The DIVINE FEMININE
in GEOMETRIC CONSCIOUSNESS
by Bethe Hagens
Bethe Hagens, Ph.D . School of Public Policy and Administration Walden University
Anthropology of Consciousness, Vol. 17, Issue 1, pp. 1-34. ISSN1053-4202, © 2006 by the American Anthropological Association. All rights reserved. Please direct all requests for permissions to photocopy or reproduce article content through the University of California Press's Rights and Permissions website, www.ucpress.edu/journals/rights.htm
Plato spoke of geometry as an extremely rare natural gift of consciousness, but also as a mode of perception accessible to anyone initiated into the tradition by a master. Such teachers were the shamans of his day, and it was these individuals who convened the mystery schools and sponsored initiations into the esoteric philosophies of geomancy that have grown from their vision. Though the history of women in ancient mystery traditions is largely lost to us, Greek mythology holds that our human capacity for geometric vision is a gift of the divine feminine- energetic sources of wisdom conceptualized as a lineage of goddesses. Born from primal Chaos is Gaia, from whose name comes "geometry"-geo (earth) + metr (measure, mother). She gives birth to Mnemosyne, goddess of Memory, from whose name comes "mnemonic." The daughters of Mnemosyne are the Muses- the arts and sciences. Memory is the legacy of the sacred Earth, and the arts enable humans to actively remember. The essence of this divine feminine lineage is sustainability of sacred place (Chora) through enduring values of order, proportion, a universal aesthetic, and connectivity. In this essay, I explore the ways in which I learned to embrace myself as a physical and spiritual geomantic consciousness and how I have used geometric vision as an interdisciplinary teaching and learning process.
Keywords : sacred geometry, divine feminine, mythology, archaeoastronomy, visualization
World mythology and religion are replete with evidence of an ancient, eclectic, integrated geometric art/science (geomancy) in which certain principles of shape, numeracy and connection unified the experience of body, mind and essence through metaphors of Earth and the elements, All Beings, and Sky. 1 I entered the tradition through Plato's text Timaeus (1965) in which he describes an ideal etheric body composed of 120 identical right triangles (known technically as a spherical hexakis icosahedron) that organizes all matter, "above and below." A vibrating, invisible, female "container for becoming," it births the five dynamic elements of creation-Fire, Earth, Air, Water and Aether. 2 Each element is a geometric shape, a color, and a place in the order of creation: Fire is tetrahedron-red-first; Earth is cube-yellow-second; Air is octahedron-white-third; Water is icosahedron-black-fourth; and Aether is dodecahedron-green-fifth. Anticipating the synergetic geometry of R. Buckminster Fuller (1975) by nearly 2500 years, Plato reveals how the 120-triangle sphere geometrically encloses each of the five elements such that their corners fall only on its "corners."
Although the etheric 120-triangle spherical Receptacle is itself never visible, the architecture of the five perfect shapes she "holds" can actually be seen at virtually ever scale imaginable-from viruses, crystals, molecules, plankton, and pollen grains to Earth itself. Plato passed on to us an anciently derived anthropic principle: the Receptacle is an unchanging constant, a template of natural structure for a carbon-based world. In Phaedo, Plato graphically imagines Earth as seen from above as a ball composed of twelve patches of skin-analogous to a modern soccer ball and identical in its structural geometry to a third form of carbon discovered in the mid-1980s and named fullerene in honor of R. Buckminster Fuller.
Figure 1. THE COSMIC RECEPTACLE - GAIA (above) composed of 120 identical right triangles "holds" the five dynamic elements of Creation: (from left) tetrahedron, cube, octahedron, icosahedron, and dodecahedron.
This Receptacle- Gaia, the divine feminine-is known in the East as the Dao or "mother of all things." It appears as well in the arts and mythologies of many indigenous cultures in the Americas , but was conceptualized as hoops rather than triangles. In the Brulé Sioux myth of Creation, for example, All was numberless hoops within hoops. Primordial Mother Earth was composed of fifteen hoops to which the Creator called the various powers and manifestations of material reality including Sun, Moon, and stars. 3 This view does not diverge significantly from the ancient view that Plato himself had been taught. A closer look at Plato's geometry reveals that 15 symmetrically interlocked equators- great circles (or hoops) which divide the sphere in half-create the 120 identical right triangles of the spherical hexakis icosahedron. Sioux mythology deviates only where it parenthetically adds "local context"-a sixteenth hoop to represent Earth's orbit around the sun (the ecliptic). The color-element symbolism in the cultures of the Americas is broadly consistent with Plato's, but references to the five shapes are obscure and often set in the context of secret initiation rituals. 4 Throughout this essay, when I invoke the divine feminine as geometric consciousness, I am referring to the 15-hoop/120-triangle Receptacle.
Figure 2. The Russian geometric map (left) symmetrically nests an icosahedron (composed of 20 identical equilateral triangles) with its "perfect dual", the dodecahedron (composed of 12 pentagons) so that the corners of each figure fall at the centers of the faces of the other. Becker immediately recognized the geometry of geodesic domes in this map, but felt it was missing "major struts" that were identified only as arrows of force. The Becker-Hagens planetary grid model (right) adds these struts and is based upon the synergetic geometry of R. Buckminster Fuller as well as Plato's divine feminine Receptacle described in Timaeus .
In 1982, designer-mathematician Bill Becker introduced me to an obscure whole-earth mapping system developed in the late 1970s by an engineer, linguist, and historian in Russia (Bird 1975). The team of researchers had used an interlocking framework of spherical icosahedron and dodecahedron (see Figure 2) to explain patterns of Earth's geography, topography, climate, animal migrations, ocean currents, and the siting of ancient civilizations (among other phenomena). It totally captivated me, and years of future teaching and research almost instantly flashed through my mind. The map's allure was its capacity to "store" information across a wide variety of fields in a single visual model. Also, it was efficient-a modular system that "packed" the Platonic solids as well as other geometric solids into a single container. It was a woman's tool, a tool for multitasking! And I had not yet read Plato . . . The whole idea seemed, literally, divinely inspired to me . . . so much so that I went to Moscow to meet one of the inventors, engineer Valery Makarov, and was astonished to learn that in traditional Russian schools geometry and geography were always taught together from the very earliest grades. Once my eyes had opened to this connection, I began to see traces of the Platonic geometries everywhere. I found my skills in pattern recognition expanding exponentially, and for the next ten years, my undergraduate anthropology and geography students used a very similar map to explore the planet and its cultures. 5
This "planetary grid project," as it has come to be known, was done in full collaboration with Becker, a professor of industrial design at the University of Illinois-Chicago. Following his engineering knowledge about how matter connects, we slightly modified the original Russian map by adding what Becker identified as missing structural supports in their "dome" over the earth. Simultaneously, we read Plato's Timaeus and I began taping the geometry onto a world globe. The addition of the "struts" revealed the 15-hoop/120-triangle pattern and we immediately realized that we had created in three dimensions the divine feminine Receptacle that Plato had struggled to describe with words.
The experience was so powerful that I assigned globes and tape, rather than textbooks, to my students. Where Bill had seen a Bucky Fuller dome, a nursing student saw a blastomere. Another saw a virus. I saw the three aspects of God in the structural properties of the three corners of the triangles. We found that we were not only communicating across disciplines as diverse as world mythology, structural engineering, and biochemistry, but we had a way to store our insights and discoveries in a relatively easily retrievable format. This worked even for the kinds of anomalous information that become relevant only years after they have been lost or forgotten. The spherical geometry proved extremely effective as a mnemonic device. I found that the students could use the corners and "struts" of the model to retrieve, compare and contrast cross-cultural and interdisciplinary data more quickly and comfortably than with any other system I'd ever tried. The Mother Sphere (as I called it then) became my template for most everything I observed, to the point of near, if not total obsession. 6
Late one evening in 1992, bleary-eyed and frustrated over a piece of writing that would not come together, I looked up from my computer and found myself staring at a dusty sphere of the constellations that sat on a shelf next to my Earth globe. It had been a gift from a friend many years before, and I kept it only because its dazzling blue color made me happy whenever I glanced over at it. Apart from that, I almost defiantly ignored it. I knew nothing about astronomy, and I had absolutely no desire to learn about the sky since my focus was the Earth. I had no idea what a celestial sphere mapped or how to use it. But this night would prove to be different when the meaning of the celestial sphere dawned on me not as cognition but as what I suppose could be labeled "embodied experiential intellection." I had been trying to force a secular, material description upon an Earth alive in my consciousness as a living crystal being whose etheric geometric skeleton could be mapped in its patterns of energy flows . . . in ocean currents, the winds, river systems, and distributions of precious minerals. It seemed to me that ancient humans had known this sacred, hidden body of Earth and had settled on it in ways that took advantage of very visceral powers of place. It was in the midst of this struggle to appropriately language my increasingly ecstatic visions of this Being, alive with cubes and music and a rainbow of colors, that everything went black. My perceptual mind shut down.
I still don't know exactly what happened, but I had the instantaneous and complete realization of being Everything and Nothing, of shimmering in the Dark somewhere between frantic desire to create and shattering loneliness. As suddenly, my being centered on the plane of Earth's orbit around the sun and my "waist" extended into infinity through the center of the galaxy. Over my "shoulder" was the brilliant Light sash of the Milky Way. I was absolutely overcome with the perfection of Earth's place in it all. It was an "adult" vision, an unbelievably personalized and intimate release from intellectual obsession and the conventional confinement of material consciousness. Even more, I remembered this experience from my childhood when I had many times found myself trembling in this blackness, terrified that it was Death. Now I realized that all along I'd been experiencing the Divine Feminine-Gaia, the Cosmic Receptacle that Plato so emotionally described in Timaeus and that I'd been using to map Earth. The living celestial sphere, Sky, opened to me and became as much an extension of myself as a friend. I went into retreat for more than a year.
