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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:
Regular polyhedron. Platonic solid: Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron; Regular spherical polyhedron. Dihedron, Hosohedron; Kepler–Poinsot polyhedron (Regular star polyhedra) Small stellated dodecahedron, Great stellated dodecahedron, Great icosahedron, Great dodecahedron; Abstract regular polyhedra (Projective polyhedron)
A chiliagram is a 1,000-sided star polygon. There are 199 regular forms [a] given by Schläfli symbols of the form {1000/n}, where n is an integer between 2 and 500 that is coprime to 1,000. There are also 300 regular star figures in the remaining cases. For example, the regular {1000/499} star polygon is constructed by 1000 nearly radial edges.
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.
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
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
For 4-dimensional skew polyhedra, Coxeter offered a modified Schläfli symbol {l,m|n} for these figures, with {l,m} implying the vertex figure, m l-gons around a vertex, and n-gonal holes. Their vertex figures are skew polygons, zig-zagging between two planes. The regular skew polyhedra, represented by {l,m|n}, follow this equation:
A uniform polyhedron is a polyhedron in which the faces are regular and they are isogonal; examples include Platonic and Archimedean solids as well as prisms and antiprisms. [3] The Johnson solids are named after American mathematician Norman Johnson (1930–2017), who published a list of 92 such polyhedra in 1966.
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