Archetypal Nature of Numbers
NUMBER AND JUNG'S
"Psychologically, the rotundum or mandala is a symbol of the self. The self is the archetype of order par excellence. The structure of the mandala is arithmetical, for "whole" numbers are likewise archetypes of order. This is true particularly of the number 4, the Pythagorean tetraktys. Since a state of confusion is generally the result of a psychic conflict, we find in practice that the dyad, the conjoined two, is also associated with the mandala."
Collected Works, Vol. 10, pp. 805
"A frequency of four events provides the three intervals that also form the base triangle of the tetrahedron apexed by the initially unpaired, angularly finite event. The insideness and outsideness of this primitively evolved tetrahedron constitute the minimum macrocosm-microcosm-differentiating system of the Universe. This tetrahedron has six angularly directional interrelationship lines interconnecting its four finite events."
In Pythagorean thought, number is considered the common ordering factor of both psyche and matter. Thus, the study of the inherent rhythms of the natural number sequence was of paramount importance. There were specific numbers which were considered divine manifestations, and in fact corresponded with various gods. The studies of geometry, music, astronomy, and color were therefore sacred arts, as shown in the Prayer to the Tetractys:
"Bless us divine number, Thou who generatest gods and men! O Holy, Holy Tetractys, thou that containest the root and the source of the eternally flowing creation! For the divine number begins with the profound, pure unity until it comes to the holy four; then it begets the mother of all, the all-comprising, the all-bounding, the first born, the never-swerving, the never-tiring holy ten, the keyholder of all."
Plato and Pythagoras both dealt with geometry as an aspect of number, (triangles, squares, oblongs, rectangles, etc.). Plato, of course, has his series of ideal regular and irregular forms which naturally had a plurality (number) of faces and vertices.
The kernal of Pythagorean wisdom is contained in a primal BASE TEN ARRAY, the divine TETRACTYS, a pyramidal array of 10 points which contain a great mystery of universal nature. The TETRACTYS is made up of the numbers 1, 2, 3, 4, which add up to 10, arrayed in the form of a perfect triangle. This figure is also reflected in slightly different form as the qabalistic Tree of Life.
The four levels of the array contain not only the basic intervals--fourth, fifth, octave, and double octave--but also point, line, plane, and solid. The unequal numbers of this cryptic formula may be incomprehensible except to the initiated.
Psychologists such as Carl Jung and Edward Edinger have explicated its psychological meaning and symbolism as an expression of the epitome of the process of personality development, with four stages and three steps of transition.
(Insert graphic of TETRACTYS containing VECTOR EQUILIBRIUM)
The matrix of the Tetractys represents a qualitative field--dynamic internal structure. In it the ontological and logical ordering of numbers is translated into rhythmical and geometric images. There is no need to invoke the numerals as elements of an infinite series, but rather we imagine them as the "ensemble" of a finite series.
According to Jungian analyst Marie Louise vonFranz in NUMBER AND TIME (19 ), "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. This 'field,' which we take to represent the structural outlines of the collective unconscious, is organized around the central archetype of the Self.
For this reason historical mandala structures deserve particular attention. In corresponding "cosmic models" and mathematical representations of God, the first four natural numbers predominate to an exceptional degree, just as they did in the systematic divinatory techniques of the past."
Plato and Pythagoras both dealt with geometry as an aspect of number. Plato's series of regular and irregular forms naturally had a plurality (number) of faces and vertices. In PQM terms, "the forms" of Plato are, perhaps, robust attractors in the super quantum gravity potential of the three-geometry of our actual expanding universe. They represent "eternal values," whose content can theoretically be imprinted upon and influence the quantum mind field.
Jung found a vast psychological significance in the number four. Jungians conclude that "mankind's repeated attempts to establish an orientation toward wholeness posses a quaternary structure, and appears to correspond to an archetypal psychic predisposition in man." vonFranz, Number and Time, p. 115.
We can equate four disciplines with four modes of being: physics = physical; music = emotional; psychology = thinking; metaphysics = intuition. They equate with Jung's four functions of cognition: sensation, feeling, thinking, and intuition.
It is a cross-cultural phenomenon from sources as widely separated as the Chinese, Native Americans, Asiatic Indians, Incan and Mayan, and pre-Christian Mediterranean--this tendency to cleave to fourness in modelling reality. Some of the Pythagoreans regarded four as the essence of justice, being four-square in one's dealings.
