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REGULAR POLYHEDRA
Because they have a high degree ofsymetry and order2
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Icosahedron
Dodecahedron
Octahedron
Tetrahedron
Hexahedron
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History
Kepler's Platonic solid model of the solar system from Mysterium Cosmographicum
(1596)
http://en.wikipedia.org/wiki/Mysterium_Cosmographicumhttp://en.wikipedia.org/wiki/Image:Kepler-solar-system-1.pnghttp://en.wikipedia.org/wiki/Mysterium_Cosmographicum -
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The Platonic solids feature prominently in the philosophy ofPlato for whom they are named. in which he associatedeach of the four classical elements (earth, air, water, andfire) 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: theheat 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, theicosahedron, flows out of one's hand when picked up, as ifit 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 breakwhen picked up, in stark difference to the smooth flow of
water. The fifth Platonic solid, the dodecahedron, Platoobscurely remarks, "...the god used for arranging the
http://en.wikipedia.org/wiki/Platohttp://en.wikipedia.org/wiki/Classical_elementhttp://en.wikipedia.org/wiki/Earth_(classical_element)http://en.wikipedia.org/wiki/Air_(classical_element)http://en.wikipedia.org/wiki/Water_(classical_element)http://en.wikipedia.org/wiki/Fire_(classical_element)http://en.wikipedia.org/wiki/Fire_(classical_element)http://en.wikipedia.org/wiki/Water_(classical_element)http://en.wikipedia.org/wiki/Air_(classical_element)http://en.wikipedia.org/wiki/Earth_(classical_element)http://en.wikipedia.org/wiki/Classical_elementhttp://en.wikipedia.org/wiki/Plato -
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Tetrahedron
Octahedron
Dodecahedron
Icosahedron
Cube/Hexahedron
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Combinatorial propertiesA convex polyhedron is a Platonic solid if and only if
1.all its faces are congruent convex regular polygons,
2.none of its faces intersect except at their edges, and
3.the same number of faces meet at each of its vertices.
Each Platonic solid can therefore be denoted by a symbol
{p, q} where
p = the number of sides of each face (or the number of
vertices of each face) andq = the number of faces meeting at each vertex (or the
number of edges meeting at each vertex).
http://en.wikipedia.org/wiki/Congruenthttp://en.wikipedia.org/wiki/Regular_polygonhttp://en.wikipedia.org/wiki/Vertex_(geometry)http://en.wikipedia.org/wiki/Vertex_(geometry)http://en.wikipedia.org/wiki/Regular_polygonhttp://en.wikipedia.org/wiki/Congruent -
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Polyhedron Vertices Edges Faces Schlfli symbolVertexconfigurat
tetrahedron 4 6 4 {3, 3} 3.3.3
cube 8 12 6 {4, 3} 4.4.4
octahedron 6 12 8 {3, 4} 3.3.3.3
dodecahedron 20 30 12 {5, 3} 5.5.5
icosahedron 12 30 20 {3, 5} 3.3.3.3.3
http://en.wikipedia.org/wiki/Schl%C3%83%C2%A4fli_symbolhttp://en.wikipedia.org/wiki/Vertex_configurationhttp://en.wikipedia.org/wiki/Vertex_configurationhttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Dodecahedronhttp://en.wikipedia.org/wiki/Icosahedronhttp://en.wikipedia.org/wiki/Icosahedronhttp://en.wikipedia.org/wiki/Dodecahedronhttp://en.wikipedia.org/wiki/Octahedronhttp://en.wikipedia.org/wiki/Cubehttp://en.wikipedia.org/wiki/Tetrahedronhttp://en.wikipedia.org/wiki/Vertex_configurationhttp://en.wikipedia.org/wiki/Schl%C3%83%C2%A4fli_symbol
