrees of freedom.
The Tree of Life emerges from an unobservable or implicate geometrical matrix. This isotropic vector matrix is dynamically inter-tensioned. The dynamic interaction and tension among the Spheres of the Tree of Life functions synergetically. We can revision the old notion of static hierarchy; the synergetic state could be called "synarchy." Though the geometry of the Tree stays the same, our notion of its dynamics can take a quantum leap forward in conceptual terms. We can keep viewing the Tree in, at best, 19th century terms, or contemporize to 21st century paradigms.
Buckminster Fuller pointed out that ancient philosophical and scientific thought was based on the notion of the cube, rather than the tetrahedron, nature's most economical self-organizing base unit. This mistake of the Greek mathematicians has been perpetuated down through the ages, but we can correctly revision the matrix of reality with a few simple adjustments in our thinking. The Synergetic Qabala includes original graphics and paintings which help make these conceptual changes clearer.
Experiential phenomena, including mystical states, are inseparable from the physical universe, and are deeply connected with the laws which govern the physical universe. Buckminster Fuller advanced a single model to describe the shape of the physical universe, the shape of energy's behavior, the shape of our thinking, and the shape of the metaphysical or philosophical universe. He considered adoption of the cube in classical times as misguided and erroneous, since it has nothing to do with nature's own coordinates.
With the cube and the square the ancient Greek mathematicians entered the world of nature and Reality by the wrong door, rather than through Nature's Way which is in triangles and tetrahedra. Without meaning to be critical, the early Qabalists perpetuated this partial truth, in what we can term their "Cube-ala," with its primary geometries of the Tree of Life and Throne-Chariot or Cube of Space.
"THE CUBE OF SPACE" by Iona Miller, c1982
Three dimensions can be modeled with perpendiculars in the cube. Four dimensions can be modeled with equiangularity in the tetrahedron. What the three axes of the cube do for three dimensions, the four axes of the tetrahedron do for four dimensions. The tetrahedron provides for the convergence and divergence of four centrally-coordinate planes. Fuller says it is erroneous to describe time as a fourth dimension; he says that all dimensions require time. Einstein (and later QM) demonstrated that time emanates from the observer. The tetrahedron is stability incarnate, a nest of principle.
In Qabala, the Spheres themselves are actually numbers, (Sephiroth means number) and in Fuller's geometric philosophy, there is a direct relationship between number and geometry. Fuller created the first explicit formulas for the area of a circle in triangular modules and for the volume of a sphere in tetrahedral modules--all without pi. In contrast to the classical XYZ coordinates, Fuller's three-way great circle grid, has 60-degree coordination and a tetrahedral matrix. This is nature's way, since there are no 90 degree angles in nature, no true perpendiculars.
We can retune or tune-in a clearer picture of the nature of the primal matrix, since we have made conceptual advances over the ancients. The reciprocal operation of the old alchemical notion of "squaring the circle" is to envisioned the old Cube of Space encompassed by its precipitating matrix, from which it emanates. an encircling Sphere. The 12-dimensional model defines not only 'top down' object-space, but also 'bottom up' self-time, with the potential for psyche/consciousness to exist in every particle in the psychophysical universe from the start.
German mathematician Georg F.B. Riemann proposed the hypersphere as a model of the cosmos. The so-called Riemann sphere is the three-dimensional surface of a four-dimensional ball, and presumes the hypersphere is not embedded in any higher-dimensional space. Einstein chose this shape as the hypothetical overall shape of the cosmos when he fomulated his first relativity model. It models a finite space without a problematic boundary.
The null directions at a point (Vector Equilibrium) have the holomorphic structure of a Riemann sphere. This holomorphic structure is implied as lying "behind the scenes" in solutions of Einstein's (vacuum) equations. We can imagine a twelve-dimensional matrix mapped onto a Sphere, representing all possible modes of psychophysical space and time.
When all possible modes of space and time are mapped onto the Riemann sphere, we see an ascending 'ladder' of dimensions (where time manifests) and we also find six complementary pairs of opposites. Though they are superimposed, a 'top-down' view of this 'ladder' is the perspective of our Left Hemisphere consciousness ("the Observer"), while a 'bottom-up' view reflects the Right Hemisphere state of Being.
These superpositions constitute our experience of psychophysical reality. The top-down information state is our control structure, psyche or soul, with its complementary quantum superposition soma, the supply structure. In a sense, life is the experience of dynamic geometrical transformations. The apprehension of the geometric properties of a 'circle,'--the 12 dimensional matrix--both intrinsically and extrinsically, are crucial for human higher-order consciousness. According to Burrow:
"...a sentient being could only sense that his or her universe was informed by a twelve-dimensional matrix, both at the extrinsic'classical' macro- as well as quantum mechaical intrinsic level, once he/she had ascended beyond the 'circle's' half-way point, i.e. as represented specifically by the seventh-dimension. This extrinsic/intrinsic apprehension of being in the pressence of a circle/sphere is best illustrated by Pythagoras' derivation of all numbers from the geometric properties of a circle. 'Mathematical Platonism', too--the notion that numbers are not "real" and that mathematical concepts exist in a timeless, ethereal realm--derives from this same experientially apprehended 'geometric source."
The invisible (or implicate) presence of the self-organizing 12-dimensional template at the quantum level is the virtual substrate of classical structure and vice versa. It yields six pairs of complementary psychophysical dimensions of space and time and space-time. This psychophysical model demonstrates a synthesis between physics and psychology. See "A Psychophysical Theory of Everything: Consciousness Beyond Complementarity," by Barron Burrow. http://www.maximus.dircon.co.uk/ It is a model for transcendence or experience of holistic "no time," and nonlocality in the 11th dimension (Daath). By definition, experiential phenomena must have a qualitative apprehension of, and relationship to, time. Due to the principles of synecdoche and self-similarity, we share the same 'arrow of time' with the psychophysical universe.
'Top down' consciousness of an external object actually alternates with 'bottom up,' behavioral superposition or identification with this same object. The 'magic' of this model is that it makes a synethsis between physics, psychology, and philosophy. It is metapsychological. A true Theory of Everything must be a synthesis of psychology, science and religion: consilience.
---------- SYNERGETIC QABALA: VARIATIONS ON THE COSMIC AXIS, TREE of LIFE, DOUBLE CUBE, & CUBE OF SPACE
"The Tree of Life"
The two competing attractors in this model are the top-down Kether, and bottom-up Malkuth. Qabalistic wisdom tells us that, in fact, "Kether is in Malkuth", so they are complementary superpositions.
