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Excluding the infinite sets there are 75 uniform polyhedra. Categories include:
They can also be grouped by their Symmetry Group , which is done below. HISTORY The Platonic Solid s 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 Solid s after the original work of Archimedes was lost. Kepler (1619) discovered two of the regular Kepler-Poinsot Solid s and Louis Poinsot (1809) discovered the other two. Of the remaining 37 were discoved by Badoureau (1881). Hess (1878) discovered 2 more and Pitsch (1881) indepentantly discovered 18, not all previously discovered. The famous group theorist Donald Coxeter discovered the remaining twelve in colaboration with Miller (1930-1932) but did not publish. M.S. and H.C. Longuet-Higgins and independently discovered 11 of these. In a seminal paper: : H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller, ''Uniform polyhedra'', Phil. Trans. 1954, 246 A, 401-50. the full list of uniform polyhedra was published, and it was conjectured that the list was complete. J. Skilling later confirmed this result. LISTED BY SYMMETRY GROUPS AND VERTEX ARRANGEMENTS All the uniform polyhedra are listed below by their Symmetry Group s and subgrouped by their vertex arrangments. Regular polyhedra are labeled by their Schläfli Symbol . Other nonregular uniform polyhedra are listed by their Vertex Configuration or their Uniform polyhedron index U(1-80). Convex forms and fundamental vertex arrangments The dihedral, tetrahedral, octahedral, and icosahedral symmetry polyhedra can be named by construction operations upon a parent form. Note: ''Dihedra'' are an infinite set of two-sided polyhedra (2 identical polygons) which generates the prisms as truncated forms. Each of these convex forms define a vertex arrangment that can be identified for the nonconvex forms in the next section. Definition of operations Note: For nonconvex forms below an additional descriptor Nonuniform is used when the Convex Hull of the vertex arrangement has same topology as one of these, but has nonregular faces. For example an ''nonuniform runcinated'' form may have Rectangles created in place of the edges rather than Squares . Tetrahedral Symmetry There are 2 convex uniform polyheda, Tetrahedron , and Truncated Tetrahedron , and one nonconvex form, the Tetrahemihexahedron which have ''tetrahedral symmetry''. The Tetrahedron is self dual. In addition the Octahedron , Truncated Octahedron , Cuboctahedron , and Icosahedron have tetrahedral symmetry as well as higher symmetry. They are added for completeness below, although their nonconvex forms with octahedral symmetry are not included here. Octahedral Symmetry There are 8 convex forms, and 10 nonconvex forms with octahedral symmetry. Icosahedral Symmetry There are 8 convex forms and 46 nonconvex forms. Some of the nonconvex snub forms have nonuniform chiral symmetry, and some have achiral symmetry. There are many ''nonuniform'' forms of varied degrees of truncation and runcination. Dihedral Symmetry There are four infinite sets of uniform prisms and antiprisms: (1) convex prisms, (2) star prisms, (3) convex antiprisms, and (4) star antiprims. They share two sets of vertex arrangements:
Prisms and star prisms are Dnh. Antiprisms are Dnd. Star antiprisms exist in both form, depending on whether ''q'' is odd or even for a given Star Polygon {p/q}. For a given star polygon {p/q}, if q>p/2, it is considered a ''crossed'' form with a center point reflection applied between the two halves. However some do not exist, such as {7/5}, {9/7}, and {10/7}, as well as any others that fail the vertex figure existence constraint: : cos(2π/n) < cos(π/n), where n=p/q which is equivalent to p/q > 3/2. Note: The Cube and Octahedron are listed here with dihedral symmetry (as a ''tetragonal prism'' and ''trigonal antiprism'' respectively), although if uniformly colored, they also have octahedral symmetry. SKILLING'S FIGURE One further nonconvex uniform polyhedron is the Great Disnub Dirhombidodecahedron , also known as ''Skilling's figure'', which is vertex-uniform, but has pairs of edges which coincide in space such that four faces meet at some edges. This has Ih symmetry. : SEE ALSO
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