My "night vision" was so total, so compelling, that I ultimately left my position as Professor of Anthropology at a state university in Illinois and began to follow an unknown path into myself and the cosmos. As I studied my desk-model celestial sphere, I believe I began to see the hoops described by Black Elk, and soon I could see them in the night sky as well (see Figure 3). I began communicating in some depth with astrophysicist Tom Weaver at Lawrence Livermore Laboratories to try to understand how the solar system was oriented geometrically to the theoretical plane of the Milky Way as well as how that plane is intersected by other theoretical "planes" including the Magellanic Stream and the Sagittarius Dwarf Stream. Because there is no material celestial sphere, only the human vision of the sky as a dome, it has been difficult to create language for the rather remarkable alignments of computer-generated data with the divine feminine geometry that I can map on my humble little globe. I will write more on this in a future article, but need now to return to the subject of consciousness. 7
Figure 3. A celestial sphere is a very ancient rendering of the sky as a globe. It supports the imaginative possibility that we can get "above" the view of the sky that we see from the Earth and look down upon it (i.e., as above, so below). The dome of the sky visible to the naked eye has remained much the same as it is now for the past 50,000 years of human history. Computers have made the celestial sphere obsolete, however, and within the past twenty years it has taken its place in planetarium museums among archaic tools. The once-standard 12'' diameter globe I use in my research is out of production, though occasionally one appears on ebay as a high-priced antique. I would never have seen the stunning sky alignments with the 15-hoop/120-triangle sphere without one. The issue is that while celestial objects may appear to form a dome over Earth, they are in fact at vastly different distances from us. There is no dome. Yet human stargazers for thousands of years visualized this geometry upon the sky and used it to create calendars of precession.
A challenge posed by my vision experience was how to be a scholar as well as live authentically in my new "receptacle." I joined the doctoral faculty at the Union Graduate School where it was then possible to break disciplinary boundaries, and I began to teach geometry as an arts-based interdisciplinary research methodology. Rather than focusing upon the usual academic burden of having to prove the scientific legitimacy of geometric modeling, I wandered across fields of inquiry with my students and simply traced and recorded compelling coincidences and harmonies of sound, shape, color, and meaning that I encountered along the way. I continue to sidestep issues of whether cultural similarities (such as between Plato and the Sioux) are products of diffusion or innovation, simply my own artistic fantasies, or possibly all of these things. What drives my work within the anthropological community of scholars is our common alignment with the principle of coherent connection. My hope is that Babel's divisive differentiation was not a decisive mythological blow to a compassionate, collective human worldview-that we do as a species constantly remember and re-language our common humanity. This inspires me to teach as I do and to want to share the sacred lineage of sustainability that my vision has taught me is geometric consciousness (geomancy).
Romance with geometric consciousness is how I experience geomancy. 8 For thousands of years, people in every part of the world have made hand-scale models of what I think of as geometric "seeds" (Plato's five element-shapes and two related rhombic figures) using wood, reeds, clay, metal, stone, string. I encourage everyone I know to do the same to see for themselves how accessible geo-mantic knowledge is for humans who seek it. The knuckles of the human hand create virtually all the proportional lengths necessary to make beautifully symmetrical forms.
From carved Neolithic "bolas" to contemporary Southeast Asian woven reed spheres, such teaching devices encapsulate the sacred dimensions of the Universe and assist the maker or user in extending awareness of Self into the infinite macrocosm and microcosm of creation. Intimate knowledge of these forms and shapes is essential to grasping the deep personal mythology of geometry, and helps the human to tap into the divine "memory" of connection.
I teach geometry (geo + metry) as an art/science of consciousness. The Greeks considered it a divine art. Linguistically, the me in geometry is the me in metaphysics, the holy spirit in the middle, the human measure. Metr is not only "measure," but "mother" and "matter" as well. I suspect a connection to Mitra, the divine Vedic principle associated with order in the dark universe.
Figure 4. Hand-held, precisely crafted geometric artifacts.
( Above ) Three models based upon identical icosa-dodecahedral symmetry include (from left) a contemporary model of a virus; a 3000-year-old bronze figure possibly associated with Hecate, the "Triple Goddess" 9 ; and a reed healing sphere from Southeast Asia. ( Below ) Carved stones from Neolithic Scotland (possibly bolas) accurately represent the five Platonic solids at least 1000 years before Plato.
Plato regarded geometry as the ultimate mnemonic (memory-aiding) device that organized primeval chaos. In Timaeus, he describes how the Demiurge (whom he calls "the god") "remembered" itself within Chora (the oldest Greek word for "place")-thereby initiating order in creation. 10 Chora (which is also related linguistically to Greek words for "chorus," "dance ground," and "the beating of a heart") is, then, almost certainly the vibrating, intelligent divine feminine Receptacle. Before there was matter, Chora (whom he also calls the "Nurse of Becoming") held the disorganized and undifferentiated aspects of shape (morphai -the perfect geometric forms), power (dynameis-fire, earth, air, water, and aether), and feeling (pathe). The Demiurge infuses Chora to materialize all that physically exists, at every scale . . . in a near-but-never-perfect replication of the Mother. 11 As above, so below.
Plato taught that the universe was constantly in motion, its elements in a perpetually fluid state of transmutation dancing and spinning into and away from connectivity. The dance ground was, of course, the eternal and unchanging body of Chora. In the stillness of meditation, the luminous spherical human 12 was the ground of the perfect cosmic dance. Plato seems to have believed that geometric mnemonics could actually link an individual to the collective memory of the human species-even to the memory of origination of human material being, autochthony, the "springing from the soil." He saw the five perfect figures not just as symbolic memory-aiding devices, but as geometric forms mathematically encoding and exemplifying real connections and the formulaic process of structural evolution in the material world.
The tetrahedron, for example, actually was (for Plato) the element of fire, the color "red," and "first" in a chain of Being. Astrophysicists who subscribe either to the Big Bang theory or to the alternate theory see fire as being present or instrumental in a first act of creation, yet Plato's thinking in this regard has often been dismissed as highly mythological and primitive. Translations of his works have generally been made by philosophers untrained in three-dimensional geometry and likely unable to recognize the sophisticated connective patterns of natural form and flow that Plato describes. 13
The legacy of contemporary Western education is that geomantic consciousness is not a taken-for-granted aspect of our worldview and therefore we do not recognize our intimate formal connectivity to the three-dimensional universe in which we live. What we call "learning" is typically accomplished quietly, without motion, in two-dimensional contexts-reading, computing, gaming, or watching media. Body wisdom is rarely integrated into this existential cognitive scheme despite the lip-service currently being paid to multiple intelligences. Thus, it isn't at all surprising that geosophic knowledge encoded in ancient artifacts, artwork, and texts has so often been overlooked. In many ways, we hold the unfortunate stereotype that a still body indicates a developed mind-and an active body, a simple mind. It is a pernicious and largely invisible ethnocentrism that denies legitimacy to the activity associated with many ancient and indigenous rituals. We do not any more easily imagine knowledge being produced by Greek, Hindu, or Egyptian scholars dancing, singing and doing yoga in their academies than we do tribal people who "dance, sing, and make that outrageous art!" as they engage in their own form of rigorous, concrete experimental research and philosophy.
Geometry is a formulaic method for the cultivation of a liminal consciousness that can bridge the mind/body gap. Common body postures in both meditation and certain forms of dance 14 are actually multi-layered geometric mnemonics. The basic seated meditative posture of yoga, the siddhasana or "adept" position also known as "perfect posture," aligns to an etheric tetrahedron (Figure 5). The base of the spine, head, and two knees approximate its four corners. This is probably the most loved and most familiar representation of the Buddha. Red, the color associated with the tetrahedron, is the color of the first (sacral) chakra at the base of the spine as well as the color of initiation and beginnings (e.g. blood at birth, circumcision, and menstruation; red ochre on a corpse as it passes to new life after death). First in the series of regular geometric forms, the tetrahedron- as fire -destroys while simultaneously creating new beginnings.
In a different way, the tetrahedron is also recognized as a primary structural archetype by science. A. G. Cairns-Smith (1985), a noted biologist, has made a very convincing case that the first organisms came into being in a bed of tetrahedral clay crystals. Clay itself is inorganic, but it holds geometric information necessary for the formation of Earth and its creations in much the same way that musical information recorded on a CD holds music by giving material structure to wave forms. He does not go so far as to declare the clay "intelligent," but argues in effect for co-creation: the tetrahedral crystals provide exactly the memory structure-very nearly a genetic code-that reproducing organisms needed to remember in order to achieve structural integrity. Buckminster Fuller used the tetrahedron as the foundation figure in all his design strategies and argued that it was the most energy-efficient figure in nature.
Figure 5. In yogic "perfect posture" or "adept position," the body aligns with the tetrahedron.
The geometric character of written Sanskrit enhances the total semantic experience of Hindu texts, which contain some of the most ancient, elegant and subtle geometric mnemonics. Again, the red/fire/tetrahedron is primary. In the Beginning, states the Bhagavad-Gita, this world was Self alone, in the shape of a person (Purusha). The name Purusha roughly translates as the purifying burning (ush) that existed before the current material Creation (pur). One of the most beloved two-dimensional images of Purusha can be easily envisioned as a tetrahedron within a cube. (See Figure 6, keeping in mind that each face of a cube is a square.) Purusha experiences loneliness and wishes a second Self into existence. Very roughly paraphrasing here: he was so large as man and wife together, he made his Self fall in two, and thence arose husband and wife. He embraced her, and men were born. This union of Purusha and Prakriti (the female half generated from himself) is the deepest, most holy Hindu symbol of Creator and Creation. 15 The divine coupling is inherently geometric in a very important way. Two (and only two) tetrahedra "nest" in a cube-the yellow element of earth (created matter), second in the Platonic series.
A very similar visual metaphor is employed in Navajo ceremonies that bring to ritual life a character named Yebitsai-"Talking God," also sometimes called "Grandfather God of the East, the Dawn"-who guides and monitors. The numinous talisman used in his ceremony is a bundle of four sticks, loosely fastened to each other end to end. Only Yebitsai's hands, which represent the white dawn (the white octahedron of air, third in the geometric series) are ever seen; they protrude through small holes in a black cloth which hides a magician who shape-shifts the sticks into a tetrahedron, a square, various diamond-like perspectives of cube faces, and an edge.
