THE GEOMETRY OF BEING: Part 4
Sacred Geometry and The New Science
By Robert Gulick
A Presentation to the Spirit of Place Symposium, 1989
University of California, Davis, California
Solar Principle: The Ancient of Days
In William Blake’s Ancient of Days the Creator is personified, setting within the sun, holding the compass, the tool of the Geometer. It is spread apart to form a triangle, symbolizing the Trinity, but here a special triangle, that which is the profile of the Great Pyramid of Giza, the temple of Initiation of ancient Egypt.
God, the Creator, is universally symbolized by the sun-source of energy for all life and all physical phenomenon on Earth. The solar principle is then the First principle of Cause-its geometrical representation is the circle, Unity, with the point, symbolizing manifestation within.(the circle without the point being the, Unmanifest-Unknowable-the Ain Soph). This then is the symbol of the Law of One-the totality in which all exists and which exists in all. To manifest in the third density requires a trinitization of the Absolute into three aspects, that is to become form-the most elementary of which is the triangle, there is an interaction of Will (purpose-impulse to be), Love (the energy-force-desire that powers the plan) and Mind (the Plan-the thought-Idea-the ordering or organization of creation). Through Trinity-point, line and circle, the Divine Geometer creates in geometry.
To be clear, there in is fundamentally no duality-that is within the “Seamless Unity”. There is only the appearance of separation-a paradox, for at each level of integration of unification of dualities there exists higher states to be unified. Thus there is no separation between the concept of divine creation and that of the evolution of natural and organic forms through an interaction of scientific laws. One no more fully explains the nature of existence than the other: science ignores spirit in nature-the impulse to exist to be alive while religion fails to explain what is plainly observable as natural history. Instead they are two sides of the same coin. The God creates through evolution and physical law, (science explains “how”) the divine in turn provides the underlying unity to science and reason or the “why” for existence. In Sacred Geometry we are dealing with a Sacred Science that connects these two. It comes from a basic premise of the reality of Spirit behind all manifestation and seeks to know the universal laws and harmonics from which all physical laws are derived and that operate to create expression in any given dimension.
Solar Principle: Symbols
Time lapse photograph of the stars turning about the pole.
The aboriginal “soak”.
The mandala literally meaning circle symbolized creation.
Zuni Indian Pot and basket.
Time in space…as the earth turns on its axis the stars appear to rotate about a fixed point, the focus of all celestial action, the Pole star.
For the Australian aborigines the Creator and first man emerged from the center of the earth through the “soak” a mystical doorway formed by a pattern of concentric circles. Manifestation occurs through the circle ritualistically re-created from the earth mixed with blood (power of life) let from the phallus (creative principle) fixing the point in physical reality.
The pot and the basket, universal utensils in all cultures, symbolized the world-the path-the Self which is why so much effort was expended n their creation. They were constant reminders-connectors to the inner dimensions of spirit, both in their design, in their decoration and in the very process of their making.
Mandalas and sacred diagrams from many traditions represent the path to the center, the abode of the Absolute, through a pattern of geometry-retracing the original process of creation or structuring the relationship of creative principles. The flower, the maze, the map one traced as one turned to scan the horizon with self in the center…these are more symbols of the solar principle.
The nesting of the Platonic solids creates a geometric matrix that gave order to the chaos of elements in nature. The five solids are said to be regular: perfect in fitting within a sphere with symmetry about their axis, all vertices touching the surface of the sphere, their faces having a single shape, the centers just touching an enclosed sphere. They are in effect “transforms” of the sphere.
The nesting of the regular solids according to Platonic cosmology
One of the traditions of sacred geometry comes from ancient Egypt, through Pythagoras to Plato, an initiate in the mysteries. Cosmology is encoded in Timaeus in geometrical metaphor. For Plato, “Purely mathematical structures are the underlying principles of matter.”
