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Dodecahedral Universe: The Ionasphere 7770 words

Synergetic Qabala
and the Qabalistic Tree of Life

by Iona Miller, O.A.K., 11/2003

“An analysis of astronomical data suggests not only that the universe is finite, but also that it has a specific, rather rigid topology (dodecahedral sphere). If confirmed, this is a major discovery about the nature of the universe.”

--George F. R. Ellis, “The Shape of the Universe,”
Nature, Vol 425, October 9, 2003

Plato informs us in his Republic: “Geometry rightly treated is the knowledge of the Eternal,” and he is reported by Plutarch to have said that “God is always geometrizing.” Nor was the conception of the Universe in the form of the Dodecahedron unknown to Plato, for in his Timaeus this idea is clearly indicated.

--Frater Achad, Anatomy of the Body of God

Sacred Geometries

Imagine that hundreds or even thousands of years ago, mystics anticipated the astounding findings of modern science by looking deeply within themselves to plumb the mysteries of the universe. The ancient doctrine of Kabbalah says that its teachings about the nature of creation were given to Abraham in the form of wisdom and sacred geometries by an angel of God.

Ten spheres or subtle dimensions of existence emanated in a geometrical pattern from the veils of negative existence, ranging from archetypal Kether through the Causal and Astral levels to Malkuth, the physical plane. Current research may bear out some kabbalistic assertions about the cosmos.

Recent discoveries by French cosmologists have led them to believe that the universe is finite with a regular dodecahedral or soccer-ball shape, which is omnitriangulated and omnisymmetrical with 20 vertices. This contrasts markedly with the flat, infinite “Standard Model”.

In the 1920’s, Hermetic magician Charles Stansfield Jones, known as Frater Achad, experimented with 3-D projections of the kabbalistic Tree of Life. The Tree of Life diagram is a key provided initiates to unlock the more comprehensive geometry of creation. When he modelled the Tree as emanating from a single point in six directions, he found the projections culminated in a dodecahedron.

Later, kabbalist Leonora Leet (1999) described the regular dodecahedron as the geometric symbol of the Earth and also that of “Perfected Malkuth”, or the realm of matter in general.

Buckminster Fuller (1975) includes this ancient Platonic solid, the dodecahedron, among the energetic tetrahedral topological relationships described in Synergetics I and II. The chords of the five spherical geometrical integrities interact energetically, pulsating through their synergetic transforms. With the dodecahedron as its basis, a geometric nest is formed which turns information into coherent structure.

Synergetics, Fuller and Applewhite

This is nature’s own way of assembly which seems to be reflected from the smallest to the largest systems, echoing the qabalistic axiom, “As Above; So Below.” This axiom is also reflected in the Holographic Concept of Reality (R. A. Miller, Webb, Dickson; 1973-75) and the fractal reiterations (self-similarity regardless of scale) found in Chaos Theory and Complexity (I. Miller, 1993a).

There is a fundamental reality which is an invisible flux that is not comprised of parts, but an inseparable interconnectedness. In this dynamic model, there are no “things,” only energetic events. The holoflux includes the ultimately flowing nature of what is, and also of that which forms therein. Patterns of interference (light waves) create matter and energy as we perceive them. This domain is not transcendental to matter but underlies it as a coherent unity.

Through the seemingly chaotic process of emergence, order appears spontaneously or instantaneously within a system. Chaos is creative at its complex “edge.” Chaos theory tells us that new kinds of laws come into play when the level of complexity of a system increases.

Complex feedback loops from subsystems constitute integration. Creation is instantaneous. The flow of energy washes life and consciousness into the world. Or, in the kabbalistic vision, consciousness washes energy and life into the world. As the kabbabalists say, “God is a verb.” Chaos is nature’s guide, the matrix of formation, cosmic “zero,” as Fuller called it. Thus, something emerges from nothing.

Our notions about ourselves and the nature of the world about us, our worldviews, are filtered through our prejudices about “the way things work”. This is particularly true when we look at complex dynamic systems. The ultimate complex dynamic system is, of course, the Universe. And whatever it is, we are that -- of the same essence. Arguably, consciousness is an intrinsic emergent property.

Our scientific theories are simply state-of-the-art models which often conflict with one another, such as the irreconcilability between relativity, quantum mechanics and string theory. Not all theories prove to accurately describe reality. The universe has always been space-time as well as matter-energy.

Relativity eliminated the newtonian illusion of absolute space and time, uniting them as one in the fabric of space-time. There is no objective point of observation in the universe. In fact, quantum theory tells us there are no “points” at all.

Counter-intuitive quantum mechanics introduced the Uncertainty Principle, mandating participation rather than observation. It shows us that at the smallest level we can only make statistical predictions about how nature will manifest from waves of probability. And the non-linear equations of chaos theory imply dynamic processes are actually deterministic though unpredictable.