"All that is physical is energetic. All that is metaphysical is synergetic." (Fuller, 1976)
Gambit: "An opening move such as that which promotes discussion." The Double Cube is the form of the magician's altar, which can be visualized as superpositioned with the body. This crystalline reflectaphor is based on a multi-faceted variation of the Greek metaphorm, the divine Tetractys, a base-10 pyramid. A variation of The Diamond Body, it is the circuitry of a dimensional teleport system. This is one of the keys to sphere-linking operations.
Combinatoric symbols represent mutually dependent, polarized functions. They reexpress, suppliment, and systematize metaphorically, across many levels. The Double Cube of Tiphareth/Daath is represented by faceted jewels of mutually synergetic, non-polarized functions. This meditational abstraction of the Middle Pillar of Equilibrium combines the silver and gold currents of the left and right pillars.
"The Sun Code of Genetic Programming," c1982 Sun Code of Genetic Substances
The so-called sun-code of genetic programming. The codones (four genetic substances) should be read from the inside-out. The four color-coded substances (G, A, U, C), combine first in 16 ways (4 x 4), then in 64 ways (4 x 16). The magic number 64 immediately reminds us of the 64 Hexagrams of the I Ching, the Chinese synergetic book of life. In this painting, the Sun-Code occupies the place of Tiphareth, surrounded by its satellite Spheres of the Tree of Life. The surrounding DNA chain in the shape of the World Egg, is a variation of the alchemical tail-eating serpent Ourobouros. Its head is formed by the Hebrew Yod, a symbol of life and sperm, the unbroken circle of life.
The Diamond Body: Buckminster Fuller and The Qabala by Iona Miller, c1992
"IO [I/O] is the cry of the lower as OI [O/I] of the higher. In figures they are 1001, in Joy. For when all is equilibrated, when all is beheld without all, there is joy, joy, joy that is but one facet of a diamond, every other facet whereof is more joyful than joy itself." -- Aleister Crowley, The Dragon-Flies
SACRED GEOMETRY:According to most creation stories, out of primal Nothingness, the All or Everything emerges or emanates. Paradoxically, everything seems to come from nothing. How does nothing become something? Energy "crystallizes" into matter in the womb of empty space, a dynamic Void. Mass is simply a form of energy. This process is structured by an underlying, invisible, geometrical lattice. Actually, it is pre-geometric. Because it has no true physical existence, it is metaphysical (beyond physics). This threshold of matter, where nothing becomes something, is of great philosophical interest.
Actually, materialism (a natural philosophy) is a theory of metaphysics. It is metaphysical thinking to consider static matter as a primary reality. In fact, any attempt to describe reality is metaphysical speculation. In its dynamic form matter cannot be separated from energy. Energy is a property of matter, which can be considered potential energy. The mystic believes in matter, but believes it is more than science has yet discovered. Even before Western science began, mystics believed that mind, consciousness, or spirit is a property of matter. It hardly matters, philosophically, if you consider it as manifesting force or manifesting spirit.
The nature of reality is that matter-energy must be taken together .The theory of relativity conceives of this single substance as a distortion of the structure of space. Physicist Ian Barbour writes that, "...in quantum theory, separate particles seem to be temporary and partial manifestations of a shifting pattern of waves that combine at one point, dissolve again, and recombine elsewhere; a particle begins to look like a local outcropping of a continuous substratum of vibrational energy." That vibrational energy is governed by the laws of probability.
But what subtle forces underlie matter-energy and space-time? All form and power are latent within the void. The Heart Sutra tells us that, "Form is not other than Void, Void is not other than Form." This implies that our human form is not other than void, and biophysics shows this to be true. Our physical makeup is largely emptiness. If we conceive of humans as being most fundamentally electromagnetic entities, instead of chemical beings, we can imagine our finer existence as wave-fronts in space. Our personal "space" is not utterly empty, but cannot be conceived apart from our matter exhibiting itself in particular ways, i.e. as "waves."
Yet, the void state, or primal matrix, is "cosmic zero," and proportionately our most fundamental reality. It is part of the surrealistic quantum realm. It lies within us all, for the relative space between our atoms is astronomical. This is the ground state of existence which mystics seek in their meditation, moving beyond mind and maya. It is that state of consciousness where outer perceptions cease, and consciousness is free to simply be.
Throughout the centuries, various geometrical forms have been revered as expressions or metaphors of higher spiritual truths. These sacred forms and symbols are a natural part of the collective consciousness which emerges in every generation. We project them outwardly from within our psyche because they are so fundamental to our existence. That apprehension is intuitive. Certain typical forms recur in meditation and ceremonial practice, worldwide.
When something emerges from nothing, it does so via non-Euclidean geometry, coming to occupy space/time. Einstein used non-Euclidean geometry to explain the relativity of time and space as the geometry that is produced by matter or matter by geometry. The perception of the transcendental or metaphysical aspects of geometry is intuitive. There are examples of philosophical geometry or geometrical philosophy from around the world. These traditions are found in India, China, Egypt, and Great Britain, to name a few.
Plato, Archimedes, and the Pythagoreans based much of their philosophical speculation around the nature of geometrical form, suggesting that mathematics and structural forms had ultimate status. Our modern science has never forsaken the tradition of seeking the understanding of forms that provide shape and meaning to physical reality. Euclidean geometry describes the nature of the human scale, but non-Euclidean models the cosmos and microcosm. More and more intricate forms of measurement became the basis of the scientific method. Eventually, this led to modern topology -- the study of those properties of geometric figures or solid bodies that remain invariant under certain transformations.
Heisenberg explained that, "The elementary particles of modern physics can be transformed into each other exactly as in the philosophy of Plato." In "sacred topology", the relationships are more than metaphorical. Metaphysical and physical reality coincide. This is abundantly illustrated in R. Buckminster Fuller's geometrical tour de force, SYNERGETICS I & II. Fuller demonstrates, via synergetics, that the whole is greater than the sum of its parts, because of the relativity of forces. In our universe, as more complex systems are built up, new properties appear that were not foreshadowed by the parts alone. These emergent properties include life, conscious awareness, and beauty.
Plato's Academy in Athens had a policy: "You are not allowed to enter here, unless you know geometry." In the dialogue, Meno, Plato describes Socrates teaching geometry to a slave. In true Socratic form, he does not instruct him directly. Rather, he elicits knowledge from the slave which he did not know he possessed. The diagrams themselves elicit the buried intuitive knowledge of a world inhabited by the gods and by the divine "Forms."
