Platonic solid

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

All its faces being congruent regular polygons
The same number of faces meeting at each of its vertices

These are in contrast to:

The Kepler-Poinsot solids, which are not convex
The Archimedean and Johnson solids, which while having regular polygons as faces, are not themselves regular

The five Platonic solids were all known to the ancient Greeks, and a proof that they are the only regular polyhedra can be found in Euclid's Elements.

1 Table
2 Limited number of Platonic polyhedra
3 Dual polyhedra
4 Origins of name
5 Ancient symbolism
6 Other symbolism
7 Inscribed Platonic polyhedra
8 Uses
9 See also
10 References
11 External links

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Table

Name	Picture	Faces	Edges	Vertices	Edges per face	Faces meeting at each vertex	Vertex configuration	Symmetry group
tetrahedron	Image:Tetrahedron.jpg (Animation)	4	6	4	3	3	3.3.3	T_d
cube (hexahedron)	Image:Hexahedron.jpg (Animation)	6	12	8	4	3	4.4.4	O_h
octahedron	Image:Octahedron.jpg (Animation)	8	12	6	3	4	3.3.3.3	O_h
dodecahedron	Image:Dodecahedron.jpg (Animation)	12	30	20	5	3	5.5.5	I_h
icosahedron	Image:Icosahedron.jpg (Animation)	20	30	12	3	5	3.3.3.3.3	I_h

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) which is equal to 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.

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Limited number of Platonic polyhedra

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

Each vertex of the solid must coincide with one vertex each of at least three faces.
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°.
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°.
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.

In fact, the problem of classifying the Platonic polyhedra is not truly geometric, but merely topological. That is, the same list of five polyhedra can be deduced using only combinatorial information about a regular polyhedron: the number v of vertices, the number e of edges, the number f of faces, the number n of edges bounding each face, and the number d of edges meeting each vertex. These numbers are related as follows:

all are positive, and n and d are at least 3, as above;
since each edge is adjacent to two faces and meets two vertices, n f = 2 e = v d;
v - e + f = 2. (This is an elementary fact of algebraic topology, that the Euler characteristic of the sphere is 2.)

Using these facts, it is not difficult to show algebraically that only the five classical regular polyhedra are possible.

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Dual polyhedra

Image:Dual Cube-Octahedron.jpg

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.

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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.

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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 (iron disulfide) 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.

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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.

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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

<math>\sqrt{\frac{5+2\sqrt{5}}{15}}</math>,

about 0.795 of the radius of the circumscribed sphere.

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Uses

Image:BluePlatonicDice.jpg

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.

Geometry of space frames is often based on platonic solids. In MERO system, platonic solids are used for naming convention of various space frame configurations. For example ½O+T refers to a configuration made of one half of octahedron and a tetrahedron.

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References

"Strena sue de nive sexangula" (On the Six-Cornered Snowflake), 1611 paper by Kepler which discussed the reason for the six-angled shape of the snow crystals and the forms and symmetries in nature.

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