The History of SACRED GEOMETRY
"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 transcendended 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. Precosmogonic 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, is still mistaking the creation for the dynamic transcendent Creator, worshipping 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.
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."
The Psychophysical 12-Dimensional Matrix;
Six pairs of complementary opposites, synergetically inter-tensioned,
reveal the implicate matrix of the Tree of Life geometry.
The surrounding Sphere is implied.
"The Diamond Body," c1981
This is an image of Tiphareth as the Cube of Space. The octahedral geometry is that of a face-centered-cubic close-pack lattice, which is that of diamond's atomic structure. The three axes formed by 3 virtual Great Circles around an implied surrounding sphere are corresponded with the three Mother letters: Aleph, Mem, and Shin. The center of the figure represents transcendence of time and space as symbolized by the letter Tau, which corresponds with The Universe. The vertices of the octahedron are planetary attributions. Each of the vectors of the surrounding cube correspond with path/letters and are color-coded to their correspondences.
"STAR OF DAVID," c1981
"Atomic V.E.M.," c1983
"He passed the flaming bounds of space and time:
The living throne, the sapphire-blaze,
Where angels tremble while they gaze,
He saw, but blasted with with excess of light,
Closed his eyes in endless night."
--Milton (1608-74), Progress of Poesy
Nested infinities of Vector Equilibrium Matrices generate and regenerate reiterations of Tree of Life vertices or nexus points. This variation on the Cube of Space combines the basic forms of the Star of David and Unicursal Hexagram. The Middle Pillar appears compressed as virtual nested spheres of white, yellow, violet. This is the synergetics of metaphysical reality, a causal plane temple of "wheels within wheels."
"DAATH: The Upper Room," c1981
Jitterbugging Vector Equilibrium Matrix is not a structure but a system, the prime nucleated system. Vector Equilibrium makes it possible to make conceptual models of 4th, 5th, and 6th dimensional omniexperience accounting by using tetrahedroning. Equilibrium between + and - is Zero. V.E. is the true zero reference of the energetic mathematics. It is cosmic zero.
Zero pulsation in the V.E. is a metaphorm of eternity and God: the zero-phase of conceptual integrity inherent in the + and - asymmetries that propogate the differentials of consciousness. V.E. is important because all the nuclear tendencies to implosion and explosion are reversible and are always in exact balance. V.E. is the anywhere, anywhen, eternally regenerative, event inceptioning and evolutionary accomodation and never seen in any physical experience.
This metaphorm (V.E.) represents the self's initial real-I-zation both inwardly and outwardly from the beginning of being "betweeness"; maximum inbetweeness. Push/pull; convergence/divergence; gravity/radiation. At zero-point, waves can pass through waves without interfering with other waves. Vectoral phase or zone of neutral resonance which occurs between outwardly pushing wave propogation and inwardly pulling gravitational coherence. Emptiness at the Center: all 4 planes of all 8 tetrahedra (i.e. 32 planes/32 paths) are congruent in the four visible planes passing through a common V.E. center, the cosmic terminal condition and nature's most economical lines of energy travel.
"Omnidirectional Philosopher's Stone," c1981
Daath is the "gateway of all inbetweeness." In this model and metaphorm, physical and metaphysical share the same design. Equanimity model where the reins of all spheres are synergetically intertensioned at Tiphareth. The center of this operational sphere is Zero-Point, Vector Equilibrium, or Cosmic Zero. This psychotronic machine is for interdimensional tuning. Thinking itself consists of filtering out macro- and micro- irrelevancies, which leaves only the lucidly relevant "con-siderations," which as Fuller points out, means "putting the stars together." This topological modeling provides a synergetic means of ascertaining the values of any system of experiences. It is the science of fundamental pattern and structural relationships of event constellations.
All paths of Circulation of the Light are contained or implied in the central hexagonal star.
Visualize Vector Equilibrium in the heart center and build up the Cube of Space from the Isotropic Vector Matrix. The center is nucleated Nothingness = Zero-Point = The Fool, Aleph/Tau. The Cuboctahedron; Triquidoid. Symmetry operations carry crystal structure into itself. Rotation and reflection operations = point operations. This metaphorm of the alchemical Circumambulatio is analogous to information transfer from vertex to vertex, creating superpositions and sphere-linking.
The dynamic activity connected with the drive to know, to penetrate, to illumine, culminates in a stillness, silence, cessation of all effort which itself dissolves in the tranquility of total negation. 0 = 2. It is only by virtue of the fact that it is Naught. All form and power are latent in the Void. Here we witness experientially the quantum propogation of radiant wave after radiant wave identifiable with given wavelengths and frequencies of enfoldment.
"Crystalline Star," c1981
Tiphareth, the Heart Center: Diamond Body plane of operations. Based on the geometry of the Necker Cube. Circulation of the Light from the Throne of Glory. Central shrine, the Throne-Chariot, is an octal code. Necker Cube star forms a diamond-shaped "plane of operations" for the consciousness of the adept. This is a metaphorm or physical analogy to Golgi cells in the brain and their hexagonal inhibitory fields. According to Karl Pribram, the golgi cell system can be considered as a "focusing device restricting or giving preferance to granule neuron (parallel fiber) activity in relatively narrow bands." This synergetic form is the Net of Artemis, an in-formation, re-call, re-membering system.
According to M.L. von Franz in Number and Time, a "mandala is the inner psychic counterpart, and synchronistic phenomena the parapsychological equivalent, of the Unus Mundus. . .attemptss have been made in the past to combine these two equivalents into a unitary reality and to construct mandalas, which via synchronicity would yield parapsychological "knowledge," (i.e. an "access sstate"). Zosimos and Bruno used them as tools for magically acquiring information about the rationally unknowable." The crucial time moment and act of personal intervention leads to qualitative, specific, time moments. They only emerge out of a latent, undifferentiated continuum when the individual confronts the continuum. Via circulation of the Light, the mystical body is formed by diamond-bodying.
"Star of Infinite Regenerations," c1981
"QBL Double Cube Gambit Grid," c1982
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.
Eliade, Mircea, A History of Religious Ideas, 1978, University of Chicago Press: Chicago.
Kaplan, Aryeh, The Bahir Illumination, 1979, Samuel Weiser, Inc., York Beach: Maine.
Meditation and Kabbalah, 1982, Samuel Weiser, Inc. York Beach: Maine.
Luminet, Jean-Pierre, Starkman and Weeks, "Is Space Finite?", SciAmer, April, 1999, p 90-97.
Wolf, Fred Alan, The Spiritual Universe: How Quantum Physics Proves the Existence of the Soul,
1996, Simon & Schuster, New York: New York.
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.
Zecharia Sitchin concludes that Sumerian science originated with the "gods from outer space." But it is not as mind-bending to imagine some more plausible alternatives to this "channelling." 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 knowledged 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 Portal
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 nconscious 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.
THE JUNGIAN APPROACH TO NUMBER
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 mthods 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).
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..
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