Figure 6. The image of Purusha (center) as Vastu Purusha (in "demon" form, trapped by the gods in a net) is used as a pattern for architectural layouts. It can be visualized as a tetrahedron (left). Prakriti and Purusha merge in the second stage of materialization; this union is represented here (right) as two interpenetrating tetrahedra whose eight combined corners define the second dynamic element of Plato's series, the cube.
In the same spirit as the above, I experience semantics as the romance of semiotics . The etymology of the word geometry is just one avenue into what appears to me to be a virtually worldwide semantic core. Greek mythology, for example, preserves it as the matrilineage of Chora- Gaia/Ge, Mnemosyne, and the Muses. Mnemonic derives from an old Indo-European root sound men- that refers to qualities and states of mind and thought. 16 In Old English, the connection between mother Gaia and her daughter Mnemosyne was preserved in the word gemynd, "memory." The geomantic sound/meanings of ge, ga, me, and metr that create the word "geometry" are neither uniquely Greek nor even Indo-European-but seem almost to be a panhuman hymn of praise to Chora that has echoed around the world from primal sources.
Ge and Gaia are name/sounds of Earth gods and goddesses in countless mythologies. 17 Gender is almost irrelevant. There is Geb in Egypt ; Geb in Papua New Guinea among the Dani; and Ge in the Caribbean . The Gaelic word for Earth is gabhal. Ga-oh and Hahgwehdiyu are Earth creators among the Iroquois. Apache mountain deities are gahe. Among the Pueblo tribes of the American southwest, they are kachinas. The sounds ka and ga are frequently interchanged. Gaokerena is the tree of life in the Iranian cosmology. Galaxy is "milk of the Mother Goddess."
In the Philippines, gaia is a model to imitate. Among the Hopi, gahopi is the "right way of living." In India, Gautama received enlightenment in Gaya, the holy city of purification in Bihar Province. Ngai is the creator of Earth among the Maasai of Kenya. Nga is the sacred sound that opens the throat in Tibetan meditation. Linguists such as Cavalli-Sforza (1991) and Sherveroskin believe that ngai may have been one of the first human utterances, a sound echoing forward from more than 200,000 years in the distant past . . . meaning "I," or possibly "I breathe."
Ganga is goddess of the Ganges, the river of eternal life. The ganas are servants of Shiva, Lord of the Dance of transformation of earthly elements. In Spanish, the fire of transformation is ganas -desire, the energy of creation. Ganeo'o is the Iroquois dance of thanks to the Creator. Our words Genesis, gene, genuine, and indigenous all flow from this same primal container of sound and meaning. To genuflect, to bend the knee as a sign of worship, is to become one with sacred geometry. Genu is an ancient sound that meant "angle." In Greek, it became gon -the root of polygon, a figure with many angles or corners.
The sounds me, ma, mr, and metr can be thought of as liminal and almost always associated with edge/dichotomies of life and death-mortality, mother, measure. In ancient Egypt, at least three separate glyphs were homonyms sounded as "mr:" (1) a plow, which symbolized love and cultivation; (2) an owl, which symbolized death; and (3) a triangle, which symbolized geometry. The nasal bilabial "m" requires only the parting of the lips to create the sound "ma," and so it is not surprising that a baby's first cry for its mother's breast is ma. In Hebrew, 'em is "mother"; 'ammah is either "mother" or a unit of measure. 18 In Hawaii, Maui brings death. Maya is the mother of Buddha, and maia is the measure of illusion. Ma'at is the ancient Egyptian Lawgiver goddess. The yoga position of ultimate awareness and surrender to Earth- Mrtasana, "corpse pose"-is, when seen "from above," identical to the most beloved representation of the Virgin Mary standing with her open arms outstretched.
As in the word geometry, ma/mr/metr is often combined with ge to sound a sacred name . Aurgelmir is the first being in the Norse universe who emerged from the great cosmic void, Ginnungagap. Mageogeo is a creator in the Tuamotus; magatama are sacred crystals of the Shinto Earth spirit religion in Japan . Zoroaster, a contemporary of Pythagoras, taught that Gayomart (also called Gaya Maretan) was the primal human, a luminous spherical being.
Also intertwined in this semantic scheme is one other persistent sound- ang or ank, from which we derive angle. The hieroglyphic symbol for "life" in ancient Egypt was the ankh-a sandal strap. The circular part of the strap went around the ankle, the primary angle by which humans walk. Linguists believe that about six thousand years ago, throughout most of Europe and India, the sounds ang and ank both meant "angle" in the sense of "bending, or turning on an axis" (Watkins 1969). The sound gel (as in angle) was related to ge, but was more specific. It implied "to form into a ball," "bright and shining." Perhaps it anticipated either Ge or the luminous spherical human. The sound an meant "aloft" and, oddly, "an old woman singing." Angh carried the meaning of "tight, painfully constricted"-the vaginal opening at birth? Angwhi was "snake." El was the bend in the arm (the elbow) or the bend in the lower leg (the knee), but it was also the elm and the alder-extremely common cosmic creation trees. In Gaelic, the very oldest and most important unit of measure was the ell. 19
Neither linguists nor etymologists currently recognize a link between angle and angel, however. Angel is thought to be a sound that originally meant "messenger," one borrowed by the Greeks from an unknown Oriental source. I believe that messenger was Hermes/Mercury, who is so often associated with earth energies and measures. What could be called the "angel of geometry"- Plato's etherically luminous sphere of hoops/triangles, aloft, spiralling, singing memory-does not exist as a documented cluster of sound/meanings. Yet I have experienced myself as this Receptacle of consciousness as I have sought her name in non-Western sources that combine the sounds ang with ge/gaia (and the related ka) in sacred creative contexts. 20
In the Qu-ran of Islam, for example, ankabut is a trickster spider who miraculously spins a huge web over the entrance to a cave in which the Prophet hides from those who seek to kill him (Surah 29 in The Holy Qu-ran). The Hopi Spider Woman is Koyangwuti. A Maori ( New Zealand ) female nature spirit is Kurangaituku. Eskimo shamans are angakut, and the primary Creator is Anguta. Tangaroa is the primary Polynesian Creator. Among the Chinese, the Creator is Panku or Pangu. The all-present creator for the Zulu of South Africa is uMvelinqangi. Yang is the creative aspect of the Tao, and altjiranga is the aboriginal Australian name for the Creative primordial. Ultimately, the ge/ga sound is the most ubiquitous. English retains it in god. In Gaelic, gael means "love" and gae is "cage" (the Cosmic Receptacle?). Gae is also "jail" (the entrapment of spirit in matter, perhaps). 21
Plato's Timaeus systematizes an entire system of geometry that was at least 1000 years old at the time he wrote, and the Demiurge that organized the wildly vibrating aspects of Chaos was known long before to the Hindus. Even the sounds in the name Bhagavad-Gita ("Song of the Blessed One") carry precisely the meanings of geometry. Gita is from gei, "to sing". Bha is "to shine," and bhag is "to share out good fortune." Bhagha is "elbow, a bending, an angle." The early voice of humanity seems, to me, to be fundamentally geomantic-a creation song, a living process of interlingual connection that is endless, turning and bending itself, in and out of empirical linguistic science and the wisdom of mythology.
The idea that the world is held in Chora (the dance ground), and continually sung into existence, is perhaps most beautifully expressed by Aboriginals in Australia who view themselves as the part of Earth responsible for remembering and chanting all the songs of creation. People who forget sacred connectivity, who do not sing the praises of the Creator, are in countless mythologies destroyed by earth, air, fire or water-by catastrophes of elements out of control, out of balance, out of tune. The contemporary science of cymatics (Jenny 2001) is built from exactly this wisdom: particles of matter assume and maintain coherent shapes as a function of very specific interference patterns, the shape effect of overlapping vibrations (harmonies). Physicists actually describe all matter in this way. In the infinitely hot universe of plasma energy, matter forms as a kind of supercool crust (or bubble wall) created by the "beats" of juxtaposed wave forms (Prokopec et al. 1995). The miracle is the universality of a limited number of "natural forms" (simulacra) that seem to pervade the "crust" at every scale. The human eye/brain/mind seems designed to see simulacra and to find great meaning in them (Michell 1979).
Simulacra are images of people and things that can be seen in rocks, clouds, smoke . . . Their "appearance" is sometimes so aesthetic-and timing so powerful- that it can be difficult to believe they are natural flow forms. Simulacra are common neo-shamanic and religious phenomena (witness the significance attached to a Madonna perceived in the rust of a screen door or burned onto a tortilla) and are often cited as evidence of divine interference in the affairs of humans. I think of this response as an example of a much larger panhuman phenomenon known as the Law of Similars-the belief that things that look alike must share an important sympathetic resonance. The principal power of a shaman or seer is, possibly, the ability to visualize simulacra in detail at extraordinary ranges of scale. This expanded power of sight brings with it the awesome responsibility of helping to keep alive (and sometimes even restoring) the shapes of cosmic harmony. 22 And sacred vision is only one of the extraordinary burdens a shaman- who is most certainly a geomancer-must take on .
The techniques used to generate such visions are clearly both scientific and meditative. They vary from culture to culture, but there are some striking parallels. The Oneida questing tradition of sitting small, standing tall is identical to the principle of magnifying clairvoyance described by Patanjali in the Yoga Sutras, numbers 27 and 28. It was possible, he said, to make oneself infinitely small such than an atom would appear large in comparison-or large enough to see the whole Earth. A kind of magnifying clairvoyance can also occur in dreams. The Hebrew tradition of prophetic dreaming, for example, produced a vision of the world tree startlingly similar to ultrasound images of a human placenta and embryo. The German chemist F. A. Kekule credits a geometric simulacrum in a dream of Oroboros (the almost unimaginably ancient image of cosmic energy as a vortex/snake swallowing its own tail) as the inspiration for his hexagonal model of a benzene ring molecule.