According to Edinger, "One of Jung's major discoveries is the psychological significance of the number four as it relates to the symbolism of psychic wholeness and the four functions. The significance of the quaternity is basic to his whole theory of the psyche, both as regards its structure and its developmental goal, the individuation process. We are thus particularly alert to quaternity symbolism as it appears in dreams and in the imagery of myth and folklore. However, there are other numerical motifs which are commonly encountered."
"Jung describes Pythagorean number symbolism which is pertinent here. THE NUMBER ONE AS THE FIRST AND ORIGINAL NUMBER IS, STRICTLY SPEAKING, NOT A NUMBER AT ALL. One as unity and totality exists prior to the awareness of numbers which requires a capacity to distinguish between separate, discrete entities. Thus, "one" symbolically corresponds to the uroboric state prior to the creation and separation of things. Two is the first real number since with it is born the possibility of discriminating one thing from another. Two symbolizes the act of creation, the emergence of the ego from the original state of unity."
The Tetractys is inherent in our biology:
"The entire microcosm is replete with harmonic occurrences. The long strings of nucleic acid in DNA are structured precisely according to the Pythagorean Tetractys, the fourfold subdivision of the octave (octave, fifth, fourth and major second). The Pythagoreans attributed magic power to the Tetractys and called it sacred. This same structure is almost ubiquitous in those mysterious processes whereby inorganic structures are transformed into organic life," according to Berendt, p. 69.
The four-fold structure holds sway in physics as Minkowski and Einstein's four-dimensional model of the universe, and the four forces of nature (nuclear, electrical, weak interaction, and gravitational). It reminds of Fuller's notion that nature operates in +4, -4 event octaves.
Sometimes psychic and physical events coincide as synchronicities, revealing a "common meaning." In this case the nature of the psyche is revealed as the transcendent unity of existence using the rhythms of natural numbers. VonFranz tells us that, "These numbers illustrated some of the most primitive and basic forms of the spirit. 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."
Number and Time, p. 303.
So, there is a common basis between numbers and the psyche. Numbers apparently order both the physical and psychic worlds.
Building on the diagram of the Tetractys, Edward Edinger imagines a +4 descent and a -4 ascent, a movement of the differentiating ego separating from undifferentiated objective wholeness into subjective experiential development, and then moving back toward the One in a recursive reunification. This echoed in the old alchemical axiom of Maria Prophetessa:
"One becomes two: two becomes three: and out of the third comes the One as the fourth."
Or another version:
"One, and it is two: two, and it is three: and three, and it is four: and four, and its is three, and three, and it is two, and two, and it is one."
These stages symbolize the transformational process:
"This first stage of the coniunctio...brings about the union of soul and spirit over and against the body; they link up as partners against the body. In the second stage, united spirit and soul reunite with the body. In the third stage, the previously united spirit-soul-body is now united with the world, and conversely (recursively) the Self, or source. This is characterized by independent autonomous being. Original unity has been differentiated into a fourfold multiplicity. After living fully in the world the task of individuation becomes the ascent back up through the number series of the Tetractys."
(insert diagram Edinger, Mysterium Lectures, p.282)
The Diamond Body:
Buckminster Fuller and The Qabala
by Iona R. Miller, copyright, 1992
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
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 QABALAThe 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.
These systems are virtually pre-geometrical. They don't really exist in nature, per se. Yet they are the invisible lattice, or barebones, 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 sightIn Aion, psychologist Carl Jung reiterates the identity of psyche and matter:
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.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.
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 outbreath 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.
- Picture of The Macrocosmic Snowflake within a Vector Equilibrium grid.
- The Preceding Article is an exerpt from the book, The Diamond Body, by Iona R. Miller, copyright 1984.
N = 50, V = 6: A 56-FACET STONEHENGE MANDALOG
by Iona Miller, c1981
24. I am Nuit, and my word is six and fifty.
25. Divide, add, multiply, and understand.
Aleister Crowley, The Book of the Law
In ancient times, and in modern metaphysics (such as qabala), letters were also numbers. Language was alphanumeric, so words carried symbolic numerical values. The ancient Egyptian sky goddess, Nu or Nuit, represents infinite space or cosmic zero, the universal source. HAD, or Hadit represents the microcosmic inner being--the Self, the still point of the axis, the center of the circle....to Nuit
The Philosopher's Stone is circular unity and its inherent dynamic +4, -4 octave wavecycle.