In Plato's view, before birth we were familiar with purely mathematical "objects" and relations, as well as moral ideals or values. He alleged that we lost this knowledge at birth, but can recover it through revelation. For Plato, the virtues paled in comparison with the highest capacity of the soul, which is the mystical beholding of the eternal. Mystics tell us the divine is formless.
Actual space is not empty, but the possibility has always been a controversial problem in philosophy. Aristotle found the concept of a total void unacceptable. Scientists still cannot make a perfect experimental vacuum. Astronomy shows us that matter is averagely distributed, with roughly as much matter between the stars as there is within them. Yet, there are large voids in space, with stars clustered on the outer edges of these bubbles of void. The preponderance is of volumes of empty space over volumes of matter. So, the characteristic feature of the universe is not matter, but empty space.
Again, most of the interior of the atom is empty, as is the space occupied by the galaxies, and the vast regions which separate them. Stars, atoms, and the vacuum are a seamless whole. The notion of the eternal nature of sacred emptiness is echoed in modern quantum theory. This is not ordinary, but dynamic and receptive emptiness. The central concept in Quantum Field Theory is that of the field, which exists everywhere and everywhen. It is a field of curved space/time. Matter is not separate from its surrounding space. The field can take the form of quanta or particles. It is the fundamental physical entity, the only physical reality.
The field contains the potentiality of all possible states or conditions in the universe. Of these states, the most fundamental is the Ground State, (or "vacuum state"). Matter is rare in the universe. Most of it consists of a very volatile, excited, ionic plasma. Only 5% of matter is neither too hot, nor too dilute, to congeal as a solid, liquid, or gas. This form of matter is so rare it has been described as 'trace contaminants.'
Yet, some form of matter is essential to all activity. All matter is in motion. Activity is the essence of being. From the "cosmic zero", everything -- the totality of "excited states" -- arises by creative processes. It is also the state into which everything subsides by absorption processes. The ground state is characterized by the fact that it stretches to infinity, uniform and changeless. It is the same everywhere and everywhen because of the identity of space/time. Within the universal field the values of the ground state and excited states are all one. An unlimited amount of particles come into being and vanish endlessly. Matter is thus a temporary manifestation of the Void.
The so-called "cosmic zero" was modeled by geometer/philosopher Fuller in the figure he called Vector Equilibrium, (V.E.). It might actually claim to be the first "Buckyball," the one Fuller himself described. This geometry is the precursor of the new elemental' Fullerenes.' The V.E. geometry has been recognized for a long time. It was one of the thirteen Archimedian solids, the cuboctahedron.
But the Greeks were fascinated with regular forms whose faces were all the same, such as the cube. They failed to understand the delicate balancing act the cuboctahedron symbolizes. They simply did not notice that Vector Equilibrium is pristine equanimity because they were looking elsewhere. The Greeks never really comprehended the energic or energetic properties of Vector Equilibrium, i.e. dynamic stability. They had a static, non-relativistic view of natural philosophy.
It remained for Fuller to assert that, "zero pulsation in the Vector Equilibrium is the nearest approach we will ever know to eternity and god." The conceptual model is the closest our minds and senses can come to that cosmic realization, short of mystical revelation. Mind alone cannot fathom the depth of this Void. Geometry is a construct of the intellect.
The V.E. center is primal "emptiness." It is a mathematical anomaly where the normal laws of the space/time continuum break down. It is not a symbol of ultimate order. It looks like a very rational, orderly system, but it is ultimately irrational. It defies logic. V.E. is the breeder of wave-particle duality, the uncertainty principle, and non-locality at the quantum level. It is a zone of neutral resonance where waves can pass through waves without interference, according to Fuller. Yet, it never physically exists as a structure, since nature abhors a vacuum.
In quantum mechanics a system can never have an energy of exactly zero. There is no such thing as absolute emptiness. However, the minimal motion of the ground state is called zero point energy, or zero point motion. Cosmic zero exists, paradoxically, in the realm of the psyche (our conceptualization) and in quantum reality in the atomic nucleus expressed as force. It is the form of formlessness, the root metaphor. It also forms the roots of the cosmic Tree of Life, since there is a fine-to-non-existent line between organic and inorganic matter on the quantum scale.
Physics is the patterns of organic energy, all of which are dynamic, alive. Mass is energy, so the subatomic world is always restlessly in motion. Inert matter is full of motion when we look closely at it . The activity of matter is its essence.
Vector Equilibrium emanates/condenses from a center in twelve fundamental directions. This emergent energy moves outward through space/time in the form of a cuboctahedron, alternating with its mathematical reciprocal, an octahedron-within-a-cube. It is a truncated cube with 50 symmetrically positioned topological features. Vector Equilibrium has the same surface area as a sphere, yet contains no volume, i.e. it contains "nothing."
The Vector Equilibrium system has 12 vertices, 8 triangular faces, 24 edges, and 32 planes. It is omnidirectional equilibrium, symbolically and physically speaking. As such, it is a perfect symbol for "holding the tension of the opposites," or "uniting the opposites." Yet, it is more than a metaphor. It is an archetypal image which bridges the macrocosm with the microcosm. It is a living example of the Hermetic Axiom, "As Above, So Below," uniting spiritual and literal reality. V.E. makes it possible to make conceptual models of other dimensions (hyperspace), mathematically and mystically.
In the V.E. figure, equilibrium between positive and negative is zero. It is the equalization of the forces of push/pull, radiation/gravitation, or tension/compression. Fuller alleged it" represents the limits of the mind's ability to conceptualize 'in'." For Fuller, all of space/time is undergird with a pre-geometric matrix, which is an infinite field of vector equilibria. An entire universe can be seeded from one V.E., self-generating to fill all space/time. There may be no ultimate physical building-block of matter, but there is one single entity that undergirds and composes everything in the universe, according to Fuller. The basic element of the universe is dynamic patterns.
This field constitutes a "cosmic blueprint" which Fuller called the Isotropic Vector Matrix (I.V.M.), a living continuum. The allocation of divinity to this "mother of all fields" marks its archetypal character. As a variation on the theme of Celestial Goddess, it symbolizes the embodiment of nature. Like a modern Isis, it iterates the theme of the underlying, inseparable cosmic web or net, connecting all.