While providing a unified vision, the holographic concept doesn’t provide equations to unite all the forces of nature. Yet it reminds us of the holistic perspective that each part contains the whole, even though at a lower level of resolution. Physics demands experimentation while astrophysics relies largely on ever-sharper observation to determine the nature of our existence.

All these disciplines interrelate in certain specialty fields such as quantum cosmology, consciousness studies, transpersonal psychology, and metaphysics - that which by its very nature emerges from a domain beyond the physical, the realm of Source.

Our models still cannot completely reconcile the physics of the very small and the very large. Untestable theories which make no predictions are essentially philosophy rather than good science. Conversely, a strong argument can also be made that scientific ‘materialism’ is essentially a philosophy.

String theory, for example, remains a philosophy or conceptual paradigm, since no experiments can yet be devised to prove its applicability. Nor are they likely to be in the foreseeable future. It alleges that the essence of all manifest reality is vibrating strings, tiny one dimensional loops (rather than the quarks of quantum mechanics). These infinitesimal strings are the building blocks of matter.

Some kinds of string theory have open strings; some have closed circles. Some have one kind of vibration while others called heterotic have different kinds of vibrations moving in opposite directions.

Several of the superstring theories and one supergravity theory all fit into a kind of ecosystem called M-theory (for matrix or magic, take your pick), which is an alternative description of the same phenomena. The string part is natural because a point particle has an infinite energy singularity whereas a tiny string isn’t a point of energy but rather a base vibration.

All these theories are supersymmetric in 10 or so dimensions, meaning each force-carrying boson is paired with a matter-forming fermion but our particles are nn-symmetric in 4 dimensions. Our fermions, like the neutrino, electron, and proton don’t pair up in any kind of ‘arranged marriage’ with the photon or graviton or any other force particle.

Schwarz calls the Second String Revolution the finding that the formerly competing versions of string theory are actually variations on a theme, and the theme is M-theory. A nagging problem, already recognized in the work of Kaluza and Klein, has been: What of these extra dimensions? Where are they? Why do we never see them?

To get from one to the other requires compactifying the 6 or so extra dimensions. There are millions of ways of doing this and one or two might produce the odd array of particles we find. So the theories aren’t just philosophy but they might as well be unless we can figure out which one if any really coincides with nature.

Strings are orders of magnitude smaller than atoms, yet condition their various subatomic manifestations, such as quarks, protons, neutrons and electrons. Observed particle properties are a reflection of the various ways a string can vibrate.

Each preferred pattern of resonant vibration appears as a particle determined by the oscillatory pattern (Greene, 1999). This pervasive resonance phenomenon is literally the Music of the Spheres of ancient philosophy.

A string field includes the number of possible configurations of a string in space. It is related to a new kind of geometry in an enormous extension of the idea of space. It is defined by all the possible configurations of a string.

Strings act like point particles. However, a stringlike particle should be thought of as a “wavelike” disturbance in that huge space, just as a graviton is a wave in ordinary space. The interference of these wavelike disturbances could, indeed, be the holographic domain.

One version of string theory posits our four manifest dimensions (3-D + time) are undergird by 6 subtle enfolded dimensions curled up unobservably in space-time, according to the mathematics. This 10-dimension theory reminds of the qabalistic notion of 10 spheres, with only the lower four being manifest while the other six are more fundamental emanations.

String theory, because of the inelegance of its multiple solutions is being superceded by M-theory or Membranes, with branes or p-branes as fundamental phenomena. M-theory relies on supersymmetry. This supersymmetry transforms the coordinates of space-time so the laws of physics become the same for all observers, implying gravity.

In M-theory, everything may arise from strings, bubbles and sheets in higher dimensions of space-time. Membranes take the shape of them all. M-theory has 11 dimensions so it is more fundamental perhaps than strings. Gravity is the geometry of space-time. “Matrix” theory is based on an infinite number of zero-branes (particles). Their coordinates are not ordinary numbers but matrices, making space-time a fuzzy concept.

A testable holistic model is the Holy Grail of science, a “theory of everything”, or T.O.E. uniting all the forces of nature, including gravity. In principle it would explain everything from the macroscopic Big Bang to the most elusive microscopic phenomena. Yet it has remained elusive. However, science is only roughly 200 years old and growing exponentially.

Some philosophies, on the other hand, such as Kabbalah claim up to a 4,000 year lineage. The Jewish kabbalists had a holistic theory of everything including mankind’s relationship to cosmos depicted in their Tree of Life diagram. It is a holographic encyclopedia of phenomenology. It is a graphic display of the unity of micro- and macrocosm, of how man and universe interpenetrate and reflect one another.

