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 regular polygons of the same size and shape, and
- the same number of faces meeting at each of its vertices.
(These are in contrast to
- the Kepler-Poinsot solids, which are not convex, and
- 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.
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Table
| Name and picture | Faces | Edges | Vertices | Edges per face | Faces meeting at each vertex | Symmetry group |
|---|---|---|---|---|---|---|
| tetrahedron | 4 | 6 | 4 | 3 | 3 | Td |
| cube (hexahedron) | 6 | 12 | 8 | 4 | 3 | Oh |
| octahedron | 8 | 12 | 6 | 3 | 4 | Oh |
| dodecahedron | 12 | 30 | 20 | 5 | 3 | Ih |
| icosahedron | 20 | 30 | 12 | 3 | 5 | Ih |
Limited number of Platonic polyhedra
The limitation to five such three-dimensional solids is easily demonstrated:
- 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 should be possible with 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
Connecting the centers of all the pairs of adjacent faces of any platonic solid produces another (smaller) platonic solid. 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. Proposition 13 describes the construction of the tetrahedron, proposition 14 of the octahedron, proposition 15 of the cube, proposition 16 of the icosahedron, and proposition 17 of the dodecahedron.
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] (http://kjmaclean.com/Geometry/Platonic.html)
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.
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.
See also
- convex regular 4-polytopes - the 4-dimensional analog of the Platonic solids
- Schläfli symbol
- list of regular polytopes
External links
- Stella: Polyhedron Navigator (http://www.software3d.com/Stella.html) Tool for exploring polyhedra
- Paper Models of Polyhedra (http://www.korthalsaltes.com/) Many links
- The Uniform Polyhedra (http://www.mathconsult.ch/showroom/unipoly/)
- Virtual Reality Polyhedra (http://www.georgehart.com/virtual-polyhedra/vp.html) The Encyclopedia of Polyhedra
- London South Bank University (http://www.lsbu.ac.uk/water/platonic.html) Water structure and behavior
- Book XIII (http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII13.html) of Euclid's Elements.
- Interactive 3D Polyhedra (http://ibiblio.org/e-notes/3Dapp/Convex.htm) in Javada:Platonisk legeme
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