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by Bethe Hagens


Geometry:  An Ancient, Universal Thinking Model?

There is a secret stone, hidden in a deep well,

worthless and rejected, concealed in dung or filth.

It is a thing which is found everywhere,

which is a stone and no stone,

contemptible and precious,

hidden, concealed, and yet known to everyone.1


So many metaphors of thinking are encapsulated in legends of this secret alchemical stone, the Philosopher's Stone, that it is almost impossible to know where to begin. Buried within the self, within the everyday world, within life itself, is a common essence—a  shape, a creative vessel in which the elements of creation are mixed and transformed. This "stone which is no stone" is intangible—as deep as Breath, as profound as Love, and as intelligent as Light. It is an ideal form, one that can be grasped by the intellect but never actually seen. Eternal and unchanging, it is the sacred container of ever-changing cosmic processes. The Rock of Ages. It is an image that sheds light on form, gives the sense of predictability and connection—memory—to pieces of thought that might otherwise seem random. Thinking is a highly mysterious alchemical process, a sublimation of heart and mind.




Plato’s ideal divine feminine receptacle, perhaps the hidden stone upon which the Demiurgos fixed his eye as he created a living World “designed to supply its own nourishment from its own waste.”    (Image © Bethe Hagens 2006)

Plato used this shape to organize his teachings about the origins of life.  He called it the ideal body of the cosmos, the Mother of Becoming.  He describes it as a sphere, composed of 120 identical triangles, that "contains" the five dynamic elements of creation: Fire, Earth, Air, Water, and Aether (Life Energy).2 Spiritual leaders among many tribes in native North America had a remarkably similar vision.  The Sioux universe, for example, is also spherical and contains the same five creative elements.  In the Beginning, they say, the universe was composed of numberless “hoops” (circles), each a kind of skeleton with no substance.3 All was orbits within orbits within orbits. Earth came into being when the Creator called together “the sixteen hoops.” Fifteen of them formed the body of Earth, and the sixteenth located her in relationship to the Sun upon a common circular plane—the ecliptic. Plato did not think in terms of circles, but in fact, it is exactly fifteen interlocked circles that creates his spherical container of 120 identical triangles.

Very broadly defined, cosmology is the legacy of human thinking about the universe as a whole—about the contents, structure, and evolution of the universe from the beginning of time to the future. Most, if not all, of the world's cosmologies assign primary dynamic roles to the same five elements Plato identifies.4 Each has a unique and indispensible role in the cycles of transformation of the living material cosmos—that is, in what we would call the ecosystem, environment, Gaia, or Mother Earth.  Fire  is purification; Earth a principle of materialization; Air a cosmic cycle of breath; Water flow; and Aether an ineffable refinement.  In the Platonic cosmos, each of these elements was symbolized by one of five perfectly symmetrical geometric shapes, the so-called Platonic solids.  These same shapes were also well-known and modeled by many earlier cultures, though geometry (the art/science of “Earth measuring” or, alternately, “Divine mothering”) is usually attributed to the Greeks.5



 Plato assigns one of five “perfect shapes” to represent each of the elements of the cosmos:  Tetrahedron (Fire), Cube (Earth), Octahedron (Air), Icosahedron (Water) and Dodecahedron (Aether/Life Energy). In Elements, Euclid explains why these “Platonic solids” are a complete and unique mathematical set.    (Image © Bethe Hagens 2006)


These perfect geometric shapes are all "contained" in Plato’s spherical cosmic container.  Each of the five shapes can be positioned inside the 15-hoop, 120-triangle sphere so that every one of its corners will fallon a corner of one of the sphere’s triangles.  The divine container organizes forms.  It isn’t visible, but it holds everything in order and is in this way the masterplan of natural structure and “found everywhere.” In the same way, it organizes thinking and perception.