The Absolute, the Unmanifest, coagulates out of the void through the vibrational desire to be…It becomes Being, the Supreme “I AM”. It is the primal point of emanation, the Bindu, the center of creation in Hindu teaching. From this center the Being expands in space ,”That”, into Becoming, “I AM”, manifested as sphere, the most perfect volume. “I Am That I Am”- the Universe, the Demiurge, is perfect manifestation, a volume whose center is everywhere at once.
Creation occurs from primal Point (Will to be) to circle moving in Time (the activity of Mind in manifestation) through Space (the relationship of All to the Center) symbolized by the line-radius of the circle. In the Trinity space is the mediator, the Christ principle in western theology, the relationship to the “Father”.
In the original creation consciousness was without form, without a “body” to experience the “self. This primal chaos of energy within the Universe was organized by geometry. Thus geometry exists in the space between Being and Becoming…within the perfect sphere…the regular Platonic solids organize the original chaos of Elements. The path from Chaos to Cosmos(order) is through geometry (what we now call the quantum field).
In Platonic terms the solids represented the Elements of matter in the sense that matter had 4 qualities or states in the phenomenal world-what science would call solid, liquid, gas and plasma; on another level they corresponded to the dimensions in which Man existed:
Earth the Cube: 6 sides physical plane
Water the Icosahedron: 20 side emotional plane
Air the Octahedron: 8 sides mental plane
Fire the Tetrahedron: 4 sides spiritual plane
A fifth element pervaded all the others:
Ether the Dodecahedron: 12 sides Akasha-Space
The tetrahedron and cube, fire and earth, were the poles of an octave of energy, between which were inserted air and water, the octahedron and icosahedron, as notes in a musical scale, such that there was a proportional harmony between the elements. The matrix of geometries was therefore the archetypal thought-form.
“Plato’s fabricator of the Universe created order from the primordial chaos of these elements by means of the essential forms and numbers. The ordering according to number and form on a higher plane resulted in the intended disposition of the five elements in the physical universe. The essential forms and numbers then act as the interface between the higher and lower realms. They have in themselves, and in their analogues with the elements, the power to shape the material world”
Plato’s geometrical ordering of energy becomes then the fundamental quantum field, the hologram or the M-field. Individually they are like fractal seed patterns which impose limits and therefore order-pattern in a chaotic system.
The Cosmic Web
Cosmic web of plazma
The spirals of the fractal universe
Tibetan Stanzas of Dyzan (Secret Doctrine): The ancient cosmology of Tibet – The numbers, principles, of Creation: the coagulation out of the Void, the insemination of the Cosmic Egg, the separation of the Absolute and the Manifest creation in vibration.
• “Father-Mother spin a web. Upper end in Spirit, the Light of the One Darkness and the Lower end in Matter. And this web is the Universe Spun out of two substances made as one, which is Svabhâvat . Stanza III, 10
• It expands with the Breath of Fire (Spirit) and contracts when
touched by the Breath of Mother (matter). Stanza III, 11
• Svabhâvat sends Fohat to harden the Atoms. Each is a part of the Web. Reflecting the “Self-Existent Lord”, like a Mirror, each becomes in turn a World. Stanza III, 12
• Svabhâvat, is the the Voice of the Word, the Numbers, for he is One and Nine. the Ten are the Arûpa (formless) Universe …And the square (matter) is circled (spirit). Stanza IV, 5
• Fohat is the Steed… thought is the Rider. Fohat (Cosmic Fire) traces spirals…lines upon it (the web). Stanza V, 2 & 4
Creation follows the activity/union, the dance of Shiva & Shakti: Idea and power-thought and fohat.
Three produced four out of the sacred sound AUM…together they are seven whose activity produces the forms is succession…fire, water, air and earth and out of these the forms of creation and the planes of consciousness.
Expanding sphere of electromagnetic energy produces spiraling lines of electro-static and magnetic waves that intersect creating nodes which when connected create a sequence of platonic solids. (see the Electric Universe).
The solids can be traced within the Kabbalistic “Tree of Life”, the cosmic pattern of ten Divine emanations of the Sefiroth, the foundational structure of the Universe. The tree then is seen as a projection of a volumetric-3-dimensional higher reality. Each solid occupies (is the transform of) one of the Four Worlds whose descending order traditionally generated the Tree. The edges of the solids define the paths through the planes, the sides of the solids, connecting the Sefiroth, the focus of energy at the vertexes.