The living Void gives birth to all phenomenal forms. It pulsates with the rhythm of creation and destruction of material particles. Another goddess, Nuit, is the essence of Infinite Space. She is infinite energy density pervading the entire cosmos. She is the receptivity of the void to the wavelength of radiation. Adjusting the geometry of the void influences the propagation of radiation. The goddess Ma-at, or Balance, is another expression of the same universal V.E. energy. The Upanishads identifies Brahman with the void:
Brahman is life. Brahman is joy. Brahman is the Void. Joy, verily, that is the same as the Void. The Void, verily, that is the same as joy.
The geometry of Vector Equilibrium is inferred from that of closest-packed spheres of equal radius. It just happens to be the geometry which underlies all matter since it is found in the nucleus of all atoms as sub-atomic force. Here, in the interior of the atoms, Newtonian physics does not apply. This is a probabilistic, acausal world. Here synchronicity prevails over chronicity.
This uniform geometrical field, with the property of Divine consciousness, is the basis for a geometrical model of reality spanning the abyss between the metaphysical and the physical. It is not the first model in history to attempt to do so. There is an ancient geometrical model which shares a common framework with the Vector Equilibrium.
THE GEOMETRY OF THE QABALA
The ancient mystical system of the Qabala is formulated around the geometrical glyph known as the Tree of Life. This sacred geometry system came down to us through Judaism and Hermetic Philosophy.
It is one of the main currents of thought in the Western Occult Tradition. A mathematically accurate image of the Tree of Life can be constructed by dividing a vertical line into four equal lengths and filling in four intersecting circles, using a fourth of the line as radius. The nexus points are the positions of the 10 spheres, and connecting paths join the centers of the spheres.
GEOMETRIC CONSTRUCTION OF THE TREE OF LIFE
1. Begin by imagining a vertical line, A-Z 2. Divide the line into four equal segments. 3. Using the segment length for a radius, inscribe four interpenetrating circles whose centers lie along the vertical line. These represent the Four Worlds, or Planes of existence. 4. The points-of-intersection are related to the numbers of Spheres on the Tree of Life as diagramed 1 - 10. 5. With vertical lines (Red) join 3-5-8 for Left-Handed Pillar of Severity; join points 2-4-7 for the Pillar of Mercy. Line 1-6-9-10 forms THE MIDDLE PILLAR. 6. The blue lines mark the paths between the Spheres. The spheres represent states of consciousness or ways of being, while the paths are ways of transition or change. It symbolizes all ways of being and becoming. Like the quantum field, it contains the potentiality of all possible states or conditions in the universe. It is just the ancient way of saying it, and it turns out to be physically provable. The ancient model coincides with state-of-the-art physics. The 10 spheres and 22 paths yield a total of 32 planes of reference, like the Vector Equilibrium.
This "tree" was first described in writing in the Sephir Yetzirah, or The Book of Formation. Legend says the book came to Abraham, from God, through an angel. Today, we might say it was "channeled" from a divine source. Jewish scholar, Gershom Scholem dates the workaround the third to sixth century. However, there is some astronomical evidence that the system does date from the time of Abraham, approximately 4,000 years ago.
Whether the patriarch was the first to receive the revelation, or not, this archetypal consciousness map encodes a system of spiritual wisdom and growth in geometrical form. It describes the nature of the universe, consciousness, and the creation. It seems to have intuitively anticipated the discovery of the physical nature of reality. The glyph itself was used as a meditation devise by students of the Qabala, an oral wisdom tradition. One aim of the qabalists was to see the Tree always, everywhere, in everything. Another essential doctrine of Qabala is that humans possess a "spirit body" that can detach itself from the physical body and ascend to higher planes.
Most modern students of Qabala are familiar with the standard 2-dimensional representation of the Tree, which is drawn "flat," even though the circles are always conceived of as spheres. There is also a long tradition of a so-called 4-dimensional Tree of Life diagram, based on The Book of Formation. This geometrical figure consists of two interfacing pyramids surrounding a central core, within an enclosing cube.
This octahedron-within-a-cube happens to be the crystal structure of diamond, a face-centered-cubic lattice. Recalling that Abraham is also the patriarch of the Moslems, it is interesting to note reference to a spiritual body, a "diamond body", in the writing of Shaikh Amad Ahsa'i from the 13th century. Speaking of a substance akin to that of the medieval Philosopher's Stone, he equated the "diamond body" with the "Resurrection Body" of the faithful believer in the Paradise of the future Aeon.
For the Qabalists, meditation on this figure provided a mystical body as a vehicle for consciousness in imaginal space. This diamond body was used during meditation to enter the inner court of Divinity, symbolized by the center of the figure. Here the Beginning and End of time are co-temporaneous, space ceases to be a hindrance, and the mystic beheld the ecstatic vision of the Lord on His Throne. The geometry was a "gateway" to another dimension of experience.
This Alpha-Omega point has all the physical qualities of VectorEquilibrium. The ancient meditation practice was known as Merkabah mysticism, and those who practiced it called themselves, "Riders in the Chariot." They claimed to see into the future. This insinuates that the figure is a "vehicle" for moving consciousness through imaginal time and space. It seems to be a metaphysical "time machine," or inter-dimensional gateway.
As in most design or mandala meditations, the aspirant pictures him or herself at the center of the figure through visualization, contemplation, and identification. Through this process of centering, these mystics came to know themselves and God. Perhaps they learned, as Fuller later discovered, of the unusual anomalous conditions of perfect equilibrium. This meditation provided a specific technique for escaping from time. Its realization meant instantaneous enlightenment, a paradoxical leap out of Time. It yields transcendence.
Occultist, Paul Foster Case published an array of this type in his book, The Tarot (1947). Aware of the ancient mystic practice, he included the figure in his study of Qabala, calling it alternately the Key of the Cosmos and Numbers, or the Cube of Space. Again, we have the form of the octahedron-within- a-cube. Vector Equilibrium and the Cube of Space are mathematical duals, or reciprocals of one another. They "jitterbug" back and forth as the figure grows outward to fill all space. They are two ways of looking at the same geometrical phenomena.
Case pointed out that its construction was based on the six-pointed Star of David, (or Shield of David), also known as "The Star of the Macrocosm." His descriptors of zero as a sacred, empty, consciousness field reiterate the qualities of Vector Equilibrium.