Does our psychobiology somehow condition or reflect our philosophies? Reality seems to depend on how you look at it. Were the ancient mystics prescient? Were they somehow able to intuitively perceive the true nature of the cosmos by looking into themselves?

Intuition is a non-linear informational source which creates “quantum leaps” in our awareness. But mystics have also always spoken of the illusory nature of consensus reality. However, without the matrix of theory, data flies by incomprehensibly, its deeper meaning lost. If so, how does what the ancients discovered relate to current scientific findings? Can we know the nature of the universe if we do not even know ourselves?

The ancient philosophers corresponded the dodecahedron with the fifth element, Ether, or the Universe. We can examine its symbolic and phenomenological manifestations at the macro- and micro levels in both science and metaphysics.

The Shape of the Universe

“An analysis of astronomical data suggests not only that the universe is finite, but also that it has a specific, rather rigid topology (dodecahedral sphere). If confirmed, this is a major discovery about the nature of the universe.”

--George F. R. Ellis, “The Shape of the Universe,”
Nature, Vol 425, October 9, 2003

In early October of 2003, a team of French astrophysicists announced that rather than flat and endless, the cosmos is shaped like a soccer ball, with curved pentagonal-shaped panels. Their discovery comes from mapping the all-pervading microwave radiation released during the Big Bang over 13 billion years ago.

Fluctuations in temperature, (like waves in the ocean), at the extreme ranges of the radiation do not match the Standard Model. There are no large waves which would be expected in an infinite universe. Data showing fluctuating density is arguably consistent with the cosmos being a Poincare dodecahedral space. This finite spherical solid universe would be contained by 12 curved pentagons.

Independent mathematician, Jeffrey Weeks of New York points out,

“A journey of 60 billion light years across a dodecahedral Universe would bring you right back to Earth. Like a circumnavigation of the globe, it would be a seamless ride: there would be no obvious point at which one ‘re-entered’ the Universe.”

“The most distant objects would be visible in opposite directions, although they would be seen at different ages. Trying to spot the same galaxy in two different places “would be like trying to recognize the same person viewed at age 50 face-on, and at the age of 7 from the top of their head, in a crowd of billions.”

Synergetics and the Dodecahedron

“All that is physical is energetic. All that is metaphysical is synergetic.” (Fuller, 1979, p. 68)

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. It bears on the nature of reality from the micro to macro level, and is the basis of atomic structure and quantum cosmology, and life itself.
If we look at the structural matrix of the universe, atoms, or life we see the same equilibriating geometrical architecture repeating over and over. The human body shares the same structural laws as Cosmos. This echoes the axiom “As Above; So Below.”

Tensegrity Structual Matrix

According to the fundamental laws of physics as described by Ian Stewart (1998), nature likes to conserve, or minimize, the energy it has to expend in order to do its work. The original proponent of this view was R.Buckminster Fuller (with E.J. Applewhite), who expressed it mathematically and philosophically in his tour de force, Synergetics: Explorations in the Geometry of Thinking (1975) and Synergetics II (1979).

Fuller discovered Nature's own rules of assembly. His vision was founded on the geometry of closest-packed spheres, which can be found in the nuclei of all atoms. In fact, there is much to link the nature of Fuller's primary modules, the self-assembling tetrahedron and the Vector Equilibrium Matrix to the virtual vacuum or quantum foam, as well as to the dodecahedron.

Nature's own economy and minimalism is the reason why: (a) "the surface of smallest area that encloses a given volume is a sphere"; (b) "Without some constraint, the area of minimal surface would be zero"; and, (c) "Minimal surface" is a surface whose area is the smallest possible, subject to the following constraints: the shape's surface must contain some given volume, and its boundary should lie on some given surface or curve, or both (p. 104).

Thus it can be said that in its material form, energy seeks to equilibrate itself into a perfectly energy-efficient, completely energy-balanced shape of a sphere. Its underlying "assimilation and accommodation" dynamics therefore all seem subsequent to this basic geometrically equilibrative law of energy conservation. Therefore, the distinctive shapes which virtual/actual particle/energy wave patterns reciprocally evoke depends upon their underlying energy-redistribution dynamics.

In a notable Scientific American article (1/1998), Donald E. Ingber revitalizes the current of Synergetics by identifying the prominence of tensegrity in geometric shapes (including the dodecahedron) in "The Architecture of Life." He relates it to complexity (chaos theory) at macro and micro levels, stating in his introduction:
Life is the ultimate example of complexity at work. An oganism...develops through an incredibly complex series of interactions involving a vast number of different components...[which] are themselves made up of smaller molecular components, which independently exhibit their own dynamic behavior...Yet, when they are combined into some larger functioning unit--such as a cell or tissue-- utterly new and unpredictable properties emerge, including the ability to move, to change shape and to grow....That nature applies common assembly rules is implied by the recurrence--at scales from the molecular to the macroscopic--of certain patterns, such as spirals, pentagons and triangulated forms...After all, [everything is] made of the same building blocks: atoms of carbon, hydrogen, oxygen, nitrogen and phosphorus. The only difference is how the atoms are arranged in three-dimensional space (p. 48).