The Lesser Maze: Vedic Cosmology
The “Lesser Maze” as construct-ed. by Gordon Plummer in the Mathematics of the Cosmic Mind.
The octave of Vedic Harmonics is defined by the Regular solids:
Do – Sphere
Re – Icosahedron
Mi – Octahedron
Fa – Star Tetrahedron
So – Cube
La – Dodechedron
Ti – Icosahedron
Do – Sphere
From Hindu-Vedic tradition there also comes a cosmology encoded in the geometry of the solids, each nested with in the others in sequence, describing not the relationship of primal energies, but instead the process of involution-evolution from seed spirit-consciousness (inner icosahedron) to monad (octahedron) to soul (star tetrahedron constructed on the monad) and body (cube, the outer form of the soul). This the full play of manifested existence sits within the Dodecahedron, seen in Hindu tradition as Prakriti, the feminine power of creation, the totality of manifestation-nature-the Universal Mother, whose principle is represents in the number 12. Projected from this in turn, untouched by manifestation within and in turn sitting unmanifest within manifestation, the Icosahedron, Brahma the Infinite creator as Divine Thought-Pure Ideation. The sequence repeats infinitely inward and outward, a harmonic progression of worlds within worlds-reality within reality: Thought -Purusha generating existence-Prakriti with all manifestation within.
The maze of nested geometries becomes a model for proportioning sacred space so as to bring one in touch with the primal creative process. The series of spheres that can be drawn just touching the vertices, edges and planes of this sequence of solids is seven in number, thus defining the whole note interval of a harmonic octave and the number of chakras in the body.
In both traditions each of the solids enfolds the others like an Implicate order…the progression defining a process of states of being in one and a relationship of principle in the other…each like a Quantum state related in whole number ratios. All together they are the Universe, the body of experience of the Creator. Progression through these geometries or their harmonic proportions in the Temple was to awaken in one a corresponding state of consciousness.
The Universal Matrix
Fuller’s Universal Matrix orders existence.
Buckminster Fuller, inventor of the geodesic dome, proposed that a universal, all-space filling lattice of tetrahedron and octahedrons called the isotropic vector matrix, was the geometrical underpinning ordering energy-matter in physical manifestation. Such a matrix is generated by connecting the centers of close-packed spheres. Statistical observation of the Universe indicates that matter is distributed as if on the surface of spheres…voids which are millions of light years across.
Geometrical Solids in Nature
Geometry in Nature
Geometry in Nature
The solids manifest in nature both in terms of physical and organizational structures. For example:
……Radialaria, marine micro-organisms , exhibit all five regular solids.
……Chlorophyll molecule: the central porphorin ring common to both blood and chlorophyll is constructed by a nested geometry of atoms defining a dodecahedron around an octahedron around an inner tetrahedron centered upon a single atom.
…….Elementary particles of matter defined by physics by the combination of constituent quarks, thought to be the most fundamental unit of matter, tend to be grouped into families described by the geometry of the regular solids.
Platonic Solids Cosmology
Permanent Atom: Koilon, Fractal of Being-Occult Chemistry
Sacred Geometry- Platonic cosmology organize chaos
Geometry of the Solar System
Johannas Kepler’s model of the solar system.
Geometry seems to demonstrate itself in pure forms at scales that are either very large, relative to man, of very small. The ancient astronomers mapped the wheel of the zodiac in a pattern of angles, trines and squares. Johannes Kepler’s who developed the laws of planetary motion, also discovered the the orbits of the planets as a proportional relationship to each other could be defined by the spheres that alternately circumscribed a series of nested solids. Significantly in this arrangement the two solids that defined Earth’s orbit are the icosahedron and dodecahedron.
John Martineau: Book of Coincidence. The ratio of orbital relationships between various planets of the solar system are defined by the ratios between the various inscribed and circumscribed poly- hedra and regular polygons.