Zero is a symbol of the absence of quality, quantity, or mass. Thus it denotes absolute freedom from every limitation whatever. It is a sign of the infinite and eternal Conscious Energy, itself No-Thing, though manifest in everything. It is That which was, is, and shall be forever; but it is nothing we can name. Boundless infinitely potential, living light, it is the rootless root of all things, of all activities, of all modes of consciousness. In it are included all imaginable and unimaginable possibilities, but it transcends them all. The Qabalists call it: (a) No-Thing; (b) The Boundless; (c) Limitless Light. Pure Conscious Energy, above and beyond thought, to us it is Superconsciousness.
With their fascination for completely regular figures, the Greeks devised a way of establishing XYZ coordinates within a cube. Our science and method of orientation has been stuck with that model ever since. However, nature's own most economical coordination is in triangles and tetrahedra, rather than squares or cubes. VectorEquilibrium is more in line with modern Quantum Mechanical models than the Newtonian XYZ. It expresses more degrees of freedom.
So, close examination of the underlying geometry of the Tree of Life reveals that the ancients were not lacking a very deep intuitive awareness of the true structure of matter and the universe. The underlying geometrical matrix of the multi-dimensional Tree is, in fact, Vector Equilibrium. To find the commonality -- the harmony --the figure of the V.E. must be subjected to a transformation process, i.e. rotation.
To bring the two figures into synch, simply rotate the V.E. to any edge of its containing cube. If the 3-dimensional V.E. is drawn flat, the inherent geometry of the Tree of Life is immediately revealed, including some of the so-called "hidden" paths.
A symbolic representation of the optimal union of opposites, it means in psychological terms, "holding the tension of the opposites." This is the essence of the Tree of Life. The theme of union of opposites has reverberated down through the ages. It was the process/goal of alchemy, creation of the Philosopher's Stone.
Now, Jungian psychology carries the torch of this philosophical pursuit. Both Vector Equilibrium and the Tree of Life express this most elegantly and economically in terms of minimal graphic elements.
Diagram of Tree of Life nested within a Vector Equilibrium
By combining the vectors of a V.E. nested within a cube, it is possible to derive the nexus points on which to construct the entire Tree of Life. In The Anatomy of the Body of God, (1925), Frater Achad, (magickal name of Charles Stansfield Jones), revealed an amplified Tree of Life projecting from a center in six directions.
Rather than a flat depiction, his revelation showed Trees radiating in the four cardinal directions, plus up and down. This is another variation on the theme of XYZ coordinates. Yet, V.E. can be shown, once again, to be the skeletal matrix on which the figure congeals. The geometrical duals yield all the necessary nexus points to form the entire Tree of Life, including the 11th mystery sphere, Daath.
Achad's advice for modern Qabalists says, ...since the "Tree" is everywhere the same in every part of space, once its general attributions are fixed in the mind, it is not well to confuse ourselves by too much attempt at progressed expansion of the idea. Rather we would return and contemplate the Centre from which All proceeds, thus obtaining the Pure Essence Here and Now.
This concept of a self-generating, self-iterating, all-space filling, crystalline system seems to herald the modern discovery of the fractal nature of the universe. Whether we look at the macrocosmic, mesocosmic, or microcosmic level, we find the same principle operating. By following any portion of the consciousness map "backwards" and "downwards" to the center, we have a conceptual means of entering that most pristine state of consciousness where All becomes No-Thing.
GEOMETRICAL IN-SIGHT: These systems are virtually pre-geometrical. They don't really exist in nature, per se. Yet they are the invisible lattice, or bare bones, of our physical and spiritual life, of all life and manifestations. In this pre-geometry we have a blueprint for the formation of all matter, all form. This is the geometry of closest-packed spheres in the nucleus of the atom.
Yet, Vector Equilibrium is more than an expression of nuclear forces. It is "an endlessly interlinked chain of atomically self-renewing links of omni-equal strength or of an omni-directionally interlinked chain matrix of ever renewed atomic links, "according to Fuller. No single configuration of matter persists indefinitely.
All is change and recombination. V.E. describes the most economical lines of movement within the atomic nucleus. And, it also is the structural matrix of a very ancient and sophisticated consciousness-changing technology. Through it we remember deep knowledge of our true nature. Noting the metaphysical aspect of physics, physicist Wolfgang Pauli said, "We should now proceed to find a neutral, or unitarian, language in which every concept we use is applicable as well to the unconscious as to matter, in order to overcome this wrong view that the unconscious psyche and matter are two things."
We have, in the V.E. model, a language or information transfer system that bridges both the physical and the spiritual, Fuller says, "In this model the physical and metaphysical share the same design." The mundane and supernatural share the same design: "As Above, So Below." It creates a mystical revelation described as ecstatic, blissful, joyful, transcendental.
Goethe's Faust opens with the mage contemplating the qabalistic, geometric design of the Macrocosm:
What jubilation bursts out of this sight Into my senses--now I feel it flowing, Youthful, a sacred fountain of delight, Through every nerve, my veins are glowing. Was it a god that made these symbols be That sooth my feverish unrest, Filling with joy my anxious breast, And with mysterious potency Make nature's hidden powers around me, manifest? Am I a god? Light grows this page-- In these pure lines my eye can see Creative nature spread in front of me. But now I grasp the meaning of the sage: "The realm of spirits is not far away; Your mind is closed, your heart is dead. Rise student, bathe without dismay In heaven's dawn your mortal head." (He contemplates the symbol.) All weaves itself into the whole, Each living in the other's soul. How heaven's powers climb up and descend. Passing the golden pails from hand to hand! Bliss-scented, they are winging Through sky and earth--their singing Is ringing through the world.
In Aion, psychologist Carl Jung reiterates the identity of psyche and matter: Psyche cannot be totally different from matter, for how otherwise could it move matter? And matter cannot be alien to psyche, for how else could matter produce psyche? Psyche and matter exist in the same world, and each partakes of the other, otherwise any reciprocal action would be impossible.
To know ourselves is to know the nature of cosmos. The yogis and masters tell us that matter is mind or consciousness at its most fundamental or gross level. From the dawn of history mankind has employed sacred geometries for metaphysical orientation and creating consciousness maps.
Most maps of the psyche imply a "journey" either to the heights or depths of experience. Long ago Heraclitus alleged, and mystic artist William Blake agreed, that the way up and the way down are one and the same. Riders in the Chariot rose to a plane above the Throne to gaze down on its resplendent beauty.
