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Concept
List of uniform polyhedra
Summary
In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent, and the polyhedron has a high degree of reflectional and rotational symmetry.
Uniform polyhedra can be divided between convex forms with convex regular polygon faces and star forms. Star forms have either regular star polygon faces or vertex figures or both.
This list includes these:
all 75 nonprismatic uniform polyhedra;
a few representatives of the infinite sets of prisms and antiprisms;
one degenerate polyhedron, Skilling's figure with overlapping edges.
It was proven in that there are only 75 uniform polyhedra other than the infinite families of prisms and antiprisms. John Skilling discovered an overlooked degenerate example, by relaxing the condition that only two faces may meet at an edge. 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)
11 Euclidean convex uniform tilings;
28 Euclidean nonconvex or apeirogonal uniform tilings;
Infinite number of uniform tilings in hyperbolic plane.
Any polygons or 4-polytopes
Four numbering schemes for the uniform polyhedra are in common use, distinguished by letters:
[C] Coxeter et al., 1954, showed the convex forms as figures 15 through 32; three prismatic forms, figures 33–35; and the nonconvex forms, figures 36–92.
[W] Wenninger, 1974, has 119 figures: 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.
Official source
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