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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:
The following list of polygons, polyhedra and polytopes gives the names of various classes of polytopes and lists some specific examples. Polytope elements
In geometry, a polyhedron (pl.: polyhedra or polyhedrons; from Greek πολύ (poly-) 'many' and ἕδρον (-hedron) 'base, seat') is a three-dimensional figure with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is a polyhedron that bounds a convex set.
List of mathematical shapes; List of two-dimensional geometric shapes. List of triangle topics; List of circle topics; List of curves; List of surfaces; List of polygons, polyhedra and polytopes. List of regular polytopes and compounds
A regular polyhedron is identified by its Schläfli symbol of the form {n, m}, where n is the number of sides of each face and m the number of faces meeting at each vertex. There are 5 finite convex regular polyhedra (the Platonic solids), and four regular star polyhedra (the Kepler–Poinsot polyhedra), making nine regular polyhedra in all. In ...
Following is a list of shapes studied in mathematics. Algebraic curves Cubic plane curve ... For example, in a polyhedron (3-dimensional polytope), a face is a facet, ...
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.
A regular polyhedron with Schläfli symbol {p, q}, Coxeter diagrams , has a regular face type {p}, and regular vertex figure {q}. A vertex figure (of a polyhedron) is a polygon, seen by connecting those vertices which are one edge away from a given vertex. For regular polyhedra, this vertex figure is always a regular (and planar) polygon.