Platonic solids locate features on planets:
Tetrahedra-19.5° Teotihuacan, Cydonia & Olympus Moons on Mars, Red Spot on Jupiter
Herpes Simplex II virus, is an icosahedral within a sphere. Perhaps the roots of disease at some level have to do with the inability of a consciousness to resonate with pure form.
At the other end of the spectrum the virus, the smallest and simplest form of known life exhibits a pure geometry icosahedral, dodecahedral and spherical forms. It is as if the poles of the octave of living systems, from virus to planet, manifest in pure geometries. In between the evolving body, emotion and mind work to interfere with the pure geometrical states.
The Planetary Grid System
The icosa-dodecahedral planetary grid system of “crystal earth”.
Sonic patterns of the sun.
There is evidence in ancient alignments of temples, the Vedic mythology of a crystal earth and the locations of cultures on the planet, and in satellite pictures, of a subtle grid system overlaying the planet . It relates geophysical as well as cultural phenomenon: plate tectonics, weather patterns and ocean currents, mineral deposits, geo-magnetic flus and gravitational anomalies of the planets field, time-space warps like the Bermuda Triangle and the sacred sites of ancient civilizations centers: Great Pyramids, Nazca Plain of Peru, the Anasazi in the 4-corners.
The grid is a spherical projection of the icosahedral-dodecahedral solids, which has been developed and mapped into more detail by recent research of dowsers and anthropologists (Hagens+Becker), combining all the regular solids that organized the elemental energies of the Platonic cosmology. Plato said that if the Planet could be viewed from above (from space) it would appear a ball sewn of 12 patches(a dodecahedron).
The grid system varies form a regular geometry, an irregular mosaic of lines and spirals to flowing curvilinear “dragon currents” corresponding to the mental, physical-etheric and emotional “bodies” of the planet. Vortexes occur where grids intersect…at the intersections between patterns, and function as nodal points of convergent energy, that is as doorways or gates between the various grids of dimensions of the planet. The nodes, or Domes, are energy transmitters and receivers-each programmed to different frequencies…facilitating the integration of energy to and from the planet. Pyramidal temples located at such nodal points, like pressure points on the meridians of a living body, were of geometries that facilitated the flow of energy within, as well as to and from, the grids. Temples located on the grids contain codes in their geometrical form…they are like memory chips in the grid computer system.
The energy patterns sensed by many in the structure of the cosmos and the human body, also operate at several levels within, on and above the body of the earth. The grids are multi-leveled. The planet is a being-alive, we live on the body of an intelligence known in traditions as the planetary Logos. The Gaia Hypothesis , by J. Lovelock, developed from research into life on mars, proposes that the earth behaves like a living organism, a being called Gaia, after the ancient Greek goddess of the earth. Grids may be the meridians of this earth-being. Other research:
……Seismographic data at MIT shows the core of the planet to be an octahedral molten iron crystal.
……The earth, like the sun, rings like a bell with sonic frequencies; the grid may be geo-sonically produced by the standing wave pattern of resonances within the core and mantle of the planet.
……Patterns of the electromagnetic field measured on the surface of the earth have 4 nodal areas where the flux flows into the planet that occur at regular points in a geometrical pattern-60º N. and S.latitude and 120º E. and W. longitude.
……High temperature material composing the majority of the mantle of the earth is compressed into a crystalline structures known as perovskites. Like silicate clays this geometrical matrix of material behaves like a superconductor in which piezoelectric generated currents may account for the earth’s magnetic field as well as the planetary grids.
…..The p;anetary grid system encodes the pattern of the stars/heavens onto the planet.i.e. star mapSee how tree of life is projected onto the grid. The function of pryramids all over the globe is as a defense system and as consciousness/DNA activatin.
Hopi Creation Myth: The 1st World
Kiva- modeled the process of evolution-world to world through a sequence of circles and squares
In Tokpela (Endless Space), only Taiowa, the Creator(Infinite) existed. He conceived the finite, Sotuknang, who was commanded to gather solid substance form endless space, mould forms and arrange into nine universes-kingdoms, one for Taiowa, one for Himself and seven for Life. He did the same with water and wind.
