The Dodecahedron has been a source of metaphysical interest for at least 2000 years. Like a crystal or gem, its facets and symmetries compel our eyes and hearts to observe life more deeply. Some have believed that the Dodecahedron represents an idealized form of Divine thought, will, or idea. To contemplate this symbol was to engage in meditation upon the Divine. Today many people believe there is a lost knowledge residing in the past, slowly being rediscovered. It is known that figures like Pythagoras, Kepler, and Leonardo, among many, were educated in these Sacred Geometries, and held many beliefs about them and their role in the Universe. This polyhedron is one of an order of five
regular polyhedra, that is, they each represent the regular division
of 3-dimensional space, equilaterally and equiangularly. These
are the Tetrahedron, Cube, Octahedron, Dodecahedron, and Icosahedron,
respectively:
The Tetrahedron represents the concept of system. When four radiating points interact, they will form the Tetrahedron. It has four triangular faces, and is the structural basis for geodesic architecture. Buckminster Fuller is the best resource for a deeper understanding of this shape. It has been associated with the element Fire. The Hexahedron or Cube represents the three-dimensional, physical, manifest world, with its four directions (expressed by squares), right angles, and polarities. It has six faces. It has been associated with the element Earth. The Octahedron has a dual relationship to the Cube. Whereas the Cube has six faces and eight vertices, the Octahedron has eight faces and six vertices. It has been associated with the element Air. The Dodecahedron will be addressed down below. The Icosahedron has twenty triangular faces and twelve vertices. It is also considered a first-frequency geodesic structure. It has been associated with the element Water. Progression
1. The Circle represents Unity and Totality. All points from the center radiate equally. It also symbolizes the void; the pre-manifest seed or germinal idea, expanding and asserting itself by will. It is the first, primordial, state of all things. The Beginning. 2. The first step in manifestation is differentiation. So, the circle will split and divide by two at the center, bilaterally. The Cosmic idea is preparing to self-replicate. This is Logos. 3. As the splitting has begun, new radiating points are formed, and new tangents created. The area of the two inner circles equals one-half of the outer one, yet the circumference of them both is identical to the latter. Thus the paradox where the two and one are united. There cannot be two without first, the One. This synthesis and interplay of opposites-as-unity is also revealed in the Chinese Yin-Yang symbol:
4. We next draw tangential arcs from the top and bottom points as centers, creating a vesica, which is also expressed when two circles each touch another at the center, like this (at 90 degrees):
This is Unity becoming Duality. The relationship created between these two circles can yield all regular polygons and in particular the geometric solution for squaring the circle, which is the most elegant geometric analogy of the infinite becoming manifest. A variant of this symbol is in the zodiac, namely Pisces:
Some of the sacred ratios are expressed in totality by the symmetrical vesica (in particular ) then by the Triangle, Square, and Pentagon, respectively.
It can be seen that the first regular divisions of 2-dimensional space emerge as the Triangle, Square and Pentagon. Then come the hexagon, octagon, decagon, and dodecagon...the Ten-Thousand things.
5. In this closed system of radiation, differentiation is a constant impulse, and each circle tangentially presses against another, pushing the envelope of the totality of the surrounding universe. One circle might begin at an open center, until it reaches the other radiating circles. 6. Here we see the first organic system, and thus pentagonal symmetry, as is often evident in many forms of life. The points created by the intersections of these basic radiations within a circular area yield the five-fold relation symbol known as the Pentagon. 7. Like all regular polygons, the Pentagon is filled with many internal correlations. Each side is opposite a point, every diagonal that unites two sides, internally, has a Golden ratio to either of the sides it unites. The instances of groups of fives and pentagonal symmetry in everyday life are many-fold. 8. This shape is known as the Pentagram. It represents the first harmonic division, again, by differentiation. Over the course of history, it has come to be associated with images of Black Magic or the Occult. Although it may indeed be an order of "secret", "forbidden", or "ancient" knowledge, it is by no means malevolent. It simply expresses archetypal geometries and relationships that are infinitely exposed all around us, by virtue of the interactivity of radiating points, which then beget infinite forms. As stated earlier, each internal diagonal line has a Golden ratio to an outer side. The Golden proportion is one of the constant, incommensurate ratios that govern manifestation. So, too, is PI. 9. This represents the second order of differentiation, the second harmonic division. Each side has been bisected, and every resulting internal point connected. Symmetry yields symmetry. A smaller, fractal child has been generated from the parent. This is also known as recursion. This process can occur at any scale and size, and still retain the same inter-relationships and proportions. This shape also reveals many regular and semi-regular polyhedra at various views. It is among my favorite as a meditative device, like a Mandala. Additionally it contains 30 scalene triangles of varying scale, which symbolize the 30° of each of the 12 houses of the zodiac, as well as the idealized circular sequence of 30 days in each of the 12 months.