Metaphysical models usually speak of an ascent to utopian heights or "inner planes." Psychological models are usually concerned with descents into the subconscious depths and use subterranean imagery. These depths were the familiar territory of shamans for millennia. Both the occult and transpersonal psychology models include heights and depths of human experience. Maslow introduced the concept of "peak experiences" to psychology.
We may have the erroneous idea that peaks, or heights, or "highs" are "good." This is mainly because we are enculturated to strive upward. This is only one of a myriad of states of consciousness, and it belongs to manically over-achieving heroic ego. Negatively directed, this same energy can produce an all-time high in a sociopath or criminally insane personality, during a heinous act. All that is "up" is not "good"; and, all that is "down" is not "evil." This misapprehension is the root of the notion of a spirit/matter duality, where all matter is inherently "evil". In this belief system, "we exist: therefore we are sinful." Yet, our heights and depths contain the first-hand experiential knowledge that 'we are one.'
This old system of orientation and modeling the universe, with its ascents and descents of the soul, was adequate for many years. However, for the modern individual, with a consciousness that is not earth-bound, the old hierarchical model is no longer the best metaphor. Today we are not confined to the spectrum of reality dictated solely by our finite senses.
We are directly aware of physical realities ranging from the sub-atomic to the cosmic. Our perceptions are amplified with technology. New ways of seeing lead to new philosophical perspectives. We can align our mystical worldview with physical reality. The true nature of physical reality remains a mystery. Even for physicists, it has an elusive quality. No one really knows just what a quantum, a photon, or an atom "really is" or what it is doing when we're not looking at it.
Physics is another interpretation of reality. We have become aware of a vast physical dimension without and an equally vast realm of psyche within. They are available for exploration. And there are ancient and modern technologies which aid us here. In terms of modern orientation and consciousness models, an omnidirectional system allows more degrees of freedom. This freedom is a conceptual, imaginal, and physical reality.
Buckminster Fuller said that in whole systems the directions are always out and in. "In and out are the relevant directions, not up and down." Models or maps based on "in and out", rather than "up and down", allow us to conceive of particular states or conditions. As a metaphor, "in and out" is a typical characteristic of the phase states of matter. It is either in or out of phase or synchronization. During observation, matter is either in solid physical existence (particle) or out of solid physical existence (wave).
This wave-particle duality exists within us all. This uncertainty surrounding substantive existence leads to paradox. We cannot grasp one part of nature without another part slipping through our fingers. Photons, neutrons, and even atoms have no definite form until they are measured. Wave-particle duality is a mystery. However, this primal characteristic of atoms, and the energy exchange in their cores, is the basis of our physical existence. The emergence of stable matter requires the balancing of tendencies toward implosion and explosion. This is the function of Vector Equilibrium.
Physicist David Bohm modeled an "in and out" universe of implicate (in) and explicate (out) order. Yogis focus on the in and out breath during pranayama, and the experience connects them with cosmic time. The psychological tendencies, introversion and extroversion, express the polarity in ways of being. In cybernetics there is input (I) and output (O), the direction of the signal determining if it is I/O or O/I. Vector Equilibrium is the most economical model using geometry to express the ultimate union of the opposites. It creates the paradoxical state that is neither in nor out, up nor down, neither this nor that.
For Fuller, the balancing of the tension of the opposites was achieved in his tensegrity sphere. It is stabilized dynamically to neither explode nor collapse. This original "Buckyball" is a tangible example of Vector Equilibrium. According to Fuller, "matter" itself is a contained explosion, and the Vector Equilibrium is its austere image. It is a system not a structure. It underlies structure or formation of something from apparent "nothingness."
Vector Equilibrium is the lattice or invisible framework, and its blueprint is the cuboctahedron. Its explicit form can enfold on itself until it compacts down into an octahedron, then a tetrahedron, the prime geometrical form. This is accomplished by doubling up on the vectors on the edges. This creates the difference, for example, between carbon, diamond, and silicon atomic structures. They are all face-centered cubic crystals. Coincidentally these elements are symbolic icons in our culture. They are the hallmarks of technology.
A simple way to visualize the Vector Equilibrium is through the closest packing of spheres which underlies the geometry of crystal formation. Imagine a cluster of ping pong balls glued together, 12 around 1, then building out further and further. If you imagine vectors connecting the centers of each ball, IN THE THIRD LAYER you can find the Vector Equilibrium.
The vectors that radiate out from the center are exactly equal to the vectors that bond the faces. No other structure can make this claim. This is the symmetry the Greeks missed, because it is an energetic process, manifesting force, not a thing. Vectors are not points in space traveling through time. Vectors map energy events. Time determines a vector's length -- the time it takes an energy event to happen. There must be time for wave functions to propagate. We know a musical note requires time to exist, because a note is nothing in an instant.
Time is a duration. If we ask ourselves "a duration of what?", we can only answer "...of nothing, or something." All "somethings" are composed of matter, and the nature of matter is consciousness. So time becomes a duration of consciousness. Energy is substance. Substance is energy at the quantum level. Our substance is energy, and consciousness is fully capable of quantum leaps.
Fuller's energy mapping uses energetic triangles, in which three lines are not just lying there but are busy stabilizing the angles opposite them. Fuller describes the dynamic domain of "reality" as a broad spectrum of energy events, across a small portion of which our senses can "tune." Vector Equilibrium allows us to conceptually, metaphorically, and spiritually bridge the abyss between the mystical and scientific perspectives through sacred geometry. It is a key to the implicate and explicate order.
THE DIAMOND BODY
THE DIAMOND BODY (c) Iona Miller, 1981 THE DIAMOND BODY is a contemporary meditation technique from secret practices with over 4,000 years of mystical tradition. The oldest-known practice of this technique of meditation was known as Merkabah Mysticism, allegedly developed by the Jewish Patriarch, Abraham. It is a regeneration meditation for immortality.
THE DIAMOND BODY is examined from perspectives in Jungian psychology, from definitions in physics and Jewish mysticism. This is then synthesized into the framework of Magick. There is an inherent spiritual aspect of matter, found in the nature of order and synchronous events. This spiritual aspect is explored through the philosophical concepts known as the Philosopher's Stone A geometrical image is developed from solid state physics, corresponding to both the Jewish mystical traditions and current concepts in Imaginal psychology. A mathematical model is generated to assist the development of anima consciousness via a visualization exercise of the diamond body. A resonance occurs between specific internal-state functions and the "outside." These resonance relationships are discussed and specific models are given to aid visualization and resonance.
