name two polyhedron
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Only five regular polyhedrons exist: the tetrahedron (four triangular faces), the cube (six square faces), the octahedron (eight triangular faces—think of two pyramids placed bottom to bottom), the dodecahedron (12 pentagonal faces), and the icosahedron (20 triangular faces).
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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 Sopov (1970) 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:
40 potential uniform polyhedra with degenerate vertex figures which have overlapping edges (not counted by Coxeter);
The uniform tilings (infinite polyhedra)
11 Euclidean uniform tessellations with convex faces;
14 Euclidean uniform tilings with nonconvex faces;
Infinite number of uniform tilings in hyperbolic plane.
Any polygons or 4-polytopes
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 Sopov (1970) 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:
40 potential uniform polyhedra with degenerate vertex figures which have overlapping edges (not counted by Coxeter);
The uniform tilings (infinite polyhedra)
11 Euclidean uniform tessellations with convex faces;
14 Euclidean uniform tilings with nonconvex faces;
Infinite number of uniform tilings in hyperbolic plane.
Any polygons or 4-polytopes
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