Cosmic trees, entwined snakes and hexagon/turtle images permeate the artwork, mythology, and dream visions of people everywhere. Cultures as remote from one another in time and place as the Hindu, Maya, Aboriginal Australian, and North American Plains Indian have all emphasized these symbols in sacred geometric contexts. Metaphors of the snake/tree (which I understand as the principle of spiralling) 23 and the turtle/hexagon (possibly representing the powerful and efficient "packing" of divine energy as matter . . . think honeycomb!) are perhaps the deepest totemic meanings in the human psyche as well as the most basic representations of geometric connectivity.
The cosmic tree is present in virtually every creation mythology. It is the axis, the orienting principle-the spine of Chora, the spinning divine feminine Receptacle. Any tree can be the cosmic tree, for branches grow from a living tree in a DNA-like snake of life that spirals in a phi ratio (1:1.618). 24 Trees have been used throughout time as a starting point for teaching the immanence of geometry in the everyday world.
I believe that the turtle, whose carapace is covered with hexagons, has remained so sacred throughout time because it represents a threshold through which a variety of shapes can connect (see Figure 7). Like hexagonal pegs, all five of Plato's perfect figures-and two other regular figures, the rhombic triacontahedron (with its 30 identical diamond faces) and the rhombic dodecahedron (with 12 identical diamond faces)-can be positioned to meet at this threshold and to enter into structural connection with each other. 25
Figure 7. The H exagonal D oor is another vision that spontaneously opened to me as I explored geometric consciousness. At the center is the tetrahedron. From upper left clockwise are the rhombic dodecahedron, rhombic triacontahedron, cube, dodecahedron, icosahedron, and octahedron.
Turtles and trees have appeared almost universally to create a picture of the cosmos as a vast whirlpool, a maelstrom-what Giorgio de Santillana and Hertha von Dechend (1969) have characterized as "Hamlet's Mill." 26 The gyroscopic spiralling of Earth's axis of rotation-known technically as precession -has been recorded in countless mythological explanations of observable, incremental and repetitive changes in the location of the sun and planets at particular times of the year against the background of "fixed stars." In many stories Earth is a turtle, and S nake (the hoopsnake or Oroboros, for example) is the path traced in the sky by Earth's axial pole (the cosmic tree or churn in the mill) over the period of 25,920 years it takes to complete a precession circuit around the ecliptic north pole . Draco (a dragon-snake) is the Western constellation that curls around the ecliptic north pole-which is located at the "heart of the dragon." The Hindu god Vishnu represents the cosmic tree . He is also a combined snake/turtle avatar, the axis of every perfect spinning sphere. The name Cuculcan , Maya ruler of the four directions and the four elements, means "to move round and round;" and cuceb (from cucul) is the spiralling of the great wheels of the Maya calendar of days. In Norse mythology, the cosmic spiral is a squirrel named Ratatosk who runs up and down Yggdrasil, the cosmic tree, carrying messages between gods and humans. Oddly, the Maya word cuceb means "squirrel," after the spiral path the animal takes as it scampers up a tree (Kenton 1928). 27
There are no fixed situational contexts to which these metaphors specifically apply. They are archetypes that float, bubble-like, through all cultures and all aspects of perception-semantic crystals that store the energy of memory and render the experience of life mystically, even magically, interconnected. Their reality is that this connectedness is not a primitive wish, desire, or belief, but turns out to be astronomical, geophysical, and biological fact. Their truth is a timeless power to evoke new knowledge by inspiring the process of search.
Over the ages, I would suggest, these geomantic images have persisted because they are reliable generative frames for perplexing, and ultimately unanswerable questions about the ancient past and its possible return as the near future. As unrecognized cultural memes, they have quietly stimulated both science and religion to develop and refine concepts needed to even ask such questions. Is there a galactic axis of rotation? Is Earth the infinite expanse of the ecliptic plane? Has Earth's axis wobbled out of "alignment" (it tilts about 23 1/2 o off the plane of the ecliptic)? Was there a cataclysmic collision with a comet or giant meteor, a catastrophic polar ice build-up, a pole shift, a technology gone too far, a sin too great . . . or is the "lack" of alignment only an illusion, a byproduct of the deep human belief in symmetry and the need to find it? Has it always been this way? What have we actually survived?
Threshold of Initiation: Through the Hexagonal Door
The scale of vision and consciousness needed to conceptualize such questions- whether within the archetypal structures of mythology, or within the observational framework of rational/empirical science-is enormous. They are true problems in sacred geometry-in geomancy, the use of sacred geometry to understand physical existence. Even more staggering are the physical and spiritual measures of the cosmos that have inspired these questions: light years in distance; eons in time; millions of degrees in temperature; blinding insight; perfect communion; burning desire. In my own geometric odyssey, I was surprised-and then again, not-that hexagonal (six-sided) simulacra at all scales in nature (from patterns in the stars to turtles, honeycombs, and diamonds) are so often chosen, however unconsciously, to symbolize vast questions and choices of human existence.
The nearly universal archetype linking the cube (which represents earth in the Platonic scheme and can be visualized two-dimensionally as a hexagon-see Figure 6) with both the turtle and Earth is one of the most fascinating and enduring. It's not hard to understand why a turtle is such an appropriate Earth symbol. There are hundreds of varieties of them, and they live virtually everywhere. Turtles are almost living picture rocks. The designs on their shells often reflect some aspect of their native environment-winding rivers, stars, snowflakes, tide pools, cultivated terraces. Parched earth buckles and cracks into hexagonal and pentagonal cells like those on a turtle's back.
The arched carapace of the turtle (or tortoise) has been used, especially in cultures of Asia , to represent both the central mountain of the universe as well as the dome of the sky. The animal is often portrayed as an immortal mediator between heaven and earth. Curiously, in Japanese mythology, the tortoise lives to 12,000 years of age-just about one-half the cycle of precession. Apollo's lyre, a gift from the trickster Hermes, is strung across a turtle's carapace, and the Greeks of Plato's time actually fashioned harps this way. Alchemists used turtle shells to cover their crucibles, and Chinese astronomers relied upon them as instruments of divination. The traditional American Indian sweat lodge is, symbolically, a turtle shell, and the North American continent is known to many tribes as Turtle Island . In the myth of the Earth Diver, an extremely common creation story known in many parts of the world, a maiden falls to primordial Earth through a cosmic door in the heavens. Magic birds break her fall as she comes to rest on the back of a turtle. A hero animal then sacrifices itself by diving deep into the cosmic waters in which the turtle swims to bring up a tiny bit of soil that magically expands into Earth when placed on the turtle's back (Cirlot 1971, Herder Freiburg 1986, Hamilton 1988, Cooper 1990).
It's surprising, nonetheless, to find how uniquely appropriate a symbol of the awesome life force of Earth the turtle really is. Turtles can withstand drought and famine by retreating into a kind of zombie-like state. They live to prodigious ages and can survive both on land and in water. In addition to breathing through a set of lungs, freshwater turtles can use their mouth cavity in gill-like fashion. They have keen eyesight and can distinguish between colors. Their shell is strong enough to resist alligator bites, and the leathery skin on their extremities wrinkles and folds to allow for extension and retraction of their head and feet. The Hawksbill turtle of the Caribbean actually feeds exclusively on crystal-the glass spines of poisonous subtropical sponges! No comparable diet has ever been described for any other vertebrate. Researchers have found that the digestive tract of a typical Hawksbill is full of razor-sharp glass shards that do no harm to the turtle. Its feces are typically concentrated into solid crystal (Weiss 1988).
By geomantic logic, then, if Earth is a turtle/cube-hexagon ("as above, so below"), there must be a corresponding simulacrum in Sky. And in fact, a hexagonal door can indeed be seen as a cluster of exceptionally bright stars in the constellations around Orion. These stars include Sirius, Rigel, Aldebaran, Capella, Castor/Pollux, Procyon, and Betelgeuse. They are spread almost equally north and south of the celestial equator and can be seen from almost everywhere on Earth. The Milky Way appears as a path through the center. Astronomers refer to these stars as the Winter Hexagon (Raymo 1982). For some reason, of the seven geometric "seed" figures, a two-dimensional representation of a cube is easiest to visualize as a three-dimensional object. Once the Winter Hexagon pops into three dimensions as a cube, it is difficult not to see it that way. Its "unseen edges" have a unique connective energy. This is an especially powerful memory-aiding aspect of geometry.
Figure 8. Astronomers refer to the pattern of bright stars surrounding the constellation of Orion as the Winter Hexagon. The stars also can be visualized as the corners of a huge celestial cube which may have been the inspiration for the vision in Revelation of a New Jerusalem descending from heaven (H agens 1997b). Many cultural traditions in South America and the Pacific see a turtle in these stars (Allen 1963), and a drawing of the Maya world tree made by Diez Lopez Cogolludo in 1640 (Kenton 1928) pictures a tree bursting upwards through the back of a turtle resting on a cube. The "incomparably mighty churn" of the Sea of Milk (The Milky Way) in the Mahabarata is traditionally illustrated as Vishnu atop a tree/snake that protrudes through the back of a turtle (Santillana and von Dechend 1977).
The cube has probably been embraced more often as a mnemonic of sacred life in the material cosmos than any other figure. Both Greek and Hindu philosophers equated the element earth with a yellow cube. The Chinese used Earth/turtle symbolism, labeled the element of earth yellow, and named their goddess of the Yellow Earth Nukua (a sound close to "cube," but I can't find evidence that Nukua was specifically linked to the cube). Curiously, though, the Indo-European root sound for "turtle" was ghelu, derived from ghel, which meant "to shine"; and the English words "yellow" and "gold" both derive from ghel. (The parent sound of both ghel and ghelu is ge!)
Many South American tribes identify the brilliant stars of the Winter Hexagon with a tortoise shell, and the Taino (a now virtually extinct tribe who were the first in this hemisphere to meet Columbus) apparently had an extremely complex creation mythology that combined the axis mundi (axial pole, cosmic tree), Orion, turtle shells, cubes, rhombs and sacred spittle (Stevens-Arroyo 1988).
