# Regular polytope

polytope whose symmetry group acts transitively on its flags
(Redirected from Regular polyhedron)

In mathematics, a regular polytope is a shape with sides and vertices that are symmetrical. Like other polytopes, regular polytopes are found in any number of dimensions.

A two-dimensional regular polytope is a regular polygon, and a three-dimensional regular polytope is a regular polyhedron. Even though these groups include different shapes, they are based on the same idea of the shapes having regular symmetry. Regular polytopes have regular facets (faces, as seen from the 3D viewpoint), and their vertex figures are regular.

Each number of dimensions has a different set of regular polytopes. However, all of them have a simple shape. The simplest of regular polytopes is the triangle in 2D, the tetrahedron in 3D, and the pentachoron in 4D.

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds