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Uniform polyhedra may be Regular , Quasi-regular or Semi-regular . 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). Categories include:
They can also be grouped by their Symmetry Group , which is done below. HISTORY
Indexing There are four major published indexing efforts from the works above. To distinguish them, they are given by indexing different letters, C for Coxeter 1954 first enumeration figures, '''W''' for the 1974 book Polyhedron models by Wenninger, '''K''' for the 1993 Kaleido solution, and '''U''' for the Maeder solution used by Mathematica and extensively reproduced elsewhere. # C" class="copylinks" target="_blank">{Link without Title} 1954: This paper listed the uniform polyhedra by figures in the paper from 15-92. Starting with 15-32 for the convex forms, 33-35 for 3 infinite prismatic sets, and ending with 36-92 for the nonconvex forms. # W" class="copylinks" target="_blank">{Link without Title} 1974: Wenninger's book ''Polyhedron model'' numbered figures 1-119: 1-5 for the Platonic solids, 6-18 for the Archimedean solids, 19-66 for stellated forms including the 4 regular nonconvex polyhedra, and ended with 67-119 for the nonconvex uniform polyhedra. # K" class="copylinks" target="_blank">{Link without Title} 1993 Kaleido: The 80 figures given in the Kaleido solution were grouped by symmetry, numbered 1-80: 1-5 as representatives for the infinite families of prismatic forms with Dihedral Symmetry , 6-9 with Tetrahedral Symmetry , 10-26 with Octahedral Symmetry , 46-80 with Icosahedral Symmetry . # U" class="copylinks" target="_blank">{Link without Title} 1993 Mathematica: This listing followed the Kaleido one, but moved the 5 prismatic forms to last, shifting the nonprismatic forms back 5, and now 1-75. DIHEDRAL SYMMETRY There are two infinite sets of uniform polyhedra with and Antiprism s. These sets both include forms with Star Polygon s. Main article: Prismatic Uniform Polyhedron CONVEX FORMS AND FUNDAMENTAL VERTEX ARRANGEMENTS The convex uniform polyhedra can be named by Wythoff Construction operations upon a parent form. Note: '' Dihedra '' are members of an infinite set of two-sided polyhedra (2 identical polygons) which generate the prisms as truncated forms. Each of these convex forms define set of ''vertices'' that can be identified for the nonconvex forms in the next section. Definition of operations NONCONVEX FORMS LISTED BY SYMMETRY GROUPS AND VERTEX ARRANGEMENTS All the uniform polyhedra are listed below by their Symmetry Group s and subgrouped by their vertex arrangements. Regular polyhedra are labeled by their Schläfli Symbol . Other nonregular uniform polyhedra are listed with their Vertex Configuration or their Uniform polyhedron index U(1-80). 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 cantellated'' form may have Rectangles created in place of the edges rather than Squares . Tetrahedral symmetry There are 2 convex uniform polyhedra, the 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 with '' Icosahedral Symmetry '' (or 47 nonconvex forms if Skilling's figure is included). Some of the nonconvex snub forms have nonuniform chiral symmetry, and some have achiral symmetry. SKILLING'S FIGURE One further nonconvex 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. It is sometimes but not always counted as a uniform polyhedron. It has Ih symmetry. : SEE ALSO
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