In approximately 1325 BCE, Tutankhamen's organs were entombed in an alabaster cube; statues of Isis, Nephthys, Horus, and Thoth stood at its four corners. According to the Lucie Lamy (1981), the ancient Egyptian Salt Papyrus (probably written at the end of the Nineteenth Dynasty in the late 15th century BCE, but named after the English Egyptologist who collected it, Henry Salt) portrays the celestial House of Life as a cube in which Osiris rests. There is fairly wide scholarly agreement that an association exists between Orion-Sirius and Osiris (e.g. Krupp 1991, Lockyer 1964). The Holy of Holies in Solomon's temple (1 Kings 6:20) is a perfect cube, and the religious scholar Josephus (1835) wrote extensively about the ways in which its measured proportions imitated the system of the created world. In the sandpaintings of the Navajo Shooting Chant, spirit medicine people emerge from a black cube over which a protective rainbow person (the Milky Way) arches protectively (Newcomb and Reichard 1975). A thunderbolt-wielding Huitzilipochtli lives at the center of a cubic Mesoamerican cosmos, illustrated in the Codex Fejervary-Mayer in exact architectural plan view perspective as it would be seen from above, without a top (see Figure 9).
Indigenous people in almost all parts of Australia hold sacred an All-Father. His name varies (Ngurunderi, Baiame, Daramulun . . .), but he is the cosmic axis who establishes the path the dead travel to join the heroes in the Sky World. He is often portrayed lying on the back of a turtle with (or as) a representation of the Milky Way (Rainbow Snake) arching over him. In his many aspects, All-Father represents every luminous arch of nature-the phosphorescence in sea waves, the pearly interior of shells, the sunlight in water droplets. Analogous to the way that the hexagonal door permits passage of all the spherically symmetrical crystalline geometries, the All-Father snake/turtle "spits" crystals and medicine spirits from his mouth (Poignant 1967).
In traditional Bora ritual (Mathews 1894), the thundering voice of the All-Father is invoked by whirling a rhomb-shaped bullroarer pierced with holes that represented the stars in Orion's belt. Swung on a string from the end of a pole, the bullroarer circles the sacred ritual space analogously to the way the constellation Orion circles the Earth, tethered to the axial pole. All-Father (whose voice is heard) is imagined as an axis/snake. He is always represented with a large phallus. The Bora saw the entire constellation of Orion as a group of young men dancing in a korobra or corroboree, an extremely important ritual dance celebrating the All-Father. Perhaps this linguistic clue sets Rainbow Snake/Turtle/All Father of the corroboree not only into the traditions of Oroboros and Chora, but also into the long line of geometers (Pythagoras and Patanjali among them) whose names literally mean "snake in the dance ground." 28
Figure 9. "Plan view" of Huitzilipochtli at the center of a celestial cube.
This drawing is adapted from the Codex Fejervary-Mayer, a deer skin screen-fold book illustrating aspects of the Mesoamerican calendar, believed to be pre-Columbian and of Mixteca -Puebla origin. The cult of Huitzilipochtli somewhat disrupted and merged with the lineage of the earlier Xiutecuhtli to create a syncretic supreme Aztec deity and lord of the calendar (Miller and Taube 1993, 93-96, 189-190; Littleton 2002). His attributes as a god of storms, sun, and death are similar to other divine figures associated with Orion.
Figure 10. An original interpretation of cubic-hexagonal geometry in rock shelter art from Arnhem Land, Australia. My drawing is adapted from a photo in Poignant 1967:137.
Drawings of Rainbow Snake such as the remarkable piece of rock shelter art from Arnhem Land sketched above (which have possibly been continuously maintained for more than 40,000 years), are to my mind tour de force representations of the shamanic consciousness of the hexagonal door. From one perspective, the male (the smaller figure) is born from the female. From another, she is the celestial cube and he the rhombic shape (the bullroarer . . . Orion) that is pulled from her side. The corners of the hexagon fall at her elbows, knees, throat, navel and vagina. I imagine her as Everywoman . . . the spinning disk of the Milky Way, the hexagonal bed upon which her lover (like Shiva) will lie, and the virginal hymen that will remain unbroken by divine connection. (The mythology of the Triple-Goddess Hecate, whose symbol is a hexagon, is virtually the same.) The male is Everyman, a son and lover. His penis is an umbilical cord. Both divine figures are embodied bullroarers, rhombic-brained beings singing out from the ends of spinal cords. 29 Together, as Rainbow Snake (who is androgynous), they dance the spiralling energies of life in a spherical cosmos. Their shared hexagonal wall is primal cleavage-cutting and clinging-the corpus callosum of our dihedral brains.
A Predictable Yet Unexpected Hermeneutic?
I want to emphasize that the interpretation above is my own ecstatic geomantic vision, one that I have shared with Aboriginal Australian friends, and an example of ways in which interpreters continue to connect across time via geomantic imagery. More than anything else, it is an expression of my personal history-the product of a quasi-Christian hermeneutic I began to absorb early in my life. I want to bring this essay full-circle by turning to an ancient prophetic text describing the celestial cube that is quite alive in Christian culture today. It contains vital, accurate information about ancient cosmic geometry and the way in which it was generated. That text is John's Book of Revelation. "Above me there was an open door to heaven," John writes (Rev. 4:1). "I also saw a new Jerusalem, the holy city, coming down out of heaven from God, beautiful as a bride to meet her husband" (Rev. 21:2). The text is ecstatic, not conventionally logical, but John's description of the city is precise . It is a perfect cube, "twelve thousand furlongs in length, in width, and in height" (Rev. 21:16-17). 30
Leaving for the moment the issue of the size of the city (12,000 furlongs is 1440 miles), many things in John's text clearly tie it to the "celestial cube" that astronomers have named the Winter Hexagon. It is entirely possible that John was inspired to write from direct observation of the sky. His vision took place on Patmos, an island in the Aegean Sea off the western coast of Turkey and the site of an ancient observatory. (I will return momentarily to the importance of this geographic location). Pythagoras, who apparently paid close attention to the sky, was born nearby on the island of Samos. Though there are multiple meanings and associations in the text, the "seven lampstands of gold" that John first sees (Rev. 1:12) can be understood in one context as the seven major stars of the celestial cube. These stars are among the fifteen brightest in the entire visible sky, and ghel is etymologically the root of both "gold" and "shining."
When commanded by the voice of the angel to write, John says that he turned around to see who had spoken and must have seen the Winter Hexagon and the constellation of Orion standing in the Milky Way. He writes:
I turned around to see whose voice it was that spoke to me. When I did so I saw seven lampstands of gold, and among the lampstands One like a Son of Man wearing an ankle-length robe, with a sash of gold about his breast. The hair of his head was as white as snow-white wool and his eyes blazed like fire . . . His voice sounded like the roar of rushing waters. In his right hand he held seven stars. A sharp, two-edged sword came out of his mouth, and his face shone like the sun at its brightest. 31[Rev. 1:12-16]
As seen from Earth, Orion's right hand stretches out toward the seven stars of the Pleiades. His entire head is in the Milky Way which streams down like a sash across his right shoulder.
The crystalline city does not come down out of heaven until after the terrors of the Apocalypse. The "coming down" of the city almost certainly refers to a specific point in the 25,920-year precession cycle (represented in ancient zodiacal calendars worldwide) when Earth and its axis of rotation align with the constellation of Orion and the stars of the celestial cube to indicate entrance into a wider galactic geometric harmony (Hagens 1992, 1999)-the death of one cycle, and the birth of a new one. Early in the prophecy of things to come, John intimates that the city is the bride of the Lord:
A great sign appeared in the sky, a woman clothed with the sun, with the moon under her feet, and on her head a crown of twelve stars . . . she was with child . . . [and] wailed aloud in pain as she labored to give birth . [Rev. 12:1-2].
Both Moon and Sun pass through the celestial cube, and so the woman may be both hexagonal door and connective lover/mother in the way of the Aboriginal Everywoman:
'Come, I will show you the woman who is the bride of the Lamb.' He carried me away to the top of a very high mountain and showed me the holy city Jerusalem coming down out of heaven from God . . . The city had the radiance of a precious jewel that sparkled like a diamond. [Rev. 21:9-11]
Only the constellations of the Lord (Orion) and the Lamb (Monoceros, the Unicorn, who is known in esoteric texts as the "Lamb of the Apocalypse" [Shepard 1956]) are actually in the celestial cube.
Around the throne was a rainbow as brilliant as an emerald [Rev. 4:3] . . . I saw no temple in the city. The Lord, God the Almighty, is its temple-he and the Lamb . . . The city had no need of sun or moon, for the glory of God gave it light [Rev. 21:22-24] . . . The river of life-giving water, clear as crystal, which issued from the throne of God and of the Lamb . . . flowed down the middle of the streets [Rev. 22:1-2].
Recall that the Winter Hexagon is the brightest area in the entire celestial sphere, that a crystal Rainbow Snake appears in legend across Oceania, Australia , and North America . John's image of the foundation of the city being composed of twelve distinct kinds of jewels has often been interpreted as the zodiacal ages-since each age is traditionally associated with a gemstone. 32 It can also be imagined as the hexagonal threshold (or crystal-spitting mouth of the All-Father) through which pass all of the geometric shapes. The text concludes with a warning and allusion to Earth's axis of rotation (the cosmic snake/tree): "If anyone takes from the words of this prophetic book, God will take away his share in the tree of life and the holy city described here" (Rev. 22:19).
John's identity has never been agreed upon. He does not seem to be the author of the Book of John, but works of prophecy are not always written "in character." Many aspects of his vision are extremely similar to that of the prophet Mohammed. 33 In Surah 17 of the Qu-ran, Mohammed is visited by an angel presumed to be Gabriel (but possibly Orion) and taken into the sky "for a journey by night from the Sacred Mosque to the Farthest Mosque." The Sacred Mosque is the Kaa'ba, the cube-shaped temple at Mecca . (Kaa'ba means "cube" in Arabic.) Recent research focused upon a golden Varanasi plate found hanging in the Kaa'ba (Oak 2004) indicates that the temple may originally have been a Hindu shrine to Shiva. Kaa'ba may even have come from the Tamil kabaa, considered one of the oldest in the world. Dravidians of the Indus Valley worshiped Shiva as their Primal Deity, and his temples in South India are called Kabaalishwaran. Kabaali refers to Lord Shiva, who lies upon a hexagonal bed! In any case, the Farthest Mosque is sometimes interpreted as the Temple of Solomon, which is also cubic, but it seems more likely that it is the celestial cube. Mohammed actually sees the angel "by the lote-tree [a fruit-bearing tree, symbolic of heavenly bliss] of the utmost boundary," shrouded with clouds, "Nigh unto which is the Garden of the Abode" of Allah, "the Lord of Sirius" (Surah 53).
