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A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry. Uniform polyhedra may be regular (if also face and edge transitive), quasi-regular (if edge transitive but not face transitive) or semi-regular (if neither edge nor face transitive). The faces and vertices need not be convex, so many of the uniform polyhedra are also star polyhedra. Excluding the infinite sets, there are 75 uniform polyhedra (or 76 if edges are allowed to coincide). * Convex * * 5 Platonic solids – regular convex polyhedra * * 13 Archimedean solids – 2 quasiregular and 11 semiregular convex polyhedra * Star * * 4 Kepler–Poinsot polyhedra – regular nonconvex polyhedra * * 53 uniform star polyhedra – 5 quasiregular and 48 semiregular * * 1 star polyhedron found by John Skilling with pairs of edges that coincide, called the great disnub dirhombidodecahedron (Skilling's figure). There are also two infinite sets of uniform prisms and antiprisms, including convex and star forms. Dual polyhedra to uniform polyhedra are face-transitive (isohedral) and have regular vertex figures, and are generally classified in parallel with their dual (uniform) polyhedron. The dual of a regular polyhedron is regular, while the dual of an Archimedean solid is a Catalan solid. The concept of uniform polyhedron is a special case of the concept of uniform polytope, which also applies to shapes in higher-dimensional (or lower-dimensional) space. ==History== * The Platonic solids date back to the classical Greeks and were studied by Plato, Theaetetus and Euclid. *Johannes Kepler (1571–1630) was the first to publish the complete list of Archimedean solids after the original work of Archimedes was lost. Regular star polyhedra: * Kepler (1619) discovered two of the regular Kepler–Poinsot polyhedra and Louis Poinsot (1809) discovered the other two. Other 53 nonregular star polyhedra: * Of the remaining 53, Albert Badoureau (1881) discovered 36. Edmund Hess (1878) discovered two more and Pitsch (1881) independently discovered 18, of which 15 had not previously been discovered. * The geometer H.S.M. Coxeter discovered the remaining twelve in collaboration with J. C. P. Miller (1930–1932) but did not publish. M.S. Longuet-Higgins and H.C. Longuet-Higgins independently discovered eleven of these. * published the list of uniform polyhedra. * proved their conjecture that the list was complete. * In 1974, Magnus Wenninger published his book ''Polyhedron models'', which lists all 75 nonprismatic uniform polyhedra, with many previously unpublished names given to them by Norman Johnson. * independently proved the completeness, and showed that if the definition of uniform polyhedron is relaxed to allow edges to coincide then there is just one extra possibility. * In 1987, Edmond Bonan drew all the uniform polyhedra and their duals in 3D, with a Turbo Pascal program called Polyca : almost of them were shown during the International Stereoscopic Union Congress held at the Congress Theatre, Eastbourne, United Kingdom. * In 1993, Zvi Har'El produced a complete kaleidoscopic construction of the uniform polyhedra and duals with a computer program called Kaleido, and summarized in a paper ''Uniform Solution for Uniform Polyhedra'', counting figures 1-80. * Also in 1993, R. Mäder ported this Kaleido solution to Mathematica with a slightly different indexing system. * In 2002 Peter W. Messer discovered a minimal set of closed-form expressions for determining the main combinatorial and metrical quantities of any uniform polyhedron (and its dual) given only its Wythoff symbol.〔(Closed-Form Expressions for Uniform Polyhedra and Their Duals, Peter W. Messer, Discrete Comput Geom 27:353–375 (2002) )〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Uniform polyhedron」の詳細全文を読む スポンサード リンク
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