Platonic Solid, Platonic Solid

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.

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.

Two common arguments are given below. Both of these arguments only show that there can be no more than five Platonic solids. That all five actually exist is a separate question—one that can be answered by an explicit construction.

The high degree of symmetry of the Platonic solids can be interpreted in a number of ways. Most importantly, the vertices of each solid are all equivalent under the action of the symmetry group, as are the edges and faces. One says the action of the symmetry group is transitive on the vertices, edges, and faces. In fact, this is another way of defining regularity of a polyhedron: a polyhedron is regular if and only if it is vertex-uniform, edge-uniform, and face-uniform.

Wythoff's kaleidoscope construction is a method for constructing polyhedra directly from their symmetry groups. We list for reference Wythoff's symbol for each of the Platonic solids.

In MERO system, Platonic solids are used for naming convention of various space frame configurations. For example O+T refers to a configuration made of one half of octahedron and a tetrahedron.

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