In the same year that Einstein discovered his famous equation E = mc2, magician Crowley channelled an occult treatise called THE BOOK OF THE LAW. We are not recommending nor endorsing such material, but it does reveal some alphnumeric qualities of the number field, and this is of interest.
56 radii form a circular unity--a mandalog which reveals interesting properties of number dynamics, and relates to the geometric construction of Stonehenge in a very empirical, demonstrable way that does not force the numbers onto the structure, but rather illuminates it.
Magician and Qabalist, Aleister Crowley, in LIBER NV, proceeded to divide, add, multiply and understand, disclosing relationships of 50 and 6 to one another:
4. Consider of six and fifty that 6 - 50 = 0.12
0 the circumference, Nuit.
. the center, Hadit
1 the unity proceeding, Ra-Hoor-Khuit
2 the world of illusion.
Nuit thus comprehends All in None.
Also 50 + 6 = 56 = 5 + 6 = 11, the key of all Rituals.
And 50 x 6 = 300, the Spirit of the Child within.
In the Commentaries of Al, Crowley corresponds Nuit with matter and Hadit with motion "in their full physical sense". If we pursue this correspondence in physics, Nuit and Hadit seem to disclose the relationship of a field to radiant wave propagation. Wave mechanics is an important fundamental from cosmological levels (Cult of the Infinite Without) to sub-atomic levels (Cult of the Infinite Within).
Matter is usually described as varying states of fluidity, going from a particle state to a wave, and then back again. David Bohm, in Wholeness and the Implicate Order, states that "Every physical situation is now characterized by a wave function." Since motion can be described as a complex wave interaction (or interference pattern), an understanding of wave phenomena in a field is fundamental for a contemporary view of the Universe.
This understanding is critical for establishing one's world-view, or conception of "things-as-they-are." Thus, we remain consistent with current scientific research. Toward this end, we may examine the natural number sequence (integers) as a qualitative field, not merely as a quantitative series.
Nuit, as the number 56, has a relationship with all numbers: "Nuit comprehends All..."
Further amplification of this relationship may be graphically represented by a spiral array of numbers along 56 radii to form a mandala. Joining the 56 equidistant points on the circumference yields a 56-facet circularity.
6. Meditate upon Nuit as the Continuous One resolved
into None and Two as the phases of her being.
This quote from Crowley's Liber XI (Liber NV) could be related to a circle or "zero"-figure (Continuous One); "Two" could be connected to an emergent, self-organizing reflection sequence where the halves of the circle mirror one another.
Had is self-defined as motion by the statement "...for it is I that go." Had is all-pervasive in the number field.
Mass is neither wave nor particle in that it is continually moving from one state into the other. Hadit, then, should not be symboplized as a static point (Crowley), but rather as a dynamic, Motion. This viewpoint is more consisten with current models of Quantum Mechanics.
Hadit should not be visualized as a "point" in space because how can one pin down in space something which, by definition, occcupies no space? Therefore, Crowley's attribution of the decimal point to Hadit as the center is somewhat antiquated, metaphorically.
Hadit and Nuit are inseparable. Nuit is the underlying field through which Hadit moves. If the field potential were non-existent, Hadit could not move. There would be no matrix.
Examination of the process described from The Book of the Law yields several important relationships between numbers, which may be seen as forming a "living body" of Nuit:
I. "DIVIDE": Divide a circle into 56 equal parts, the 56 radii of the proposed mandala. The reciprocal operation divides
56 - 360 = 0.1555555 x 2 = 3.111111 (polygonal pi)
A circle with 56 equal units to its faceted circumference has a diameter of 18 equal units, yielding rational pi:
56 - 18 = 3.1111111 = 'rational pi'
Rational pi makes the perimeter of the circle an integral number of times the length of the radii. Rational pi is involved in the essential structure of number.
3.1111111 - 2520 = 0.001234567
II. "ADD": Adding the sum of the first nine multiples of 56 yields the first number divisible by all base digits--2520.
The equation 7 x 360 = 2520 shows an important relationship of seven to the circumference of a circle and to the inherent structure of number. There are numbers with interesting properties which may be found by adding columns from the table of sevens, to form reflective numbers or palindromes. The sums of the colums of multiples of seven, when added to their reversal, form such reflective numbers. For example, 315 + 513 = 828; 630 + 036 = 666; 1260 + 0621 = 1881; 2520 + 0252 = 2772; 5040 + 0405 = 5445.
