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This is a degenerate uniform polyhedron rather than a uniform polyhedron, because some pairs of edges coincide. Not included are: The uniform polyhedron compounds. 40 potential uniform polyhedra with degenerate vertex figures which have overlapping edges (not counted by Coxeter); The uniform tilings (infinite polyhedra)
Coxeter, Longuet-Higgins & Miller (1954) published the list of uniform polyhedra. Sopov (1970) 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Wenninger published his book Polyhedron models, which lists all 75 nonprismatic uniform polyhedra, with many previously unpublished names given to them by Norman Johnson.
image of polyhedron; name of polyhedron; alternate names (in brackets) Wythoff symbol; Numbering systems: W - number used by Wenninger in polyhedra models, U - uniform indexing, K - Kaleido indexing, C - numbering used in Coxeter et al. 'Uniform Polyhedra'. Number of vertices V, edges E, Faces F and number of faces by type.
This category was created to reference the full set of 75 nonprismatic uniform polyhedra, as well as prismatic forms. It is a subset of Category:Polyhedra.. It is a union of 5 Platonic solids, 4 Kepler–Poinsot solids, 13 Archimedean solids, and the infinite prismatic sets in Prismatoid polyhedra, and adds 53 non-convex, non-regular uniform polyhedra.
Quasi-regular polyhedra Johnson solids (92, convex, non-uniform) Bipyramids Pyramids Stellations: Stellations: Polyhedral compounds Deltahedra (Deltahedra, equilateral triangle faces) Snub polyhedra (12 uniform, not mirror image) Zonohedron (Zonohedra, faces have 180°symmetry) Dual polyhedron: Self-dual polyhedron Catalan solid
These two uniform polyhedra cannot be generated at all by the Wythoff construction. This is the set of uniform polyhedra commonly described as the "non-Wythoffians". Instead of the triangular fundamental domains of the Wythoffian uniform polyhedra, these two polyhedra have tetragonal fundamental domains.
Quasi-regular polyhedra Johnson solids (92, convex, non-uniform) Bipyramids Pyramids Stellations: Stellations: Polyhedral compounds Deltahedra (Deltahedra, equilateral triangle faces) Snub polyhedra (12 uniform, not mirror image) Zonohedron (Zonohedra, faces have 180°symmetry) Dual polyhedron: Self-dual polyhedron Catalan solid
A polytope is a geometric object with flat sides, which exists in any general number of dimensions. The following list of polygons, polyhedra and polytopes gives the names of various classes of polytopes and lists some specific examples.