Ingber goes on from there to describe this emergent phenomenon as a process of "self-assembly" (p. 48) into increasingly complex hierarchies of life forms. He states his observation that nearly everything in our world, including the human body, is constructed using a form of architecture known as tensegrity (p. 48).

He explains, "The term refers to a system that stabilizes itself mechanically because of the way in which tensional and compressive forces are distributed and balanced within the structure" (pp. 48-49).

The key point here seems to be that the stability (or resilience) of a tensegrity structure comes not from the strength of its individual member-parts, but from the way that its mechanical stresses are balanced and distributed across all of the parts of the whole.

Ingber describes two categories of tensegrity structures, the first of which is made up entirely of "rigid struts," each of which is able to bear either tension or compression, and the second of which is composed of "prestressed" structures which bear either tension or compression even before being subjected to external forces.

“The compression-bearing rigid struts function to stretch, or tense, the flexible tension-bearing members, or "cables," while the tension-bearing cables, in turn, compress the struts. Thus, these "counteracting forces, which equilibrate throughout the structure, are what enable it to stabilize itself" (p. 49).

A closer look at some of the other interesting and relevant features of tensegrity shows that:

(a) It is the constructive, architecturally equilibrative use of gravity which gives most structures their stability by taking advantage of its continuous compression forces;

(b) Tension-bearers "map-out" the shortest path between adjacent members, resulting ideally in highly resilient geodesic-like shapes;

(c) Tensional forces, in turn, follow these accommodative shortest routes between points so that their tensional stresses become adaptively assimilated as new functions of the structure's resilient, ever equilibrating form (Ingber, p. 49-50).

When studying the functions of tensegrity structures involved in the make-up of cells, Ingber found that when attached to a flexible substrate material, cells contract and become more spherical, thereby "puckering" the material beneath them. So it seems that the tensegrity dynamics of any given structure, even a living one, can have a significant rippling effect on the dynamics of its surrounding neighbors, much like mass distorts or curves space at the macro level.

More significantly, it was seen that pushing down on a tensegrity structure forces it into what appears to be a flattened, disordered state. But as soon as the pressure is removed from it, "the energy stored in its tensed filaments causes the [structure] to spring back to its original, roughly spherical shape" (p. 50). This demonstrates that when tension and compression ratios are evenly distributed across a structure's member-parts, the structure will resiliently rebound from traumatic stressors (Miller, 2002).

As shown above, in cellular tensegrity structures all the way up to and including the human body, all interconnected structural elements rearrange themselves as needed in response to local stressors. So, in effect, the body varies the stiffness or flexibility of its bones, joints, tendons, and muscles in response to demands made upon it.

Also, cellular structures stiffen or relax their various cytoskeletal parts through contraction and extension of their minute microfilaments, microtubules, and intermediate filaments in response to its structural integrity needs. This is important because some research shows this bears directly on our own consciousness, (Miller, 2002, Part IV).

Furthermore, molecules too rearrange their shapes in order to communicate their reactions to the electro-chemical dynamics which influence their structural integrity. Thus, all levels of the human body's structure are simultaneously and continuously increasing their tensegrity states as much as possible.

Reciprocal limitations set the corresponding state of the "structural matrix" to which and within which they are attached. Elsewhere in science, the notion of reciprocal limitations is referred to as "rein theory."

Geometric Philosophy

We can expect these same tensegrity principles to apply at the cosmic level, astrophysics. 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 or curving of the structure of space-time.

Physicist Ian Barbour writes that, " 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 microcosmic 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 curved 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, Southeast Asia, Egypt, Europe, and Great Britain, to name a few. It is expressed in monuments, tombs, temples, church and mosque construction, astrology, the arts, mandalas, yantras, stupas, magical circles and symbols, etc.

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.

Pythagoras considered the geometrical nature of the dodecahedron so secret he kept it from the uninitiated. He realized the dodecahedron is related to PHI, the Golden Mean proportion. Their relationship is fractal, meaning self-reiterating at all levels, disclosing a profound interrelationship. The dodecahedron, nested within an icosahedron, nested within yet another dodecahedron, exemplifies Golden Mean ratios. Ratios established can be used to infinitely extend the figure to any scale, small or large.