These products, which read the same forward and backward, are part of the binomial reflection sequence. That these numbers have peculiar qualities has been known since ancient times. Buckminster Fuller refers to them as Scheherezade numbers as they figured prominently in the 1001 tales of the Arabian Nights.
These mathematical operations demonstrate certain rhythmical configurations inherent in the natural number sequence. Movement through them may be viewed as a wave or pulse. The reflective numbers themselves may be seen as a particle phase as the wave/particle moves in quantum leaps.
Examination of these numbers shows some interesting integers recurrent in QBL. These numbers are considered significant in many of the world's philosophical systems. As the universses of philosophy and mathematics coincide, one begins to perceive the field underlying them. A pattern is becoming recognizable.
III. "MULTIPLY": The binomial reflection sequence is also shown by multiples of 56. This reflection sequence is bounded by the number 2520. When numbers, spiralled along 56 radii are graphically represented, they form a visual example of how the Formula of Nuit, 0 = 2 works out in a beautiful mandala. The coded pattern reverses itself at the midpoint, 1260, and forms its perfect reciprocal reflection, or mirror image--the circle yields two conjoined reflective sides. The graphic design mirrors itself precisely.
(Binomial reflection sequence table, here)
We may examine NV = 56 by the equation 7 x 8 = 56. This equation is important in the magickal tradition because:
(a). Connecting every 7th radii forms the 8-pointed Star of Regeneration, a symbol of the unity of the Spirit. It also forms an octagon, the symbol for the central shrine of the transformed self.
(b). Connecting every 8th radii forms the 7-pointed star, whose interlacing lines generate the Seal of Babylon in its center.
45 degrees of arc x 56 = 2520 - 7 = 360 degrees
IV. "And Understand". This archetypal order shown by the reflection sequence has been the object of philosophical speculation since the time of Plato. Plato spoke of the number 5040 (2520 x 2) in his last work, The Republic, Book 5. In fact, this number comes at the end of a passage long considered to be a fragment of unfinished work. It may be viewed, however, as a cryptic insight which left room for one to discover "understanding." Five times in the Book of Revelations, 1260 (2520 - 2) is alluded to cabalistically. It also figured priminently in the works of Ptolemy and Hipparchus.
Commenting on Eastern philosophical systems in The Myth of Invariance, Ernest McClain states:
"1260 = 22 x 32 x 5 x 7 shows a musical theorist how the sacred number 7 generates along with the 'human' 5 and the 'divine' 3; thus it reveals a basic lesson in mathematical harmonics suppressed in the Rig Veda except by allusion, yet essential to the understanding of 'Tenness' in all ancient cosmology."
The ten spheres of the Tree of Life show QBL as a base-10 system. Remember also that Had (h = 4, a = 1, d = 5) = 10.
2520 is a special number since it accomodates or synchronizes the maximum possible amount of consecutive numbers from 1 onwards. There are larger numbers, but none with as high a degree of factors of divisibility.
We can conjecture that Crowley was presented this concept in a very abstruse, symbolic form through channelling. Despite this arcane source, it is clear that precisely what the passage from The Book of the Law requires from us is a process of deciphering. This offers the possibility of understanding the symbolic on the conceptual level, and taking up a relationship to it. Practical knowledge of later generations often presents itself in intuitive, symbolic form to earlier generations. For example the relationships of the metaphors, imagery andsymbolism of alchemy to chemistry, and natural science to physics.
V. THE AURIC KEY: Current work in graphic representation of this ancient understanding of relationships among numbers is being researched under the name SYNDEX, by Robert Marshall, et al. The first concern of synchrographics is the maximum information expressed via minimal graphic elements. As in QBL, correspondences, such as those between geometry, numbers and colors, form an integral part of the system.
Graphic representation with a 56-facet mandala discloses ordered matrices inherent in the relationships among rational numbers. They are self-organizing. For instance, the 12 "synchrostats," (numbers divisible by 5,6 and 7), fall into perfect quadric symmetry along the axes of the circle. This division of the circle into four parts is analogous to the four-directional orientation of the Banishing Ritual, for creating sacred space. The synchrostats are multiples of 210 and correspond with the 12 zodiacal signs: 12 x 210 = 2520.