The use of these geometrical forms as Magickal tools offers intriguing possibilities. They allow experimental interaction with archetypes of God-forms. They also grant access to those specific altered states where one experiences the eternal aspect of sacred time: A consciousness of immortality.
THE DIAMOND BODY as an exercise is a meditation form of the future. This practice of meditation develops a common ground, uniting psyche with matter. Through the stabilization and equilibrium fostered in the aspirant during mediation sessions, the individual is brought into increased harmony with the environment. More important, a "repair function" is now available, one able to reverse those entropic functions related to libido.
THE DIAMOND BODY is a breakthrough in meditation techniques, something for the 21-Century. Steeped in mystery, the Merkabah Chariot of Sepher-Yetzirah, the BOOK OF FORMATION is also Ezekiel's vision: the Cube of Space. For the first time, a contemporary description of THE DIAMOND BODY is formulated using synergetics, cognitive sciences, phsycis, and metaphysics. Physical and psychological anologs are offered to reveal the timeless experience available through visualizing the true Philosopher's Stone.
A. Vector Equilibrium Lattice Constants. B. Transliteration: Hebrew to English, (with numerical values). C. Graphic Depictions of the Merkabah, or Throne Chariot of God. D. The Isotropic Vector Matrix. BIBLIOGRAPHY
THE SUMERIAN LEGACY
The Sumerian legacy is an integral aspect of Synchrographics. Historically, the first synchrograph could be considered their division of the sky into the 12 divisions of the Zodiac.
An ancient "Newton," "Leonardo," or "Einstein" could have bootstrapped his mathematical system from ages of pre-historical experience, 50,000 years of human observation of the heavens and earth.
Just like the nameless creator of the Phoenician alphabet, the name of the source was lost, but the useful knowledge retained. Steeped in legend and myth by Babylonian times it was attributed to a divine source just as we attribute our own moments of inspiration or genius to a higher source, beyond our ordinary selves. This nameless genius created an oral tradition whose tables and methods were eventually written down. At first, one had to be an initiate (priest or scribe) to use the methods. Later they were adopted in everyday life.
The ancient systematic observers noticed the regularity of the passages of the planets through the constellations of the fixed stars. They conceived the grandiose idea of a mathematically determined cosmic order of greatness with lesser ever-evolving cycles of celestial manifestation, disappearance and renewal. Man sought then to harmonize with these cycles through the timing of religious festivals and astronomically based calendars in imitation of heavenly circumstances.
The Precession of the Equinoxes was first noticed as a slow but steady slippage through the Zodiac of 1 degree every 72 years. To complete one cycle of the zodiac--a "Great" or Platonic Year--requires 25,920 years. Dividing this sum by 360 yields the number 432, the root of the mythological count of 432,000 years. However, it is not only mythological, or archetypal--it was discovered by centuries of controlled astronomical observation, even prior to written record-keeping.
In India, the Kali Yuga is supposed to have begun on February 17, 3102 BC. The astronomical aspect of a yuga begins with the sun, moon, and planets in conjunction in the initial point of the ecliptic. Everything returns to the same point at the end of the age. This belief originated way before Hindu astronomy and is cross-cultural. But 3102 BC is a good approximation of the invention of the arts of writing, mathematics, and astronomy--all of which are a remarkable effort at translating celestial mathematics into the ordering principle of life on earth.
This is the echo of the old Mesopotamian doctrine which reverberated through Greece and Rome (Berossos to Seneca) as well as India (Yugas). It spread all over the known world into Egypt, the Zoroastrians, and traces are even found in China, Mexico, and South America. The old Sumerian tradition of astronomical observation was the basis of all intellectual culture, and originator of the myths of eschatology, or end times.
The sacred place, the center of transformation, has always been a refuge from the laws of the temporal world. Sacred space is the visionary gateway which opens communication with the transcendental reality of the divine. Here, as Jung states, "man is no longer a distinct individual but his mind widens out and merges into the mind of mankind--not the conscious mind, but the unconscious mind of mankind, where we are all the same.
When the concept of such a holy site or center is joined with a mathematically structured universe, derivation from ancient Mesopotamia must be suspected. It is the archetypal source, the navel of the world. In many cases, the center is conceived as an axis extending vertically upward and downward, with the center at the conjunction of the four cardinal direction.
This is the ancient model of sacred space, which corresponds in Synchrographics with the form of the T.R.I., the Triaxial Retrograde Interface. The three intersecting axes of Euclidean space with a shared coordinate.
The Hindus had a version of this centering mechanism called the regents of the directions. Brahma was in the overhead position, Vishnu below, with Shiva as the vertical axis. Each directions is attributed to a god and quality.
The essence of this image of the axial point or pole is that it symbolizes the way or place of passage from motion to rest, time to eternity, separation to unity; but then also, conversely rest to motion, eternity to time, unity to multiplicity. The realization of the nonduality of heaven and earth--even of being and nonbeing--is assimilated in the sacred center. The ego is sacrificed in the primal waters of deathlessness, and released to be carried in all directions. This is the mystical-psychological sense of sacrifice in all great religions.
A solar hero is the most frequent embodiment of this process of purgation, illumination, and unification. He unites the religious significance of the sun, the zodiac, and the seasons with circular or cyclic determinism. Man sought to rise and share the great cycle with the sun and stars, to climb beyond the material universe to the immaterial realm of the world-sustaining sun. The ageless concept of the new dimension transcending linear, historical time echoes Mesopotamian cosmology. We ourselves are already that light of consciousness, that ground of being.
Jung revealed how mythological images and numbers have always been associated with each other. Here, we find a correspondence between the Universe, #108, and cycles of the Sun.
Von Franz summarizes her Jungian view:
"The concept of natural numbers rests on an archetypal foundation. It represents a preconscious pattern of thought common to all human psyches, and therefore constitutes the basis for transmitting knowledge to a greater degree than mythological images, which exhibit more ethnological variations.
"Those aspects of the number archetype which present-day Western mathematics has made conscious in no way exhaust all its aspects...The preconscious aspect of natural numbers points to the idea of a numerical field in which individual numbers figure as energic phenomena or rhythmical configurations. The "field," which we take to represent the structural outlines of the collective unconscious, is organized around the central archetype of the Self (which corresponds with the Sun, which corresponds with 108, and Brahma). For this reason historical mandala structures deserve particular attention. In corresponding "cosmic models" and mathematical representations of God, the first four numbers predominate to an exceptional degree.
