The dodecahedron has quite a few facetings. You should be able to see that this polyhedron's vertices are the same as that of a dodecahedron this is all faceting means.
Some of its facetings are compounds of other Platonic solids, such as this compound of 5 cubes. The compound of 5 tetrahedra is another well-known faceting. The compound of 10 tetrahedra is yet another well-known faceting. Subsymmetric stellations are also possible. These are stellations that have less symmetry than the dodecahedron itself.
The polyhedron shown here is an example. Connecting those five points creates a pentagon. You can then see how the dodecahedron forms around this compound. Could a dodecahedron be a single spinning tetrahedron that pauses in 5 positions? Like the Five Tetrahedra compound, it also can be seen as a faceting of a regular dodecahedron.
Each of the ten tetrahedra here are seen in a different color. It is interesting to see how they naturally interweave. A cube has 12 edges. The cube will have one edge along each of the twelve faces of the dodecahedron, on which the edge is a diagonal. Each of the colored lines below represents a different cube found in the dodecahedron. There are 5 total: red, yellow, green, blue, and black. The compound of five cubes that forms a dodecahedron is discussed below. Chamfering a cube is the same as cantellation.
The edges are truncated instead of the corners being truncated. The cube transforms into the dodecahedron if the edges of the cube are pushed into planes. The compound of five cubes is composed of 5 cubes as its name suggests.
It is dual to the compound of five octahedra. The Icosahedron and dodecahedron are duals. Connecting the centers of the faces creates the dual.
Purusha is the anthropocosmic, paradigmatic Man or Seed that projects Prakriti, the eternally enchanting Feminine, in order that her womb may give birth to his own embodiment in the world of form. The Hindu tradition associates the icosahedron with Purusha — the seed-image of Brahma, the supreme creator. Purusha is envisioned as unmanifest and untouched by creation — just as in the following drawings, the icosahedron is untouched by the other forms.
All other volumes arise naturally out of the icosahedron, making it one obvious choice for the first form. Within this womb of creation, all shapes and forms are present in potentiation.
The star born within its pentagon is the configuration of Cosmic Man, the perfecter of life, the Golden Proportion. Both Prakriti dodecahedron and Purusha icosahedron have phi proportions. The whole coagulation is begun by the secret seed which contracts the circle, the infinite, undifferentiated spirit, into the icosahedron. The seed is phi, the fire of spirit. The golden spiral is shown in black. The sequence of root 2 ratios is shown in red.
The star tetrahedron, or stellated octahedron, is seen as the yin and the yang due to its upward pointing tetrahedron and downward pointing tetrahedron. The tetrahedron is a volume of threeness — a primary symbol of function accompanied by its reciprocal. Recall that there is an octahedron at the center of the star tetrahedron. The octahedron symbolizes the crystallization, the static perfection of matter.
It is the transformed and clarified lens of light — the double pyramid. The result of the harmonic interaction of the star tetrahedron gives birth to the cube.
The triakis icosahedron is the dual of the truncated dodecahedron. It is a Catalan solid. The triakis icosahedron is a Kleetope of the icosahedron.
This means that it is an icosahedron with triangular pyramids added to each face. The Snub Dodecahedron is an Archimedean solid that has two distinct forms that are mirror images enantiomorphs of one another. The Snub Dodecahedron has the highest sphericity 0.
For the four Platonic solids built out of squares or equilateral triangles — the cube, tetrahedron, octahedron and icosahedron — mathematicians recently figured out that the answer is no.
Any straight path starting from a corner will either hit another corner or wind around forever without returning home. Now Jayadev Athreya , David Aulicino and Patrick Hooper have shown that an infinite number of such paths do in fact exist on the dodecahedron.
Their paper , published in May in Experimental Mathematics, shows that these paths can be divided into 31 natural families. The solution required modern techniques and computer algorithms. The project began in when Athreya, of the University of Washington, and Aulicino, of Brooklyn College, started playing with a collection of card stock cutouts that fold up into the Platonic solids. Together with Hooper, of the City College of New York, the researchers figured out how to classify all the straight paths from one corner back to itself that avoid other corners.
In all these problems, the basic idea is to unroll your shape in a way that makes the paths you are studying simpler.
So to understand straight paths on a Platonic solid, you could start by cutting open enough edges to make the solid lie flat, forming what mathematicians call a net. I followed her pattern, making one small change to the final round in her diagram :. Crochet a total of 12 motifs. With a 3 color scheme, this came out to two of each motif scheme. Join the pentagons according to this drawing:.
I started by joining 5 pentagons around one pentagon 6 pentagons total to make 2 of these:. Then I joined the two pieces together. This looks easier than it is.
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