A dodecahedron can be constructed by taking three golden rectangles and assembling them at 90 degree angles. The figure formed is a 3D shape with twelves corners which are also the twelve centers of each of the twelve pentagons that form the faces of the dodcahedron. Our modern science has never forsaken the tradition of seeking the understanding of forms that reveal beauty, shape and meaning in physical reality.

Euclidean geometry describes the nature of the human scale, but non-Euclidean models describe 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 Fuller's 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." Since Plato’s main injunction is “Know Thyself”, we can infer that to some extent to know oneself one must know geometry, or, conversely, to know geometry somehow helps us know ourselves and cosmos. Plato was clearly a Pythagorean. He fled to Sicily and studied there around the time of the politically-enforced death of Socrates. 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."

Synergetic Topology

Fuller’s Synergetics, a dynamical topology, is not easy to describe or conceptualize. He calls his combination of topology and vectoral geometry, a “complex hierarchy of nuclear system intertransformabilities with low-order numerical and topological relationships.” It is a whole-systems geometrical model of energy configurations  vectoral modeling.

In other words, they are nature’s most economical forms and motions, based on the tetrahedron and great circles. They model nature and sensible reality; synergetics is the coordinate system of nature. Fuller contends, “forces in both macrocosmic and microcosmic structures interact in the same way, moving toward the most economic equilibrium patternings.”

Synergetics is energetic geometry, identifies energy with number, and employs 60-degree coordination based on closest-packed spheres. This includes the lattices of all atoms. Energy has shape. The synergetics of primary systems based on closest-packed spheres is based on the following:

Symmetrical omnitriangulated forms: tetrahedron, octahedron, icosahedron;

Omnitriangulated and omnisymmetrical forms: cube, diagonal rhomic dodecahedron, rhombic dodecahedron, dodecahedron, tetraxidecahedron, triaxidecahedron, enenicontrahedron.

The prime unit of synergetics, the Vector Equilibrium or cuboctahedron, is locally mixed symmetrical and asymmetrical.

Since the physical is always special case, Fuller spells Universe with a capital U. He called the physical Universe, “a discontinuity of such islanded events.” He called them “energetic-synergetic events.” Comprehensive Universe combines both the physical and metaphysical Universe. Asymmetry and disequilibrium are physical; symmetry an equilibrium are metaphysical.

Synergetics is concerned only with physically demonstrable phenomena. Matter never ceases its motion, and when it moves it does so synergetically. Only the metaphysical can “designedly organize the phyical.” Only the physical is alterable; the metaphysical is unalterable. Synergetic advantage flows from the macro to the micro, but not the other way around.

Fuller described how “nothing” becomes “something” through triangulation and geometric energy transforms based on the tetrahedron, octahedron, icosahedron, dodecahedron, and vector equilibrium from the smallest to the largest “things”. Size depends on frequency which is cyclic.

Conceptuality is metaphysical and independent of size. “Universe is, inferentially, the biggest system,” says Fuller. In this comprehensive system of energy processes, Universe is a nonsimultaneously potential vector equilibrium. This is a metaphysical assertion.

According to the French astrophysicists, Universe is physically a pentagonal dodecahedron. We might add, at least that is arguably true for now, as dynamic transforms seem typical of all such matrices.

According to Fuller’s theory, it may be an infinitely transforming propagating pulsivity. “Universe is not a system. Universe is not a shape. Universe is a scenario.” He calls our Universe “finite but nonsimultaneously conceptual.” He says we cannot simultaneously conceptualize Universe except as a scenario of interacting macro and micro systems. We may get an intuitive perception of its magnitude through our human fractal relationship to the greater whole. But it is more than mere analogy, for we are that in the holographic sense that the part contains the whole only in lower resolution.

The containment of somethingness by uncontained nothingness is modeled by the five omnisymmetrical Platonic polyhedra. These equiedged Platonic solids are the icosahedron, octahedron, cube, tetrahedron, and the regular dodecahedron.

“The chords of these five spherical geometric integrities all interact to produce those well-known equiedged polyhedra commonly associated with Plato. The intervolumetric quantation of these five polyhedra is demonstrated as rational when referenced to the tetrahedron as unity. Their surface values can also be rationally quantized in reverse order of magnitude by the 48 spherical triangles tiles in whole, low-order, even numbers. These hierarchies are a discovery of synergetic geometry.” (1975, 1053.21)

Anatomy of the Body of God

We have therefore discovered in the “Tree of Life,” properly Multiplied and Projected according to the Art of the Wise, what may well be looked upon as the Anatomy of the Body of God, or the natural structure of Matter under the Influence of the Concealed Spirit.” (Frater Achad, 1924).