"...These synchronistic and parapsychological aspects of number...can only be fathomed when we take into account the unconscious emotional setup and preconscious fantasies of the abserver along with his conscious mental situation and outlook. The description of such phenomena will of course no longer produce universally valid theories, but rather transmittable realizations that can exert a mind-releasing, community-building effect, just as scientific advances did in the past. The common denominator in mankind's cognitive processes thereby shifts from the level of doctrinaire intellectualism onto another plane. It centers instead on the realization of an a priori psychic structure common to all men. Depending on the epoch and an individual's creative abilities, the basic substratum becomes clothed in the most varied shapes and conscious formulations, progressively transforming 'ancient, eternal truths' into more highly differentiated conscious patterns of realization.
"As the ultimate verification of these processes stands the objective psyche and its synchronistic manifestations, which contain the mystery of the sporadic conjunction of psychic and physical events, revealing a common 'meaning.' . . .When we take into account the individual characteristics of natural numbers, we can actually demonstrate that they produce the same ordering effects in the physical and psychic realms; they therefore appear to constitute the most basic constants of nature expressing unitary psychophysical reality."
The Development of Mathematics According to Singer's A Short History of Scientific Ideas, something of the nature of mathematics must be much older than the earliest documented examples. In tribal pre-history mankind watched the movement of the heavens and kept tallies of the passage of cycles of both lunar and solar nature. The importance of the Sun grew with the rise of agriculture and the importance of dating planting and harvest times accurately.
By Sumerian times, numbers were represented in a system which combined a decimal with a sexagesimal notation. It embodied the principle of place-value, but not as we now know it. Shifting a number one place to the left multiplied its value sixty-fold, successive shifts to the right corresponded to repeated divisions by sixty to form sexagesimal fractions. In later Babylonia of the Seleucid period, the texts employ a 'zero' to indicate an empty sexagesimal place between two other figures. Remnants of the sexagesimal system survive with us in the 360 degrees of the circle, etc.
The mathematical texts usually consist either of tables for multiplication, squaring of numbers, etc. or of worked examples illustrating the solution of elementary geometrical or algebraic problems. The geometry amounts to little more than estimations of the areas of fields, though the special property of the right-angled triangle was known. The ratio of the circumference of a circle to its diameter, pi, was taken as equal to 3. This is the value adopted in the Old Testament, perhaps under Babylonian influence.
The algebra of the Old Babylonians could solve quadratic equations by a procedure equivalent to evaluating the modern formula, which gives the roots in terms of the co-efficients, thought the known texts nowhere quote or prove this rule. They also handled linear equations in several unknowns and even attempted to solve cubic and biquadratic equations.
The Old Babylonians astronomy amounted to little more than recognition of bright stars, arbitrary demarcations of the heavens, and often undated observations of striking celestial or atmospheric phenomena. There are also records of omens drawn from these, whose significance marks the beginning of astrology. Originally used to predict the fortunes of contending kingdoms, horoscope astrology subsequently developed into complex procedures for foretelling the destinies of individuals.
The later or Seleucid texts, on the other hand, embody complicated systems of theoretical astronomy. These were elaborated by the temple priests who observed the heavens from characteristic stepped watch-towers or temples, of which the Tower of Babel is a reminder.
The periodicity exhibited by the planets, and more particularly the revolutions of the sun and moon, were utilized for the measurement of time. The monthly changes of the moon are more obvious than the annual travel of the sun. So the lunar calendar was retained for religious purposes, while the solar was adopted for agriculture.
No natural numerical relation exists between the lunar month and the solar year. But by the fifth century BC, it was established that 19 solar years are equal to 235 lunar months (125 of 30 days and 110 of 29 days each) to within a fraction of a day. These 19 years, comprising 12 of 12 months each and 7 of 13 months each, were combined in a certain order to form what has been called, after the alleged Greek inventor Meton, the Metonic Cycle. The sequence in which 29-day and 30-day months followed one another was seen to be affected by the variations in the rates of motion of sun and moon, by the latitutde of the moon, and by the inclination of the ecliptic to the horizon.
In the tables for predicting the dates of successive new moons, separate columns indicated the corrections to be separately applied for these various factors affecting the length of the month. They represented fluctuating discontinuity between upper and lower limits in a characteristic manner. The Babylonian tables, which extended also to the prediction of planetary phenomena, have been classified into two main systems, according to the methods employed to represent the variation of the sun's rate of motion through the course of a year. (adapted freely from The Short History of Science). _____________________________
Indian Mathematics The Indus civilization also learned its first lessons in mathematics from astronomy, the gateway to time reckoning and temple building. In the arithmetic of trade, the merchants of India were the equals of those of Mesopotamia.
Until about 2000 years ago, they probably used numerals made up of horizontal strokes. When they began to use dried palm leaves as writing material and developed a flowing style of writing, they also began to join up these strokes, so that became and became .In this way they gradually built up different signs for each number up to nine. Each sign could be conveniently used to indicate the number of pebbles in any groove of the abacus.
The Indians learned how to tell not only how many pebbles are in a groove, but also which groove they are in. The far right stood for units, the next to the left for tens, then hundreds, and so on. An empty column used a dot, as we now use zero. Thus each value meant only that one figure.
This system does away with space-consuming repetition. We can record the same number on any groove of the abacus by using the same sign. Saving space is only a small advantage. The great advantage of the Hindu system is that it enables us to calculate with numbers.
Other ancient systems of writing all relied on the use of different symbols for the same number of pebbles in different grooves of the abacus. Before you could do written or mental calculations with them, you would need to learn a different table of addition and multiplication for each groove. When there are only nine different signs, each of which can show the number of pebbles in any groove, and a zero indicates empty grooves, you need learn only one simple table, once and for all. You can carry over in your head because there is only one simple table to remember.
The Hindu number language quickly led to a revolution in the art of calculation. The mathematicians of India began to think in fractions and write them in the modern way. By 500 AD, Indian mathematicians had solved problems that baffled the greatest scholars of antiquity. The mathematician Varahamihira calculated how to forecast the positions of the planets; Aryabhata wrote a rule for finding square roots and gave the value for pi as 3.1416.
Around 800 Ad, this numerical system was exported to Baghdad on the age-old caravan route. The Muslims used trigonometry. Because they had mastered the new arithmetic of India, they could make much fuller use of the geometry of Euclid. Improved navigational equipment emerged from the observatories which also had improved equipment. Knowledge took a big leap forward between 800-900 AD, when East met West in Baghdad, and the baseten system became the standard.