The five perfect shapes provide “frames” for perception that are so easily remembered and recalled that we “see” them at virtually every scale imaginable, from the topography of planet Earth to crystals, pollen grains, plankton, and molecules 6.  The microscopic protein shells of many common viruses, for example, are described by scientists as a structural meshing of two of the shapes—the icosahedron (Water) and the dodecahedron (Life Energy).  These viruses can be thought of as crystalline versions of the perfectly spherical Philosopher's Stone.  Even Plato's elemental symbolism applies!  Viruses are currently being identified as primary agents of the evolutionary process (Life Energy) that can proceed only within the fluids of a host cell (Water). The words virus and environment stem from an identical linguistic root, vir.


The unique compatibility of the icosahedron (20 faces, 12 corners) and dodecahedron (12 faces, 20 corners) symbolizes the sacred relationship of water to life. Within the sphere, the corners of each figure fall at the centers of the faces of the other. This geometric relationship is fundamental to both ancient and modern ways of constructing reality: (Clockwise from upper left) reed sphere from South East Asia; soccerball; human papilloma virus, and ancient Etruscan bronze die.   (Image © Bethe Hagens 2006. The Etruscan die was retrieved from http://home.tiscali.nl/~mwduga90/Dodecahedron/ and has been slightly graphically altered.)


Plato arranged the five elements in order of increasing geometric complexity, from the most basic (the tetrahedron) to the most complex (the dodecahedron). Scientists today think in terms of an identical geometric hierarchy to explain principles of molecular and cellular growth and bonding.  The most basic molecule, for example, is modeled as if it consisted of four atoms spaced equidistantly from each other at the corners of a tetrahedron (which is Plato's first element, Fire). Identical thinking describes the miraculous growth of a fertilized egg cell. The initial division of the egg into interconnected halves of a sphere is followed immediately by a second division that creates a blastomere, a tetrahedral cluster of four cells.

The Periodic Table of the Elements is an entire system of thought organized around these same five geometric shapes. Elements with the same molecular base shape are thought to—and do—have many broadly analogous physical properties.  For example, the molecular base shapes of gold, copper, and silver are all cubes. Chemical bonding of molecules with different base shapes is possible because of their unique structural transformability and ability to "link" within the corners of the 120-triangle perfect cosmic container, the invisible spherical Philosopher's Stone.

At a macroscopic level, Earth itself is now being modeled as a kind of giant aggregate molecule, a composite body gravitationally bonded or "fused" together out of numerous "planetesimals" that once orbited the sun.  Scientific thinking is that the intense heat of the Earth at the time of its birth (plasma or Fire) drove lighter elements (solids or Earth) upwards to form the crust of the planet. Over time, gases (Air) and bubbled up through cracks in the crust. As the planet cooled, liquids formed (Water) and water carried in by comets seeded biological transformation and evolution (Life). This order of elemental transformation is identical to Plato's.

The ancient Chinese conceptualized a cosmos of Fire, Earth, Metal/Air, Water and Wood using a pentagonal Chart of Phases 7. Each dynamic phase in this system also represented a color, a function of an organ in the human body, and a position (rather than a shape) in the universal scheme of creation and destruction. The Creative Order (symbolized by the pentagon) led to increasing material complexity. The Destructive Order (symbolized by the pentangle) led to progressively more subtle states of matter. No direct evidence survives to indicate whether or not ancient Chinese philosophers also used geometric shapes to represent the five phases in the Chart, but it seems almost certain that they did. Though most ancient Chinese thinking is now long lost to book burnings, Plato again provides a point of linkage, for the colors he uses to represent each of the five elements is identical to those of the Chinese. In addition, the Creative Order in the Chinese chart mirrors Plato’s hierarchy of geometric complexity.


The philosopher’s stone is a universal alchemical vessel within which elements of thought vastly separated in time, space and intent can be mixed and sublimated. (White arrows – creative, generative. Black arrows – destructive, overcoming.)    (Image © Bethe Hagens 2006. Permission may be needed for the DaVinci image?)