The Great Work of the Qabalists has always consisted in finding the equivalence of the Macrocosm and the Microcosm, heaven and earth. Among the ten spheres of the Tree of Life these are represented respectively by the spheres Kether at the top of the glyph, and Malkuth at its bottom. There is a qabalistic saying about the essence of godhead or spirit in matter: “Kether is in Malkuth.” (See figure above).

Medieval texts describe the creation of the universe as an emanation from the negative veils of existence beginning with Kether, down the ten spheres of the Tree of Life through the archetypal, causal and astral planes to the material world of Malkuth. The prehistory of this traditional Tree diagram is unknown, but may be presupposed to come from the oral mystical tradition.

Sephir Yetzirah describes cryptic directions for constructing a cube extended through the diagonals of its dualing octahedron. This system reflects that of Cartesian XYZ coordinates we use for orienting in 3-dimensional space. This minimal form of dualing octahedrons and cubes is the lattice from which both single two-dimensional and cubic six-fold Tree of Life diagrams are formed.

Elsewhere (Miller, 1992) I have shown how this figure of interpenetrating cubes and octahedrons is related to Synergetics and kabbalistic geometries through the geometry of Fuller’s Vector Equilibrium Matrix.

The ancient Kabbalistic text, Sephir Yetzirah hinted that the Tree of life could project not only in one direction, but also to the heights and depths and in the four cardinal directions. This is the basis of the kabbalistic meditative geometry known as The Cube of Space (Case, 1947) or Merkabah Throne-Chariot (Work of the Chariot, 1970) [see below]. The Jewish kabbalists did not proceed, as far as anyone knows, to project the Tree further than a single repetition, except in a very linear, ladder-like manner.

Frater Achad, following his intuition, was the first to discover such precise geometric properties in the glyph as the Vesica Pisces and its tetrahedral nature. He proceeded to create nesting, all-space filling two dimensional depictions of the Tree (Macrocosmic Snowflake), then went on to model crystalline nested 3-dimensional Trees (Garden of Eden; see Introduction). The true Kether is concealed within the Malkuth at the center of this solid all-space filling figure.

The conception of the universe as a dodecahedron appears to have originated with Plato. Without any awareness of the quantum universe or synergetic energy transformations, Frater Achad was accustomed to thinking of hyperspaces in terms of the qabalistic Cube of Space and Euclidean geometry. His model united Greek (Platonic; Pythagorean) and Hebrew (Qabalistic) models.

He summed up the main result of his research as a dodecahedron within a perfect sphere. If he thought synergetically, he would have spoken of curved pentagonal surfaces. Plato and Plutarch (Republic, Books II and II) imply that the god Vulcan erected twenty tripods in the heavens, which are the twenty solid angles of the dodecahedron formed from the junction of three lines. This framework was bounded by twelve equal and equilateral pentagons.

By starting with a finite center point of Malkuth, or the Sphere of the Earth, Frater Achad projected and multiplied the Tree out until it had 20 vertices, each representing a Kether. He identified this “Triangular Crystal Prism” with “The Stone of the Wise”, and considered it the true representation of the Philosopher’s Stone.

But it may turn out to be more than symbolic if scientists confirm that the universe is, indeed, shaped like a dodecahedron. It may be more than metaphorical, containing the true essence of the universe, from the infinitely small to macrocosmic dimensions. Elsewhere (Miller, 1992) I have shown how the Tree of Life geometry reflects actual microcosmic atomic structure.

Four of Achad’s blocks of Five Trees form a solid capable of progression in all directions so as to fill every dimension of known space. The nature of the whole figure is that of a perfect dodecahedron or twelve-fold figure, each side of which is a perfect pentagon. At the Center of All is Malkuth -Achad’s “Garden of Eden” figure (Anatomy, Chapter 9) represents the first or simple prismatic "Tree" branching out in 20 directions from its central Malkuth. If the second progression of the solid is taken to Tiphereth it exactly encloses this in a perfect dodecahedron comprised of 20 solid angles forming 12 equilateral pentagons. The Fifth progression of the original Malkuth exactly encloses all this in a perfect sphere.

When connected, the points of the 20 Kethers form twelve perfectly regular pentagons. Thus the whole structure indicates a perfect dodecahedron and, when enclosed in the circumscribing sphere, touches it at eighty points.

The whole solid is capable of expansion by means of the progression of the unit "Trees" as explained before. Likewise, it is possible to imagine the infinitely small reduction or microcosmic contraction of the whole figure.

Achad (Anatomy, Ch.10) points to references about the dodecahedron in Madame Blavatsky's Secret Doctrine, the foundation text of Theosophy, and unacknowledged basis of most New Age thought:
"The most distinct and the one prevailing idea, found in all ancient teaching, with reference to Cosmic Evolution and the first 'creation' of our Globe with all its products, organic and inorganic. . . is that the whole Kosmos has sprung from the Divine Thought. This Thought impregnates Matter, which is co-eternal with the One Reality; and all that lives and breathes evolves from the Emanations of the One Immutable Parabrahman-Mulaprakriti, the Eternal One-Root.

