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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 wikiPageRevisionID "606315917".
- List_of_uniform_polyhedra_by_Schwarz_triangle b "s".
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- List_of_uniform_polyhedra_by_Schwarz_triangle p "r".
- List_of_uniform_polyhedra_by_Schwarz_triangle subject Category:Uniform_polyhedra.
- List_of_uniform_polyhedra_by_Schwarz_triangle comment "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).".
- List_of_uniform_polyhedra_by_Schwarz_triangle label "List of uniform polyhedra by Schwarz triangle".
- List_of_uniform_polyhedra_by_Schwarz_triangle sameAs Q6644217.
- List_of_uniform_polyhedra_by_Schwarz_triangle sameAs Q6644217.
- List_of_uniform_polyhedra_by_Schwarz_triangle wasDerivedFrom List_of_uniform_polyhedra_by_Schwarz_triangle?oldid=606315917.
- List_of_uniform_polyhedra_by_Schwarz_triangle depiction Degenerate_uniform_polyhedra_vertex_figures.png.
- List_of_uniform_polyhedra_by_Schwarz_triangle isPrimaryTopicOf List_of_uniform_polyhedra_by_Schwarz_triangle.