There is a faint trace of the Platonic shapes in Buddhist stupas (or sotobas), traditional grave stones that are still used throughout India, China, and Japan.8 These markers are stylized miniature replicas of the great architectural dome stupas that contain purported physical remains of the Buddha and that symbolize the eternal return of matter to the real world of spirit. The base of these stupas is a cube that represents Earth, the stable foundation upon which all is built. Stacked upon the cube are, in order: a sphere, which represents Water; a triangular shape representing Fire, the symbol of elements in transformation; a crescent symbolizing Air, the inverted vault of the sky and the wind;  and finally a tapered sphere—Aether dissipating into perfect space. The elements of the stupa are always stacked in this order, one identical to the Destructive Order of the Chinese cosmos.


 The Native American stone medicine wheel is a two-dimensional representation of the three primary spiritual hoops which create Earth’s spherical body. Geometrically, this shape is a spherical octahedron. Inset: an identical octahedral sea creature, the microscopic radiolarian, drawn by Ernst Haeckel in the late 1800s.  (Image © Bethe Hagens 2006. The radiolarian is from Dover’s copyright-free set of images.)         



We can only guess at what must have encompassed the incredibly refined geometric thinking in the pre-Columbian Americas. Contemporary North American medicine wheels, for example, are cosmological thought processes. Large circles of sacred stones continue to represent the transformative, healing powers of the elements and their geographical orientation in the universal scheme of creation.  Like Stonehenge and other ancient megalithic astronomical observatories, these much smaller scale stone constructions functioned as calendar-clocks.  Every medicine wheel was intentionally designed to honor and to maintain the regenerative energies of a position in the cosmos unique in time, space, and transformative significance.  Each was built in harmony with the sacred, hidden, ideal order.  In addition, the stones were road signs that could be “read” by the rising and setting positions of the sun, moon, planets, and stars as they traversed their hoop-like paths around the Earth 9.

The Sioux holy man and visionary, Black Elk, used colors identical to those of the Chinese and Greeks to symbolize the five elements, as did many other tribes; but position in time and space, in the context of endless cyclic flux, seems to have far outweighed any importance that might have been attached by early American cultures to a progressional hierarchy of the elements. The thinking behind the medicine wheel merged element, season, direction, color and life form.  It served as a plan for ceremonial lodges and a compositional framework for sacred art.  It was a totality, an idea of life fully and properly lived.  The responsibility and privilege of being human was, over the course of a lifetime, to embrace and know each of the elements, thereby closing the hoop and completing an individual sacred circle.     


End Notes

1.  This poem is created from fragments of alchemical texts cited in Johannes Fabricius,           Alchemy: The Medieval Alchemists and their Royal Art (Wellingborough: The Aquarian Press, 1989), p. 21.

2.  Plato, Timaeus and Critias (New York: Viking Penguin, 1971).

3.  Ed McGaa, Eagle Man, Mother Earth Spirituality: Native American Paths to Healing Ourselves and Our World (San Francisco: HarperCollins, 1990).

4.   Bethe Hagens, “The Divine Feminine in Geometric Consciousness,” in Anthropology of Consciousness 17:1 (2006), pp. 1–34.

5.   Keith Critchlow, Time Stands Still: New Light on Megalithic Science (New York: St. Martin’s Press, 1982).

6.   Ernst Haeckel, Art Forms from Nature (New York: Prestel-Verlag, 1998).  Michael Schneider, A Beginner’s Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science (New York: HarperCollins Publishers, 1994).

7.   “Five Elements (Chinese Philosophy),” Wikipedia, the free encyclopedia.

8.   Adrian Snodgrass, The Symbolism of the Stupa (Ithaca: Cornell University, 1985).

9.   The hoops and “good Red road” of the medicine wheel are, according to Dr. Leslie Gray of Woodfish Institute http://www.woodfish.org/, an oral and visual mnemonic tradition that, even today, passed on to all who will listen but are rarely set down in print.

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