“The former of these, in its aspect of the Central Point turned inward, so to say, into regions quite inaccessible to human intellect is Absolute Abstraction; whereas, in its aspect as Mulaprakriti the Eternal Root of All, it gives one at least some hazy comprehension of the Mystery of Being.

"Therefore, it was taught in the inner temples that the visible Universe of Spirit and Matter is but the Concrete Image of the Ideal Abstraction; it was built on the Model of the First Divine Idea. Thus, our Universe existed from eternity in a latent state. The Soul animating this purely Spiritual Universe is the Central Sun, the highest Deity Itself.

“It was not the One who built the concrete form of the idea, but the First Begotten; and, as it was constructed on the geometrical figure of the dodecahedron the First Begotten 'was pleased to employ 12,000 years in its creation.' "

It would seem, from the above, that the "Model of the First Divine Idea" may have been very much in harmony with the ideal Formative Principle we have been studying.

Plato says we may liken the receiving principle to a mother, and the source or spring to a father, and the intermediate nature to a child. It reminds immediately of the mythopoetic union of Earth and Heaven and the Life of Nature, which is its offspring.

Plato describes three kinds of being:

) that which is uncreated and indestructible, changeless, eternal, imperceptible to any sense, open only to the contemplation of the intelligence, and this is the principle of the Father [spirit], the ideal or formal essence of the world;
2) that which is sensible and created and always in motion, the Child [matter], the world of change and life;

3) the Mother [spacetime], like the Father, is eternal and admits not of destruction. She provides a home for all created things, and is apprehended 'without the help of sense, by a kind of spurious reason, and is indeed hardly real.'
This mothering space is the cause of the determinism of nature. Plato identifies it as the material element of being. As pure matter, it is purely indeterminate, but it is receptive. The four elements, earth, air, fire and water, are formed from it, for 'the mother substance becomes earth and air, insofar as she receives the impressions of them.'

Plato's conception of the formation of these elements from the original substance was as purely mathematical as are our modern physical notions. 'God fashioned them by form and number,' he says: and the forms which he assigned were the forms of the regular solids. Thus the form of the fiery element is the pyramid, of air, the octahedron; of water, the icosahedron; of earth, the cube.

The fifth solid, the dodecahedron, is the form of the universe as a whole, or perhaps one might say the scaffold upon which the spherical universe is constructed.

Further, these elements are themselves compounded of simpler mathematical forms, the tetrahedron, octahedron and icosahedron of equilateral, the cube of isosceles triangles; so that if we regard the elements as molecules, we may view the triangles as atoms of the material substrate.

This geometrical account of matter gave rise to the saying ascribed to Plato that 'God always geometrizes'. God, says Plutarch in his commentary on the saying, made the world in no other way than by setting terms to infinite and chaotic matter.
Achad’s complex solid contains those elements mentioned by Plato. It should be remembered that the Qabalists attribute the four elements to Malkuth, which is often called the Sphere of the Elements. Malkuth has remained a perfect sphere in our plan and in fact represents the material substance of the Universe.

Earth, as Matter, has always been symbolized by the Cube, or the Cube within the Sphere, and we may well consider this Cube to be concealed in the Sphere of Malkuth. When the progression is made to include Tiphereth, the twenty equilateral planes become points, or tripods which mark out the dodecahedron, while the Second Progression of the Tree exactly encloses this. The points of the twenty Kethers indicate a similar but larger solid.

A Mandala of Perfected Earth

In the symbolism of both Hermetic and Jewish Kabbalah, “earth” represents not only the planetary sphere we live on, but also the whole of the physical plane, the realm of manifest matter. It is a way of representing divinity and the Godhead in matter through sacred geometry.

Leet (1999) speaks in kabbalistic terms of many of the astral plane properties of such a figure and how it is related to and derived from the Lurianic Tree. Thus the dodecahedron represents the divine personality immanent within both Earth and the cosmos. In this way, the earthy element can mate with her more transcendent partner. The transcendent element contains the entire creation, and is reflected in it at every level.

The kabbalists have an axiomatic expression which is their prime formula: “As Above; So Below,” and what could better represent this continuity than this ideal geometry. Even mystic William Blake depicted the Ancient of Days as a geometer.

Leet presents this figure to demonstrate “the only true orthogonal projection of the dodecahedron that can be geometrically allowed in relation to the Sefer Yetzirah,” which is contemplated as well as its final dualing with the icosahedron. She says it “can represent both the Yichud of those Male and Female Partzufim and also the icosa-dodeca crystalline pattern of this earth.” (p. 287).

