Any class of polyhedron which is comprised of identical faces, usually a triangle*. There are surprisingly few polyhedra which fall into this category, five in fact, namely tetrahedron, octahedron, hexahedron (cube), icosahedron, and dodecahedron. These were described by Plato (hence the name), and were used to denote the five elements (earth, water, air, fire, and the heavens). BinarOne knows the mappings better than me.

Also, a solid which decides to abstain from sexual relationships.

*Okay, okay, there's a few more conditions involving edge/vertex transitivity which lead to other side-effects such as being convex, which ariels decided to point out, which I personally think are just in retrospect to make it so that Plato's five solids are the only platonic solids so that there's less catalogging to be done. Why doesn't ariels come and give us the full definition, hm?

The five regular polyhedra: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

Named for Plato, these shapes are the only convex polyhedra which have identical regular polygons for all their faces, and the same number of faces meeting at each vertex.

Closely related to these are the Archimedean solids which have all regular polygons for faces and the same faces in the same order at each vertex, but the faces are of 2 or more types.

Also related are the Johnson solids, all the other convex polyhedra with regular polygons for all their faces but no particular regularity at their vertices.

A couple of additional curious properties of the Platonic solids and their faces and vertices:

The icosahedron has 20 faces and 12 vertices.
The dodecahedron has 12 faces and 20 vertices.

The octahedron has 8 faces and 6 vertices.
The cube has 6 faces and 8 vertices.

The tetrahedron has 4 faces and 4 vertices.

Each of the pairs above are dual polyhedra, i.e. if one connects the centers of each face of one polyhedron, the other polyhedron of the pair results. Doing this with a tetrahedron yields another tetrahedron.

Imagine a line drawn from any vertex to the center of the solid. If you were to 'slice off' the vertex on a plane perpendicular to the imagined line, without slicing past any of the other vertices, the new face would be the shape of a face of the next solid 'down' in the sequence.

The triangle-faced icosahedron would yield a pentagon.
The pentagon-faced dodecahedron would yield a triangle.

The triangle-faced octahedron would yield a square.
The square-faced cube would yield a triangle.

The triangle-faced tetrahedron, oddball that it is, would yield a triangle, since it is the last in the line and sufficient unto itself.

Plato associated the tetrahedron, cube, octahedron and icosahedron with fire, earth, air, and water. He linked the dodecahedron with the universe in its entirety, as that "which god used for embroidering the constellations on the whole heaven." Plato's followers were not comfortable with the disjunction between the four elements and the five solids. Aristotle proposed a fifth element (quintessence), the aether (to associate with the dodecahedron) which filled the cosmos and served as a medium for change and motion (this idea held, in various forms, until the Michelson Morley Experiment in 1887.)

Plato also had a system for explaining the interactions between the four elements by performing various arithmetical operations using the number of faces of the associated solids. For example, heating water produces two 'particles' of air and one of fire, or 20= (2x8)+4 (1 icosahedron=2 octahedra + 1 tetrahedron). Considering that water is H2O, this particular example is oddly prescient.

Information about association of solids with elements from: The Golden Ratio, Mario Livio, Broadway Books, 2002

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