To create life, Sotuknang created Kokyanqwati, Spider Woman, who he instructed to create life. Out of saliva and earth, she made two beings which she covered with a cape made of wisdom (tantra weaving) and over which she sang a creation song. Thus (through vibration-the matrix) did the twins come to life: Poquanghoya-to keep order on the earth by putting hands (geometry) on it-to keep it solid, and Polonqawhoya- to keep order on the earth through sound, so he was called Echo.
Poquanghoya traveled the earth-made mountains and the lower reaches. Polonquawhoya traveled through the earth sounding his call as he was bidden. All the vibratory centers along the earth’s axis, form pole to pole resounded to his call. The whole earth trembled-the universe quivered in tune. Thus he made the whole world and instrument of sound and sound and instrument for carrying messages, resounding praise to the Creator of all.
The twins were then sent to the earth’s poles, Poquanghoya to the north and Polonqawhoya to the south, to keep the earth properly rotating. (A world created and sustained by sound-its centers-chakras along the central axis.
The Kiva represents the third world where man was still in connect with the conscious forces of nature. Its form is a circle, the wold, entered from the earth below and to the fourth world above though the square doorway, incarnation in the body. According to Hopi prophecy the coming global changes portend the entering into the fifth world.
Islamic patterns demonstrate all 32 crystal symmetries… molecular SO2-quartz, here for example. It is an unfolded pattern of a cuboctahedron, a 3-dimensional Archimedian solid contained within the Universal lattice of the matrix.
Crystal earth exhibits a quartz crystal effect…a geometrical matrix able to entrain and amplify electromagnetic energy (such as quartz tuners are in radios). Quartzes and other crystals are being used and investigated for their healing and thought amplification properties. Their ability to resonant and therefore entrain information and energy projected from the human energy field results from a similarity in the geometries, ie the structural affinity, of thought-form and quartz.
The lattice of molecules in quartz crystal layer such that they spiral. In the entrainment of thought energy this spiral matrix acts like a venturi effect, a vortex, to amplify the energy…drawing in subtle energies in the process.
The crystal lattice of the earth’s grids functions in a similar fashion to resonate the evolutionary DNA codes which are mirrored within all life. It amplifies the collective thought of a critical mass of focused human beings to attune all of humanity in a quantum leap of consciousness. Crystal Earth entrains and amplifies the thought and emotion of humanity.
The process of growth mirrors the universal creative process. Human beings begin life-a fertilized ovum, as a Unity, a sphere. The sperm having traveled through multiple layers of the ovum to bring its DNA geometry codes in union with those waiting at the center. The fertilized cell springs into activity in resonating geometries which lie at the heart of being. In this state its first act as an individualized entity is to establish two magnetic poles between which an axis forms along which cell division occurs and eventually the spinal column forms. From this state of unity, like the Irrational Source, Being undergoes an internal act of division-creating multiplicity within and out of itself (for there was nothing outside of the Absolute with which to create). It is an eternal mystery, paradox, that multiplicity arises from an internal division.
The whole of Being exists between these two original polarities as an octave exists between 1 and 2. All phenomenon and growth arise out the the continual alternation of these polarities-each tending to become the other in cyclic activity. Duality is an essential condition of existence-everything in the universe maintains its continuity through polarization. Sex, male and female, is a reflection of the original polarization…the mystical union being a path back to the Source.
Cell division proceeds form a unity through a succession of primary geometries…2, 4, 8 these groupings of cells define first a tetrahedron then cube. Uniquely then the human zygote divides asymmetrically…13 (12 around 1) an icosahedron, then 21 (20 around 1) a dodecahedron…the developing being recapitulates the primal geometries which ordered creation. The number of cells follows the Fibonacci sequence whose ratios define the Golden proportion.
Tibetian Buddhist mandala called the ‘Mahasiddha Maitripada: the great powers of the Divine Mother.
