# Platonic solid

In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with

(These are in contrast to

The five Platonic solids were all known to the ancient Greeks.

## Table

Name Picture Faces Edges Vertices Edges per face Faces meeting
at each vertex
Symmetry group
tetrahedron Tetrahedron
4 6 4 3 3 Td
cube (hexahedron) Hexahedron (cube)
6 12 8 4 3 Oh
octahedron Octahedron
(Animation)
8 12 6 3 4 Oh
dodecahedron
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12 30 20 5 3 Ih
icosahedron Icosahedron
20 30 12 3 5 Ih

The number of faces times the number of edges per face (or the number of vertices per face) is equal to the number of vertices times the number of edges meeting at a vertex (or the number of faces meeting at a vertex): twice the number of edges (corresponding to the fact that an edge connects two vertices and that two faces meet at each edge). This product is equal to the order of the rotation group: 12, 24, 24, 60, and 60, respectively.

The order of the symmetry group is twice that number, i.e. four times the number of edges, hence 24, 48, 48, 120, and 120, respectively.

## Limited number of Platonic polyhedra

The limitation to five such three-dimensional solids is easily demonstrated:

1. Each vertex of the solid must coincide with one vertex each of at least three faces.
2. At each vertex of the solid, the total, among the adjacent faces, of the angles between their respective adjacent sides must be less than 360°.
3. The angles at all vertices of all faces of a Platonic solid are identical, so each vertex of each face must contribute less than 360°/3=120°.
4. Regular polygons of six or more sides have only angles of 120° or more, so the common face must be the triangle, square, or pentagon. And for:
• Triangular faces: each vertex of a regular triangle is 60°, so a shape may have 3, 4, or 5 triangles meeting at a vertex; these are the tetrahedron, octahedron, and icosahedron respectively.
• Square faces: each vertex of a square is 90°, so there is only one arrangement possible with three faces at a vertex, the cube.
• Pentagonal faces: each vertex is 108°; again, only one arrangement, of three faces at a vertex is possible, the dodecahedron.

## Dual polyhedra

File:Dual Cube-Octahedron.jpg
A dual cube-octahedron.

Connecting the centers of all the pairs of adjacent faces of any platonic solid produces another (smaller) platonic solid. The number of faces and vertices is interchanged, while the number of edges of the two is the same.

The relationship between such a pair of polyhedra is called being each other's dual.

• The dual of a tetrahedron is another tetrahedron.
• The dual of an octahedron is a cube, and vice versa.
• The dual of a dodecahedron is an icosahedron, and vice versa.

## Origins of name

The Platonic solids are named after Plato, who wrote about them in Timaeus. Plato learned about these solids from his friend Theaetetus. The constructions of the solids are included in Book XIII of Euclid's Elements. Propositions 13 through 17 describe the construction of the tetrahedron, octahedron, cube, icosahedron, and dodecahedron in that order.

## Ancient symbolism

Plato conceived the four classical elements as atoms with the geometrical shapes of four of the five platonic solids that had been discovered by the Pythagoreans (in the Timaeus). These are, of course, not the true shapes of atoms; but it turns out that they are some of the true shapes of packed atoms and molecules, namely crystals: The mineral salt sodium chloride occurs in cubic crystals, fluorite (calcium fluoride) in octahedra, and pyrite in dodecahedra (see uses below).

This concept linked fire with the tetrahedron, earth with the cube, air with the octahedron and water with the icosahedron. 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 breaks 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" (Timaeus 55). He didn't really know what else to do with it. Aristotle added a fifth element, aithêr (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.

## Other symbolism

Historically, Johannes Kepler followed the custom of the Renaissance in making mathematical correspondences, and identified the five platonic solids with the five planets – Mercury, Venus, Mars, Jupiter, Saturn which themselves represented the five classical elements.

Two-dimensional images of each of the Platonic solids are found within Metatron's Cube, a construct which originates from joining all the centres together from the Flower of Life.

## Inscribed Platonic polyhedra

When the Platonic polyhedra are inscribed in a sphere, they occupy the following percentages of that sphere's volume:

• Tetrahedron: 12.2518%
• Cube: 36.7553%
• Octahedron: 31.8310%
• Dodecahedron: 66.4909%
• Icosahedron: 60.5461%

The Platonic solids may be seen as increasingly better approximations to that sphere. (The Archimedean solids and geodesic domes are in many ways even better approximations to the sphere).

However, either the dodecahedron or the icosahedron may be seen as the Platonic solid that "best approximates" a sphere.

On one hand, the icosahedron has the most sides and the flattest dihedral angle. This may be the source of the common assumption that the icosahedron is the Platonic solid that gives the closest approximation to the sphere.

On the other hand, the dodecahedron occupies significantly more of the sphere's volume than the apparently more spherical icosahedron. The corners of the dodecahedron are less sharp than the corners of the icosahedron, and therefore fit closer to the circumscribing sphere.

The dodecahedron is also most like the sphere in the sense that it has the smallest central angle (ratio of the strut length to the radius of the circumscribed sphere), and the greatest surface area. [1]

In terms of the "variation in altitude" (the ratio between the radius of the circumscribed sphere and the radius of the inscribed sphere), the Platonic solid that best fits the sphere is a tie between the icosahedron and the dodecahedron. Both have an inscribed sphere whose radius is

${\displaystyle {\sqrt {\frac {5+2{\sqrt {5}}}{15}}}}$,

about 0.795 of the radius of the circumscribed sphere.

## Uses

The shapes are often used to make dice, because dice of these shapes can be made fair. 6-sided dice are very common, but the other numbers are commonly used in role-playing games. Such dice are commonly referred to as D followed by the number of faces (d8, d20 etc.).

The tetrahedron, cube, and octahedron, are found naturally in crystal structures. The dodecahedron is combinatorially identical to the pyritohedron (in that both have twelve pentagonal faces), which is one of the possible crystal structures of pyrite. However, the pyritohedron is not a regular dodecahedron, but rather has the same symmetry as the cube.

In meteorology and climatology, global numerical models of atmospheric flow are of increasing interest which employ grids that are based on an icosahedron (refined by triangulation) instead of the more commonly used longitude/latitude grid. This has the advantage of evenly distributed spatial resolution without singularities (i.e. the poles) at the expense of somewhat greater numerical difficulty.