The fruit-bearing trees of life in New Jerusalem can be imagined as branches of the lote-tree grown up into the celestial cube. This tree, like Yggdrasil, the tree in Norse creation mythology, has its roots in the brightness of hell at the Central Sun (the center of our galaxy) . . . in Ginnungagap, the great cleavage in the brain of our galaxy known as the Milky Way. Yggdrasil grows up through the solar system and branches out through the top of the celestial cube. An early iteration of the Maya Tree of Life in the Codex Peresianus (see Kenton 1928) shows a crucified man-tree growing between the horns of a cow-possibly Taurus. The later version published by Cogolludo in 1640 has the tree piercing a turtle shell and a cube and growing up through the arch of the Milky Way out of the galaxy (Kenton 1928). The indigenous people of Brazil still know the tree of paradise as Mira, and Mohammed's "night journey" to the lote-tree is known as the mi'raj. Through this one simple sound- mr -these traditions can all be tied back to the Egyptian hieroglyph homonyms and their meanings of love, death, and geometry.
One remaining geomantic interpretation I have made of John's vision will be, I hope, a contribution to ongoing scholarly investigations of empirical measures in ancient geometric consciousness (e.g. Michell 1981, 1988; Neal 2000; Ruggles 1988). The shortest distance between any two points on a sphere is a segment of a meridian (another mr). On Earth, for example, lines of longitude are meridians; they connect north pole to south pole, and each measures approximately one-half the circumference of Earth. A meridian that continued on around to complete a circle would cut the sphere of Earth in half and would be what geometers refer to as a "great circle." Each of the 15 hoops of the divine feminine Receptacle is a great circle which divides the sphere exactly in half, thus all of the edges of the 120 triangles are segments of meridians.
Given the principle of "as above, so below," it's highly probable that empirically-minded stargazers (wherever they were) measured distances between locations on Earth and between different objects on the dome of the sky along great circle meridians. In the manner of Chinese ivory puzzle balls, Earth (as Chora) would be imagined as a geometric seed inside a larger but identical celestial container. An arc in the heavens of equivalent length to an arc on a model of the divine feminine Receptacle provided a scale for measures transferred to Earth . Using very simple calculations based upon the average circumference of the earth, Becker and I determined the lengths of the three sides of the basic right triangle in Plato's model: 1440 miles, 2160 miles, and 2592 miles (using today's mile of 5280 feet). 34
In Revelation, "by the unit of measurement the angel used," the wall of New Jerusalem measures 144 cubits (Rev. 21:17). But this could be no ordinary cubit since John had also given the measure of the wall as 12,000 furlongs. A standard cubit is the average length from elbow to fingertip, about 18-22 inches. In Greek, the same measure is the pekhys and in Hebrew ammah. The answer to this riddle came to me in the weeks after my first visionary experience. I had marked the 15 hoop/120 triangle geometry onto both my Earth globe and my identically sized celestial sphere, yet it took months of staring before it dawned on me that I had replicated the ancient system and could proportionally compare distances on any sphere. I had trained my eye to recognize equal arcs. Ironically, I had forgotten to go outside and look at the dome of the sky!
One night near Christmas at midnight I did go out to look at the stars, and there was Orion standing center sky. I could see the whole Winter Hexagon and realized that the great circle arc between Aldebaran and Betelgeuse-the "wall" I was proposing for the New Jerusalem-was proportionally the same as the short side of Plato's triangles of the Receptacle. Inside, I compared these arcs on my equal-sized globes and found them to be virtually identical. Since John was living in Patmos (37.5° North), he would almost certainly have been using the Northern-value Greek furlong of 633.6 feet. Using simple arithmetic: 633.6 feet multiplied by 12,000 furlongs gives 7,603,200 feet. Dividing this number by the 5280 feet in a mile, the measure is 1440 miles-the length in miles of the short side of the triangle on the Earth grid. Reducing this number (for simplicity) by a simple factor of 10, a common device in ancient metrology (Michell 1988), gives the angel's measure of 144 "cubits" for the wall of New Jerusalem.
The etymological derivation of the ancient "cubit" is the Latin cubitum, "the elbow," which brings to mind not only the semantics of el -the bend in the knee or elbow-but the arms and legs of the celestial cubic female in the Arnhem Land drawing (Figure 10).
The implications of this tiny clue are enormous. Not only does it suggest very ancient origins of the foot and the mile as we know them today, but it reveals a more or less lost consciousness of design ecology as the root of geomancy -a system of logic in which common measures were observed in cosmic divine feminine Receptacles at all scales of being. We may currently find ourselves in the position of being unable to grasp sacred interconnections of natural phenomena because we try to measure them in units of time, space, and/or frequency that are more quickly determined on calculators. In the rush to embrace the metric system as the "global measure of unity" in fields such as archaeology and biophysics, we may actually be making it impossible to see the measured wisdom inherent in the objects and beings we are trying to understand.
John's ecstatic vision of the crystal city only begins to suggest the awesome perceptual energy of consciousness that can be stimulated by three-dimensional geometries. This power has long been known to indigenous cultures whose highly ambiguous geomantic art suggests to me an intentional cultivation of liminality as "ordinary reality" in a living universe of movement and interconnection. 35 Ancient and contemporary shamanic art forms snap in and out of shape and perspective- and apparently trigger an enhanced capacity for multiple meaning and symbol usage. 36 They exemplify a fundamental (and easily forgotten) principle of consciousness in Chora's sacred realm: geometric shapes will generate visions that are sacred illusions, myths of what "was," "is," and "will be"-myths of position in time. This is geomancy: the science of the trickster, who often-to his own peril-confuses vision with ultimate reality. Nonetheless, it is out of these illusions that the human future is and apparently has long been created.
1. The first surviving, integrated written statement of this tradition is Plato's Timaeus and Critias. Guthrie 1987 contains invaluable primary texts that set Plato's work into a much broader esoteric context. For readers looking for an entry point into this vast realm popularly referred to as "sacred geometry," see Schneider 1994. Both Bierhorst 1972 and Sullivan 1996 provide a wealth of ethnographic examples of geometric vision. Heath 2004 proposes a model for prehistoric consciousness founded in integer-based arithmetic and geometry.
2. William Becker and I were the first contemporary researchers to make this connection between Plato's spherical "Receptacle of Infinite Becoming" (Gaia) and the hexakis icosahedron (the only spherical configuration possible for 120 identical right triangles). See also Cornford n.d. and Becker and Hagens 1992. An internet search for Plato+Gaia+Receptacle will bring up a wealth of cached sites for further research.
3. The most extensive an widely available rendition of this vision of the cosmos as hoops is found in Neihardt 1961. A more popular version, including the conceptualization of the universe as a sphere with woman holding spirit at the center, is McGaa 1990 . I am indebted to Barbara Lane Giammarino, a Penobscot hoop dancer, for her confirmation that the so-called "crossed circle" is a two-dimensional representation of three symmetrically entwined hoops that created an octahedron. Joseph Meeker introduced the Brulé Sioux creation story in a seminar entitled "Environmental Mythology" that we convened at Mesa Verde National Park , September 10, 1989. See also Bierhorst 1972 and the etymology of Viracocha in Sullivan 1996.
4. Occasionally, the color of blue is used instead of black as the color of the element water. The photographing of sacred ceremonies around the world was commonplace early in the 20th century, and the results were often featured in popular collections published by corporations such as Time-Life Books and Readers Digest. This is rare today. As a child, I studied pictures of trickster figures such as the Navajo Yebitsai manipulating a straight line of connected sticks into and out of various three-dimensional geometric shapes. Many of the tools are illustrated in Culin 1975. Essays in Kroeber 1994 examines a context of resurgence within which these objects may again be appearing.
5. The field of global geometric mapping is full of land mines including Atlantis, ancient global cultures, and extraterrestrial travel. Nonetheless, much of the "fringe" data we considered in the early 1980s-for example, maps and anomalous artifacts discovered, published or speculated upon by dedicated lay researchers such as Harris (1975), Mertz (1972), Corliss (1978), and Hapgood (1979)-are now being reconsidered in light of more sophisticated techniques for evaluating their authenticity. As an example, historians in Beijing have identified two maps procured in a South Korean antique shop in 1972 by Harris-then a third-generation Baptist missionary-as Ming Dynasty. A major contribution to metrology and ancient mapping research, largely ignored by academia, is Stecchini n.d.
6. See Hagens 1992, Hagens 1997a, Hagens 2004 and Becker and Hagens 1991 for applications and reflective critical assessment of this model . An internet search of "Becker Hagens Grid" will reveal how, for better or worse, it took on a life of its own .
7. See Hagens 1997a and 1999 for a preliminary conceptualization of this model. Also, see Sullivan 1996 for an important indigenous conceptualization from Peru .
8. A classic, readable introduction to the field is Pennick 1995.
9. Identification of this object was verified by Valery Makarov shortly before his death. Kilford 2004 discusses similar artifacts and includes a comprehensive bibliography.
10. For ease of understanding, I will be using Anglicized spellings of Greek words.
11. Interpretations of the confusions of and similarities between Receptacle, Gaia, Chora, and Nurse of Becoming are introduced in Parada 1997 and Opsopaus 1998 (particularly Part II). At one level of analysis, the astronomical interpretation of mythology proposed in Santillana and von Dechend 1972 is foundational in my thinking: animals are stars; gods and goddesses are planets, Sun and Moon; and topographical references are metaphors for locations on the celestial sphere. Sullivan 1996 is a tour de force application of these principles. The Demiurge can be interpreted as the planet Saturn, "giver of measure and time," owing to observable geometric properties of its recurrence at specific locations and times-particularly its regular conjunction with Zeus-Jupiter
12. Pythagoras is thought to have adopted this idea of a primal, spherical luminous human essence from Zoroaster (various essays and perspectives in Guthrie 1987).