The zodiacal mandala
Like the sperm entering the ovum, the consciousness traces the evolutionary path of integration to the central point of Absolute Being, the Bindu, through the structure of the universe represented by the mandala pattern. The meditative process leads one through a sequence of geometries, circles, triangles and squares, resulting is a psycho-cosmic integration of the mind with the Infinite. The journey inward to Source in the mandalas of eastern traditions is mirrored by the journey outward to God in heaven in western traditions.
The pattern of the zodiac is a pattern of relationships: triplicities (qualities), quadiplicities (the elements: earth. water, air and fire), the 12 signs and houses of experience, the angular relationship of planets and sign. Ones natal horoscope is a personal mandala symbolizing the journey through life in terms of pure geometries of triangle-square-circle
Square-Triangle-Circle: Elements of Sacred Geometry
Zen painter, Sengai’s “Universe’
The interplay of Square-Triangle-Circle represent in many traditions the relationship of the essential qualities of existence and in themselves the Laws of One, of Three (formation) and of Four (manifestation). Material world -the Body, the square of matter, is connected to the world of Spirit, the circle, by the triangle, which has a multitude of meanings-the Soul, the Anthropos-the perfect Man, the trinity, form itself:
Square Triangle Circle
Manifestation Formation Creation
Body Form-Soul Spirit
Generation Diversity Transformation
Intelligence Love-Wisdom Will-Purpose
Holy Spirit Son Father
In alchemy the fusion of the masculine and feminine principles, as with sulphur and mercury, within the alchemical crucible brought about a transmutation of matter into spirit. The catalyst for this, the Philosopher’s Stone, was created in a mystical geometry:
“Make a circle of a man and woman, out of this a square, and out of this a triangle. Make a circle and you will have the Philosopher’s Stone”.
Perfect being, Androgyne, whose symbol was the triangle, is the principle of mediation between heaven and earth
The Four Worlds: Quadriplicity
The Four Worlds of William Blake
Fourness…the square, universally symbolizes the physical world. The concept of the four worlds represents the descent into matter from the creator to spirit through form as with the Four World of the Kabbala , the four Shakti’s (powers) of Shiva, the four directions in native traditions and others. The nature of matter is in quadriplicity:
…..the elements of fire, air, earth and water.
…..the dimensions-three in space and one in time.
…..the planes-spirit, mind, emotion and body-energy.
…..the states of matter-solid, liquid, gas and plasma.
…..the forces-gravity, electromagnetism, strong and the weak nuclear.
Circle in the Square: The Plan of the Temple
The Byzantine cupola is an archetype of temple form relating dome to the square through the triangle.
The fundamental geometry of the temple, according to architectural texts of the Chinese Sung Dynasty, was the circle in the square (heaven on earth) and the square in the circle (earth in heaven). In western tradition the archetypal form of temple is the dome over the square which can only be accomplished through the mediation of triangular forms called pendentives. This seems to have been a canonization of the way traditional peoples raise a tent (triangle) over a carpet (square or rectangle) to connect one in place to the vault of the heavens. The square is fundamental to the plan of all temples as it represented perfection and unity in physical dimension, while the circle, an irrational form, was embodied in vertical dimension. Thus the square prepared one to contact perfection in higher dimension and then grounded the transcendent experience.
In Islamic tradition the Mosque was based on the octagon, in a sense a double square representing the eight-fold path through material existence. The Domes were adorned with highly complex geometries depicting the path to Allah.
The Great Pyramid: The Archetypal Temple
The geometry of the Great Pyramid.
The Great pyramid of Gizeh epitomized the Temple in the perfection of its design and construction: In its forms and measures it modeled the earth, placing it in the solar system and in the great cycles of time. In its geometry the circle is squared, Spirit and matter are equated through the golden proportion which relates the base to the height, horizontal to vertical. In design then it was resonantly attuned to the energies of earth and cosmos, capable of receiving, transforming and transmitting these to and from the earth grids. As a Temple of Initiation then it was a doorway between dimensions.
Russian Pyramid near Moscow
Pyramids around the World
The Planetary Grid System
Planetary Grid System