210 = Aries
420 = Taurus
2520 = Pisces, completing the zodiacal circle
Thus, we see that the 'mandala of Nuit,' the 56-facet circularity, discloses and encloses the profound patterns of relationship inherent in the natural number sequence and can display it as a field, graphically. It gives us a feeling for "number as Field." Numbers have a frequency of modulation, and a type of pervading wave phenomena. We could visualize this mandala as a 'philosophical eye'. Or, alternatively, we might view it as a matrix, the all-containing 'womb of Nuit.'
A mandala is the empirical equivalent of the alchemical concept of the Unus Mundus ("One World"). The Unus Mundus is one of the many ways of referring to the experience of unification of body/soul/spirit also known as the creation of the Philosopher's Stone.
Synchronistic phenomena are the parapsychological equivalent of the Stone. Jungian psychologist, Marie-Louise von Franz recounts in Number and Time how "...attempts have been made in the past to combine these equivalents into a unitary reality" and to construct mandalas, which via synchronicity, would yield parapsychological "knowledge." This mandala, used as a magical tool, creates the possibility of an altered state of consciousness known as an "access state."
Zosimos and Bruno, Hermetic magicians, used mandalas as tools for magically acquiring "information" about the rationally unknowable. By means of this tool, the individual confronts the continuum to produce intuitive, qualitative experiences, from an otherwise latent potentiality.
Number essentially pertains to the behavior of archetypal dynamics, or "ordered sequences." The 56-radii spiral-arrangement to 2520 is the minimum necessary to disclose patterns of relationship in the qualitative field. Total archetypal order of the Unus Mundus and all its contents may be represented this way as there is an equation between archetypal images or ideas, and numbers. This is the foundation of QBL and its correspondence system. All patterns are mathematically ordered and there is rhythm in the natural number sequence..
Buckminster Fuller refers to this rhythm in Synergetics when he speaks of the pulsative octave. He states that the interaction of all numbers other than 9 creates wave phenomena, i.e. "self-invertable, self inside-outable octave increasing and decreasing pulsatively, fourfoldly, and tetrahedronally IS ALL THAT EVER HAPPENS!" This is Fuller's peculiar geometric lingo for speaking of carrier-wave order phenomena.
While this concept is fairly simple to grasp mathematically or graphically, it is difficult to express in words. Consequently, the importance of fields and wave phenomena have been understood by only a few. This is why the graphic mandala is such an important tool. It is a visual aid for understanding the relationship formed between field (Nuit) and carrier-wave order (Hadit).
In the field or manifold of psychic contents, the nucleus of the field is the most imperceptible aspect -- this is the circle, Naught. The preconscious aspect of natural numbers points to the idea of a numerical field in which individual numbers figure as energetic phenomena or rhythmical configurations.
This field, which we take to represent the structural outlines of the collective unconscious, is organized around the central archetype of the Self or Transcendent Function. This function is represented in Magic as the Holy Guardian Angel, another analogy for the experience of the Philosopher's Stone.
A very literal example of "number as field" may be found on a meadow in England at Stonehenge. Here we find a striking example of the use of a 56-radii mandala as a magical tool. The geometry of Stonehenge is a remarkable verification from the neolithic past and an ancient spiritual tradition which reiterates the importance of this arrangement.
To make the perimeter of the circle an integral number of times the length of the radii, the Stonehenge programmer used the array of 56 equidistant points. The remnants of these points are still visible and are termed "Aubrey Holes" in literature about Stonehenge. That a realization of the essence of 'circular unity' was necessary to decode the message of Stonehenge is indicated by certain key features.
1. There are 56 Aubrey Holes in the circumference of Stonehenge.
2. Joining every seventh Aubrey Hole generates the outer diameter of the circle of Sarcen stones. 7 x 8 = 8-pointed star.
3. Joining every 8th hole generates the 7-pointed star and the Star of Babylon, and determines the thickness of the Sarcen stones.
4. Joining every 6 holes equals an equilateral triangle whose apex touches the center of the Sarcen stones.
Bob Marshall refers to these properties of Stonehenge as a "Mandalog" or "Mandalometer."
In all of these operations, 56 or 5 + 6 = 11 is indeed the "Key" to decoding the Rituals.