13. See, for example, Lee 1986.
14. The Laban school of dance, for example, explicitly incorporates geometric shapes as movement icons. See Ness 1992.
15. See Arnold n.d., especially Chapters VII and XIII, for the original text translated into English. For an interpretation, see Hinduwebsite n.d. The purusha and prakriti
are not strictly male and female. Purusha is "the cause and experiencer of the pleasures and pains of the body," and the prakriti is "the energy and material form of the experience"-among many other interpretations. Mehta n.d. discusses in detail the architectural layout system based on Vastu Purusha's body.
16. Throughout this essay, I draw from and expand upon the etymological interpretations of linguist Calvin Watkins of Harvard University (Watkins 1969).
17. When I originally noticed that the Egyptian Geb and Greek Gaia shared so much in common, I began searching for connective morphemes between Greek, Hebrew, Egyptian, and Gaelic using Strong 1988, Budge 1978, and Maclennan 1979. The caveat: you will find what you look for. I find that certain morphemes jump out at me no matter what I read, much the same as geometric shapes spring from two-dimensional images. Encyclopedias, children's books, and compendiums of world mythologies and related craft traditions have been especially fertile grounds for confirming what I think I see-especially Leach 1984, Hamilton 1988, O'Neill et al. 1994, Ratajczak 1998, and Littleton 2002. Online etymology sites have surprisingly sophisticated discussions of these kinds of ideas, for example, http://www.etymoline.com .
18. Saddam Hussein warned that the Gulf War of 1991 would be "the mother of all battles." He may well have meant that it would be "the measure" of what was to come.
19. In a private communication, A.M. Davie, Grand Pryor of the Temple of Sion of the Knights Templar in Scotland , stated that the name of the traditional Scotch measuring staff known as the ell is virtually identical to the royal Egyptian staff known as the tcham. The name ell is derived from the Egyptian leg hieroglyph (foot, ankle, and lower leg), which eventually became a written letter "L."
20.I want to emphasize that I have drawn examples from a variety of academic and non-academic sources, including children's books, museum displays, and so-called "coffee table books." There are many other names which I have not cited and which don't follow the patterns that I am proposing to exemplify geomantic memory.
21. Any of these names can be Google-searched and a variety of sources identified to further research their characteristics.
22. See Besant and Leadbeater 1919. This was my initial inspiration for this idea.
23.I would suggest that the principles of symbolic connection between human, cosmos, and spiralling are most clearly articulated in rituals and symbols associated with the bullroarer and the spinner or "magic wheel." See especially Hagens 2005a and 2005b. Additional materials, including replica artifacts, highly visual PowerPoint presentations, and information for multicultural curriculum planning will soon be available at http://www.WorldGeometry.com
24. For detailed examples of these principles, see Doczi 1981, Blair 1976, Critchlow 1982, Andrews 1966, Adams and Whicher 1982, and Schwenk 1976. Each of these works on the geometry of natural form has a unique vision and poetry.
25. Becker and Hagens 1992, an unpublished manuscript presented to the Program in Industrial Design at the University of Illinois-Chicago, discusses and illustrates our theory of structural interconnectivity via the hexagonal threshold. It will soon be available at http://www.WorldGeometry.com
26. Santillana and von Dechend 1977 is probably the most comprehensive, well-documented discussion available of the impact of astronomical events upon the development of world mythology.
27. Much of this lore is from Kenton 1928, unfortunately no longer in print. It is a humble, enormously insightful book by an educated lay researcher.
28. Mathews 1894 and Mathews 1896-1897 contains descriptions of initiation ceremonies among the Keeparra and Bora tribes collected just as these rituals were phasing out of regular practice. I have suggested the tie to Oroboros (the self-defecating hoopsnake) based upon these reports . Mathews writes that the most sacred initiatory ground was referred to as the "place of urination." It was here that the boys drank urine, ate feces, and were instructed in the repetitive cycles of the cosmos of which they were a part. In preparation for returning home, each of initiate was smeared with feces and was said to have been "swallowed" and then excreted by the All-Father. This ritual is reminiscent of the reverence accorded to the Egyptian divinity Khepera, who represents the scarab-a beetle that raises its larvae in dung. It echoes Plato's teaching in Timaeus that Earth was designed to supply its own nourishment from its own decay" (Plato 1965:44-45).
29. See Hagens 1991 for a discussion of rhombic symmetry and Paleolithic representations of animal and human limbic brains as mature women.
30. We might envision the ubiquity of that cubic structure, on the micro and macro scale, as the salt of the Earth.
31. See Hagens 2005b for a summary of the nearly universal association of Orion, the solstice sun, and the bullroarer/double-edged flint blade. This essay also includes a discussion of representations of a tongue-bullroarer on the Aztec Sun Stone (popularly known as the Sun Calendar) and the circular zodiac at Denderah.
32. Two thousand years ago, the stars of Orion were known to Arab civilizations as Al Jauzah -"the giant," a female divinity unsuccessfully courted by Canopus . John's description of "The Lord" and "The Woman" appear to re-state Plato's distinction between Demiurge and Receptacle. An excellent, readable text interpreting the Book of Revelation in terms of (in the author's words) "taboo subjects such as numerology, sacred geometry and astronomy-astrology" is Strachan 1985. A professor of biblical studies at Edinburgh University , Strachan writes: "I did not deliberately set out to get involved in such suspect disciplines; it just happened that way. Biblical cosmology is like that. It is all about geometric proportions, the cycles of heavenly bodies, and numbers. I'm sorry that it should be so. It upsets me to have to accept it. I would much rather it were otherwise, because I do not want to be thought of as a pariah for having drawn attention to certain unacceptable aspects of holy writ. However, that is the risk I must take, since it was these 'unpresentable parts' that 'found me' in the course of my studies."
33.I am deeply indebted to my recently deceased colleague at Governors State University , Dr. Mohammed Kishta, for his scholarship and patience in working with me on the astronomy and revealed wisdom in the Qu-ran.
34. We were astonished by the coincidences of these measures with those in ancient metrology. Our preliminary conclusions are summarized in Becker and Hagens 1991.
35.Gisela Wendler has suggested its return as postmodern shamanic reality.
36. See Hagens 1991 for a treatment of intentional ambiguity in Paleolithic representations of the Goddess.
Adams, George and Olive Whicher - 1982 The Plant Between Sun and Earth. Boulder : Shambhala.
Allen, R. H. - 1963 Star Names: Their Lore and Meaning. New York : Dover Publications.
Andrews, Donald Hatch - 1966 The Symphony of Life. Lee's Summit , MO : Unity Books.
Arnold, Edwin (trans.) - N.d. The Bhagavad-Gita. Electronic document, http://caltechc.caltech.edu/ ~caltechc/clibrary/CD%20092/ts092024.pdf, accessed November 15, 2005.
Becker, William and Bethe Hagens - 1991 The Rings of Gaia. In The Power of Place. James Swan, ed. Pp. 257-279. Wheaton , IL : Quest.
Becker, William and Bethe Hagens - 1992 Ancient Futures and the Geometry of Life. Unpublished MS, School of Art , Architecture and Urban Planning, University of Illinois at Chicago .
Besant, Annie and Charles W. Leadbeater - 1919 Occult Chemistry: Clairvoyant Observations on the Chemical Elements. London : Theosophical Publishing House.
Bierhorst, John, ed. - 1972 In the Trail of the Wind: American Indian Poems and Ritual Orations. New York : Farrar, Strauss, and Giroux.
Bird, Chris - 1975 Planetary Grid. New Age Journal (May):36-41.
Blair, Lawrence - 1975 Rhythms of Vision: The Changing Patterns of Belief. New York : Schocken Books.
Budge, W A. Wallis - 1978 An Egyptian Hieroglyphic Dictionary in Two Volumes. New York : Dover
Publications, Inc. Cairns-Smith, A. G. - 1985 The First Organisms. Scientific American 252(6):90-101.
Catholic Biblical Association of America - 1970 The New American Bible. St. Joseph edition. New York : Catholic Book Publishing Co.
Cavalli-Sforza, L. L. - 1991 Genes, Peoples and Languages. Scientific American 265(5):104-111.
Cirlot, J. E. - 1971 A Dictionary of Symbols. 2nd edition. Jack Sage, trans. New York : Philosophical Library.
Cooper, J. C. - 1990 An Illustrated Encyclopaedia of Traditional Symbols. London : Thames and Hudson .
Cornford, Francis M. - N.d. Plato's Cosmology: The Timaeus of Plato. Bobbs-Merrill reprint.
Corliss, William R. - 1980 Ancient Man: A Handbook of Puzzling Artifacts. Glen Arm, Maryland : The Sourcebook Project.
Critchlow, Keith - 1982 Time Stands Still: New Light on Megalithic Science. New York : St. Martin's Press.
Culin, Stewart - 1975 Games of the North American Indians. New York : Dover Publications.
Doczi, Gyorgy - 1981 The Power of Limits: Proportional Harmonies in Nature, Art & Architecture. Boulder : Shambhala.
Fuller, R. Buckminster - 1975 Synergetics: Explorations in the Geometry of Thinking. New York : Macmillan Publishing Company, Inc.
Guthrie, Kenneth S. (trans.) - 1987 The Pythagorean Sourcebook and Library: An Anthology of Ancient Writings Which Relate to Pythagoras and Pythagorean Philosophy. David R. Fideler, ed. Grand Rapids , MI : Phanes Press.
Hagens, Bethe - 1991 Venuses, Turtles, and Other Hand-Held Cosmic Models. In On Semiotic Modeling. M. Anderson and F. Merrell, eds. Pp. 47-60. The Hague : Mouton de Gruyter.
Hagens, Bethe - 1992 Paradise and Precession (Keynote Address). In Proceedings of the 24th Annual Meeting in Finland : Environment-A Challenge for Adult Education. Pp. 46-57. Helsinki : Association of Finnish Adult Education Associations.