The Sefer Yetzirah Tree enclosed within an orthogonally projected dodecahedron may be read as “the consummation of the cosmic mating of Ze’ir Anpin and the Nukvah toward which the entire process has, in Lurianic cosmology, been progressing.”

This repeats our theme of cosmic mating as described by Plato, the kabbalists, and Mme. Blavatsky’s Theosophy. Could this “mating” include the interactive dynamic of Fuller’s tensegrity?

“In this cosmology, such a Yichud is dependent upon the commencement of the Tikkun, and this is also borne out by the geometry of the interconnected Sefer Yetzirah Tree and the Sabbath Star diagram. For if this final dodecahedron is to be given a tonal name, it can be seen to be contained within the inner hexagon of the Fi Sabbath Star that marks what can be considered the turning point of the cosmic octave between Tzimtzum and Tikkun.”

According to Leet, this mandala of the earth is being perfected by the Tikkun. What does it mean to perfect or restore the world of matter, or the physical plane? Some schools of Kabbalah (Lurianic, Sabbatian) contain a controversial doctrine of Universal Correction (Tikkun) that restores the existential break or gap in Absolute Being which was broken by the shattering of perfection in the Big Bang.

This restored world is not a pre-creation potentiality, however, but a concrete unity of differences and actuality. Raising the “holy sparks” dispersed during creation is the purpose of mankind working in partnership with God on the redemption of the universe. Malkuth is the immanent aspect of divinity.

As the kabbalists say, “Kether is in Malkuth.” Spirit suffuses matter. Malkuth is the turning point between Emanation and Return. The whole endeavor of mystical aspiration is called the Path of Return, and involves working one’s way philosophically back up the experiential Tree of Life to Kether by intentional participation in the Great Work.


No matter which geometrical description of nature and cosmos we start with, from Buckminster Fuller’s Synergetics to ancient and modern Qabalists, secular or sacred geometry, we seem to wind up with the dodecahedron as a fundamental element.

Much as the alchemists searched for the Godhead in matter (Miller & Miller, 1994), modern scientists seem to have found it there, whether specifically looking for it or not. We could actually argue that they have found the Philosopher’s Stone in the dodecahedral universe.

This strongly suggests that the archetypal character of the dodecahedron tends to repeat in intuitive thinking, whether spiritual or scientific. This image, arising in different contexts, comes up in different cultures periodically as a mandala of wholeness. Jung (1964) spoke of the significance of regular geometries and mandalas as images of the Self, part of the pattern of psychic growth.

Scientists have “redeemed” or restored this polygon in philosophical form through a modern projection or model of the Universe as a Dodecahedron, whether their scientific hypothesis proves out and holds, or not.

"For geometry will lead the soul toward truth and create the spirit of philosophy."

Iona Miller:


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Case, Paul Foster (1974). The Tarot. Macoy Publishing Co.

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(1979). SYNERGETICS II. New York: McMillan Publishing Co.

Greene, Brian (1999). The Elegant Universe: Superstrings, Hidden Dimenions, and the Quest for the Ultimate Theory. New York and London: W.W. Norton & Co.

Jung, Carl (1964). Man and His Symbols. London: Aldous Books, Ltd.

Leonora Leet (1999). The Secret Doctrine of the Kabbalah. Rochester, VT: Inner Traditions, Intl.

Luminet, J.-P. et al (2003). “Dodecahedral Space Topology as an Explanation for Weak Wide-Angle Temperature in the Cosmic Microwave Background.” Nature, 425, p. 593-595.

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Miller, R. A., Webb. B., “Embryonic Holography,” Psychoenergetic Systems, Stanley Krippner, Ed. Presented at the Omniversal Symposium, California State College at Sonoma, Saturday, September 29, 1973. Reprinted in Lyttle's journal Psychedelic Monographs and Essays, Vol. 6, 1993. 137-156. Also in JNLRMI 2003 at

Miller, R. A., Iona Miller, and Burt Webb (2002). Quantum Bioholography: A Review of the Field from 1973-2002. JNLRMI, Oct. 2002.

Miller, Iona and Miller, Richard (1981). The Diamond Body: A Modern Alchemical View of the Philosopher’s Stone. Grants Pass: O.A.K.

Miller, Iona (1992). “The Diamond Body: Buckminster Fuller and the Qabala.” Grants Pass: O.A.K.

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For more on Achad's geometric vision see
or google other sources.

  Vector Equilibrium Matrix entwined with Achad's Macrocosmic Snowflake

Macrocosmic Snowflake & VEM

  Trees of Life radiate omnidirectionally, filling all-space and their Kethers form a dodecahedron universe, according to Achad and recent French cosmological theories. x

Achad's Garden of Eden figure

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