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- List_of_uniform_polyhedra_by_Schwarz_triangle abstract "There are many relationships among the uniform polyhedra. The Wythoff construction is able to construct almost all of the uniform polyhedra from the Schwarz triangles. The numbers that can be used for the sides of a non-dihedral Schwarz triangle that does not necessarily lead to only degenerate uniform polyhedra are 2, 3, 3/2, 4, 4/3, 5, 5/2, 5/3, and 5/4 (but numbers with numerator 4 and those with numerator 5 may not occur together). (4/2 can also be used, but only leads to degenerate uniform polyhedra as 4 and 2 have a common factor.) There are 44 such Schwarz triangles (5 with tetrahedral symmetry, 7 with octahedral symmetry and 32 with icosahedral symmetry), which, together with the infinite family of dihedral Schwarz triangles, can form almost all of the non-degenerate uniform polyhedra. Many degenerate uniform polyhedra, with completely coincident vertices, edges, or faces, may also be generated by the Wythoff construction, and those that arise from Schwarz triangles not using 4/2 are also given in the tables below along with their non-degenerate counterparts.There are a few non-Wythoffian uniform polyhedra, which no Schwarz triangles can generate; however, most of them can be generated using the Wythoff construction as double covers (the non-Wythoffian polyhedron is covered twice instead of once) or with several additional faces (see Omnitruncated polyhedron#Other even-sided nonconvex polyhedra). Such polyhedra are marked by an asterisk in this list. The only uniform polyhedra which still fail to be generated by the Wythoff construction are the great dirhombicosidodecahedron and the great disnub dirhombidodecahedron.Each tiling of Schwarz triangles on a sphere may cover the sphere only once, or it may instead wind round the sphere a whole number of times, crossing itself in the process. The number of times the tiling winds round the sphere is the density of the tiling, and is denoted μ.Jonathan Bowers' short names for the polyhedra, known as Bowers acronyms, are used instead of the full names for the polyhedra to save space. The Maeder index is also given. Except for the dihedral Schwarz triangles, the Schwarz triangles are ordered by their densities.".
- List_of_uniform_polyhedra_by_Schwarz_triangle thumbnail Degenerate_uniform_polyhedra_vertex_figures.png?width=300.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageExternalLink polyhedra-neu.htm.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageExternalLink polyhedra.htm.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageExternalLink pqr.htm.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageExternalLink schwarz.htm.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageExternalLink uniform.pdf.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageID "35367651".
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageLength "57814".
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageOutDegree "714".
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageRevisionID "652050369".
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Antiprism.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Category:Uniform_polyhedra.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Coxeter–Dynkin_diagram.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Cube.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Cubitruncated_cuboctahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Cuboctahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Cubohemioctahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Cupola_(geometry).
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Decagonal_prism.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Decagrammic_prism.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Degeneracy_(mathematics).
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Density_(polytope).
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Digon.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Dihedral_group.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Dihedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Ditrigonal_dodecadodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Divisor.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Dodecadodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Dodecagonal_prism.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Dodecagrammic_prism.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Dodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Edge_(geometry).
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Expansion_(geometry).
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_complex_icosidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_cubicuboctahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_dirhombicosidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_disnub_dirhombidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_ditrigonal_dodecicosidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_ditrigonal_icosidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_dodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_dodecahemicosahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_dodecahemidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_dodecicosahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_dodecicosidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_icosahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_icosicosidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_icosidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_icosihemidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_inverted_snub_icosidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_retrosnub_icosidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_rhombidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_rhombihexahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_snub_dodecicosidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_snub_icosidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_stellated_dodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_stellated_truncated_dodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_truncated_cuboctahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Great_truncated_icosidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Harold_Scott_MacDonald_Coxeter.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Hemipolyhedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Hexagonal_antiprism.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Hexagonal_prism.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Hosohedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Icosahedral_symmetry.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Icosahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Icosidodecadodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Icosidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Icositruncated_dodecadodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Inverted_snub_dodecadodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Nonconvex_great_rhombicosidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Nonconvex_great_rhombicuboctahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Octagonal_prism.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Octagrammic_prism.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Octahedral_symmetry.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Octahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Octahemioctahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Omnitruncated_polyhedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Pentagonal_antiprism.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Pentagonal_prism.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Pentagrammic_antiprism.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Pentagrammic_crossed-antiprism.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Pentagrammic_prism.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Quadrilateral.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Quasiregular_polyhedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Rational_number.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Regular_polyhedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Rhombicosahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Rhombicosidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Rhombicuboctahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Rhombidodecadodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Schläfli_symbol.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Schwarz_triangle.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Small_complex_icosidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Small_complex_rhombicosidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Small_cubicuboctahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Small_ditrigonal_dodecicosidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Small_ditrigonal_icosidodecahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Small_dodecahemicosahedron.
- List_of_uniform_polyhedra_by_Schwarz_triangle wikiPageWikiLink Small_dodecahemidodecahedron.