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Platonic Solid, Dodecahedron

In the context of solid geometry the word regular is implied and usually omitted. The word irregular is also used to clarify that a polyhedron is not regular, although still assumed to have the same topology as the regular form.Other fully different topological forms, such as the rhombic dodecahedron which has 12 rhombic faces, or nonconvex star polyhedron, like the great dodecahedron, are never given with shortened names.

Plato wrote about them in the dialogue Timaeus c .360 B.C. in which he associated each of the four classical elements (earth, air, water, and fire) with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. There was intuitive justification for these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one's hand when picked up, as if it is made of tiny little balls. By contrast, a highly un-spherical solid, the hexahedron (cube) represents earth. These clumsy little solids cause dirt to crumble and break when picked up, in stark difference to the smooth flow of water. The fifth Platonic solid, the dodecahedron, Plato obscurely remarks, "...the god used for arranging the constellations on the whole heaven". Aristotle added a fifth element, aithr (aether in Latin, "ether" in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid.

Propositions 13–17 in Book XIII describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order. For each solid Euclid finds the ratio of the diameter of the circumscribed sphere to the edge length. In Proposition 18 he argues that there are no further convex regular polyhedra. Much of the information in Book XIII is probably derived from the work of Theaetetus.

The six spheres each corresponded to one of the planets (Mercury, Venus, Earth, Mars, Jupiter, and Saturn). The solids were ordered with the innermost being the octahedron, followed by the icosahedron, dodecahedron, tetrahedron, and finally the cube. In this way the structure of the solar system and the distance relationships between the planets was dictated by the Platonic solids. In the end, Kepler's original idea had to be abandoned, but out of his research came the discovery of the Kepler solids, the realization that the orbits of planets are not circles, and Kepler's laws of planetary motion for which he is now famous.

The icosahedron has the largest number of faces, the largest dihedral angle, and it hugs its inscribed sphere the tightest. The dodecahedron, on the other hand, has the smallest angular defect, the largest vertex solid angle, and it fills out its circumscribed sphere the most.

One of the forms, called the pyritohedron (named for the group of minerals of which it is typical) has twelve pentagonal faces, arranged in the same pattern as the faces of the regular dodecahedron. The faces of the pyritohedron are, however, not regular, so the pyritohedron is also not regular.

These all have icosahedral symmetry and may be obtained as stellations of the dodecahedron and the icosahedron.

Source: Wikipedia > Platonic Solid