Hagens, Bethe - 1997a A Sustainable Geometry for the Planet and the Celestial Sphere. In Proceedings: Symposium on Design Tools for Placemaking. Boston : North East Solar Energy Association.
Hagens, Bethe - 1997b Etain: A Winter Solstice Fantasy. Play produced by Seaglass Performing Arts, Kennebunkport , ME. Archived at http://www.WorldGeometry.com .
Hagens, Bethe - 1999 The Celestial Basket. Paper presented at the Oxford VI International Archaeoastronomy Conference, June 17, 1999.
Hagens, Bethe - 2004 Geometry: An Ancient Universal Teaching Model of Unity. In Educators Leading the Challenge to Alleviate School Violence. In Rose Duhon-Sells and Leslie Agard- Jones , eds. Pp. 99-115. Lewiston : Edwin Mellen.
Hagens, Bethe - 2005a Bullroarers. In The Berkshire Encyclopedia of World History, vol. I. William H. McNeill, Jerry H. Bentley, David Christian, David Levinson, J. R. McNeill, Heidi Roupp, and Judith Zinsser, eds. Pp. 276-278. Great Barrington , MA : Berkshire Publishing Group.
Hagens, Bethe - 2005b Timbres of the Spheres: The Bullroarer and the Magic Wheel. In Proceedings: Conference Internationale Musicologique. Caroline Traube and Serge Lacasse, eds. (CD) Montreal : OICM/UMontreal.
Hamilton , Virginia - 1988 In the Beginning: Creation Stories from Around the World. New York : Harcourt Brace Jovanovich, Publishers.
Hapgood, Charles H. - 1979 Maps of the Ancient Sea Kings: Evidence of Advanced Civilization in the Ice Age. New York : E. P Dutton.
Harris, Hendon M. - 1975 The Asiatic Fathers of America . Self-published.
Heath, Richard - 2004 Matrix of Creation: Sacred Geometry in the Realm of the Planets. Rochester , VT: Inner Traditions .
Herder Freiburg - 1986 The Herder Symbol Dictionary: Symbols from Art, Archaeology, Mythology, Literature, and Religion. Boris Matthews, trans. Wilmette , IL : ChironPublications.
Hindu website - N.d. Hinduism and Creation: The 24 Principles of Creation Samkhya Yoga. Electronic document, http://www.experiencefestival.com/a/Hinduism_and_Creation/id/54135 , accessed November 15, 2005.
Jenny, Hans - 2001 Cymatics: A Study of Wave Phenomena & Vibration. San Francisco : Macromedia Press.
Josephus, Flavius - 1835 The Works of Flavius Josephus, The Learned and Authentic Jewish Historian and Celebrated Warrior. William Whiston, trans. Baltimore : Armstrong and Plaskitt.
Kenton, Edna - 1928 The Book of Earths. New York : William Morrow & Company.
Kilford, L. J. P. - 2004 A Mathematical Tourist in Germany . Electronic document,
http://www. maths.ox.ac.uk/~kilford/dodecahedra.shtml , accessed November 2005.
Kroeber, Karl, ed. - 1994 American Indian Persistence and Resurgence. Durham : Duke University Press.
Krupp, E. - 1991 Beyond the Blue Horizon: Myths and Legends of the Sun, Moon, Stars and Planets. New York : HarperCollins.
Lamy, Lucie - 1981 Egyptian Mysteries: New Light on Ancient Spiritual Knowledge. New York : Crossroad.
Leach, Maria, ed. - 1984 Funk & Wagnall's Standard Dictionary of Folklore, Mythology, and Legend. New York : Harper & Row, Publishers, Inc.
Lee, Desmond - 1986 Introduction. In Plato: Timaeus and Critias. Desmond Lee, trans. Pp. 7-25. New York : Viking Penguin.
Littleton , C. Scott, ed. - 2002 Mythology: The Illustrated Anthology of World Myth and Storytelling. London : Duncan Baird Publishers.
Lockyer, J. N. - 1964 The Dawn of Astronomy. Cambridge : MIT Press.
Maclennan, Malcolm - 1979 A Pronouncing and Etymological Dictionary of the Gaelic Language. Aberdeen , Scotland : Acair and Aberdeen University Press.
Mathews, R. H. - 1894 The Bora, or Initiation Ceremonies of the Kamilaroi Tribe. Journal of the Anthropological Institute 24410-430.
Mathews, R. H. - 1896-1897 The Keeparra Ceremony of Initiation. Journal of the Anthropological Institute 26:320-340.
McGaa, Ed - 1990 Mother Earth Spirituality: Native American Paths to Healing Ourselves and Our World. San Francisco : HarperCollins Publishers.
Mehta, Vistasp - N.d. Rational Vastu in a Post Modern World. Electronic document, http://www.btsquarepeg.com/vastu/book/index.html , accessed January 5, 2006.
Mertz, Henriette - 1972 Pale Ink: Two Ancient Records of Chinese Exploration in America . Chicago : The Swallow Press, Inc.
Michell, John - 1979 Natural Likeness: Faces and Figures in Nature. New York : E. P. Dutton.
Michell, John - 1981 Ancient Metrology: The Dimensions of Stonehenge and of the Whole World as Therein Symbolized. Bristol , England : Pentacle Books.
Michell, John - 1988 The Dimensions of Paradise : The Proportions and Symbolic Numbers of Ancient Cosmology. San Francisco : Harper & Row, Publishers.
Miller, Mary, and Karl Taube - 1997 An Illustrated Dictionary of The Gods and Symbols of Ancient Mexico and the Maya. New York : Thames and Hudson .
Neal, John - 2000 All Done With Mirrors (Opus 2): An Exploration of Measure, Proportion, Ratio and Number. London : The Secret Academy .
Neihardt, John G. - 1961 Black Elk Speaks: Being the Life Story of a Holy Man of the Oglala Sioux as told to John G. Neihardt (Flaming Rainbow). Lincoln : University of Nebraska Press.
Ness, Sally - 1992 Body Movement and Culture. Philadelphia : University of Pennsylvania Press.
Newcomb, Franc J. and Gladys A. Reichard - 1975 Sandpaintings of the Navajo Shooting Chant. New York, Dover Publications, Inc.
Oak, P. N. - 2004 Was the Kaaba Originally a Hindu Temple ? Electronic document, http://www.hinduism.co.za/kaabaa.htm , accessed September 1, 2005.
O'Neill, Cynthia, Peter Casterton and Catherine Headlam - 1994 Goddesses, Heroes, and Shamans: The Young People's Guide to World Mythology. New York : Kingfisher.
Opsopaus, John - 1998 Holodemirgia vel Conditus Universi Orphicus (Annotated). Electronic document, http://www.cs.utk.edu/~mclennan/BA/Holodemiurgia/ann/#copyright , accessed November 15, 2005.
Parada, Carlos - 1997 Greek Mythology Link. Electronic document, http://homepage.mac.com/cparada/ GML/SearchGML.html, accessed November 15, 2005.
Pennick, Nigel - 1995 The Ancient Science of Geomancy: Living in Harmony with the Earth. Sebastopol , CA : CRCS Publications.
Plato - 1986 Timaeus and Critias. Desmond Lee trans. New York : Viking Penguin.
Poignant, Roslyn - 1967 Oceanic Mythology: The Myths of Polynesia , Micronesia , Melanesia , Australia . London : Paul Hamlyn.
Presidency of Islamic Researches, IFTA, Call and Guidance, eds.- N.d. The Holy Qu-ran. Al-Madinah Al-Munawarah, Saudi Arabia : King Fahd Holy Qu-ran Printing Complex.
Prokopec, Tomislav, Robert Brandenberger, Anne C. Davis, and Mark Trodden - 1995 Dynamical Breaking of CPT Symmetry in Defect Networks and Baryogenesis. Electronic document, http://www.lns.cornell.edu/public/CLNS/1995/CLNS95- 1368/clns95-1368.ps, accessed January 2003.
Ratajczak, Richard - 1998 Annotated String Figure Bibliography. Electronic document, http://www.isfa.org/ rataj.htm, accessed June 2005.
Raymo, Chet - 1982 365 Starry Nights: An Introduction to Astronomy for Every Night of the Year. New York : Simon & Schuster.
Ruggles, C. L. N. - 1988 Records in Stone: Papers in Memory of Alexander Thom. Cambridge , England : Cambridge University Press.
Santillana, G. de and H. von Dechend - 1977 Hamlet's Mill: An Essay Investigating the Origins of Human Knowledge and its Transmission through Myth. Boston : David R. Godine.
Schneider, Michael S. - 1994 A Beginner's Guide to Constructing the Universe: The Mathematical Archetypes of Nature, Art, and Science. New York : HarperCollins Publishers.
Schwenk, Theodor - 1978 Sensitive Chaos: The Creation of Flowing Forms in Water & Air. Olive Whicher and Johanna Wrigley, trans. New York : Schocken Books.
Shepard, Odell - 1956 The Lore of the Unicorn. Winchester , MA : Allen & Unwin, Inc.
Stecchini, Livio Catulo - N.d. Metrology: The Forgotten Science. Electronic document, http://www.metrum.org , accessed November 15, 2005.
Stevens-Arroyo, Antonio M. - 1988 Cave of the Jagua: The Mythological World of the Tainos. New Haven : Yale University Press.
Strachan, Gordon - 1985 Christ and the Cosmos. Dunbar , Scotland : Labarum Publications Ltd.
Strong, James - 1988 The Exhaustive Concordance of the Bible. Peabody , MA : Hendrickson Publishers.
Sullivan, William - 1996 The Secret of the Incas: Myth, Astronomy and the War Against Time. New York : Crown Publishers, Inc.
Watkins, Calvert and the Department of Linguistics at Harvard University - 1969 Appendix. In The American Heritage Dictionary of the English Language. William Morris, ed. Pp. 1496-1550. Boston : Houghton Mifflin Company.
Weiss, R. - 1988 Glass Eating Turtle Fills Unique Niche. Science News 133:70.
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