10. One of at least two possible resultant 3d projections of polyhedra, namely the Dodecahedron. 2-dimensional differentiation yields 3-dimensional manifestation. 11. Here, removed from the grid, is the Dodecahedron. Historically it has symbolized the concept of a fifth element, Ether (Aether) or Universe. It represents the perfect mediation of the infinite and the finite, the sphere and the cube, analogous to the circle and the square. It is reciprocal with the Icosahedron; whereas the Icosahedron has 20 faces and 12 vertices, the Dodecahedron has 12 faces and 20 vertices. Each shape can easily be transformed into the other by truncation or stellation. 12. And as each group of 2-dimensional circles yields 2-dimensional polygons which in turn yield 3-dimensional polyhedra, so each group of 3-dimensional spheres yields 4-dimensional polyhedra, also known as Polytopes. The Fourth dimension is an angle perpendicular to the other three, X, Y, and Z. Often Time is considered to be this angle. Another possible interpretation is a spatial one, representable in terms of Scale. 4-dimensional polyhedra can be rather complex when realized in 3-dimensional space due to the perceptual constraints of our 3d world. The famous example is the Hypercube, which might be rendered as a smaller cube within a larger, or a cube projecting cubes out of each of its faces, and two out of the bottom, like a Cross.
Humanity is represented by the Dodecahedron
and the Pentagon:
If you own an iPhone, you can explore this symmetry directly It has been shown that the Icosahedron, when viewed by certain angles internally, bears resemblance to the Enneagram:
G.I. Gurdjieff is the best resource for a deeper understanding of
this symbol.
It naturally follows that many have associated visual harmonic division and proportion with music and sound, music in particular being something of an aural geometry. Kepler tried to associate the polyhedra with the relative orbits of the planets known at that time (planet Earth was related to the Dodecahedron), and in turn to correlate these to rather sweeping musical frequencies, the sound of which create a harmony, into which we are born, always hearing, indeed a part of it. Likewise, many have believed that the soul or spirit essence of the material body exuded a harmony, much like the notion of a visual aura, and this harmony resulted from the harmonic vibrations of energy patterns, holding the molecules of the material in place to manifest in the form of structure. Quantum Physics and many Eastern mystical religions hold apparently similar views about the existence of energy fields which harness molecules (matter) by vibrating them, bestowing manifest shape, all sharing one common source. To observe things in this way can inspire reverence for all of reality, as the totality of the Universe may appear to be a collection of dormant molecules, woven together like a vast matrix, shaken and vibrated by energy fields which give them cohesion and consistency, namely mass, a medium for energy to travel by, like a wave upon the ocean, illuminated in time-space by light. The unity of all things becomes revealed. speaker design by JVC, 2007 The QJ Dodecahedron Puzzle (after Rubik's Cube)
This information has made its way around multiple sites and news sources over the last several years. Below is a screen snapshot of one such article which was published on CNN.com on October 9th, 2003. I republish it here under the fair use section 107 of United States copyright law. It suggests (in minimal detail) that the universe itself may very well be in the shape of a dodecahedron.
postage stamp from Monaco, 2000
postage stamp from Spain, 1964
"The Sacrament of the Last Supper"
"Order and Chaos"
M.C. Escher, 1943
Albrecht Dürer, 1514 Leonardo Da Vinci illustration for Pacioli's De Devina Proportione, 1509 Roman Dodecahedron, ca. 2nd century AD a natural formation of Pyrite |
Visit Shapeways.com to explore 3d geometric forms, available for purchase.
Copyright ©2015 Jason Martineau