Explore: Polytopes

In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any

There are six convex and ten star regular 4-polytopes, giving a total of sixteen. The convex regular 4-polytopes were first described by the Swiss mathematician

cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853. The two-dimensional analogue of a 4-polytope is a polygon

regular polytopes in Euclidean, spherical and hyperbolic spaces. This table shows a summary of regular polytope counts by rank. Only counting polytopes of

Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) Wikimedia Commons has media related to Cross-polytope graphs. Weisstein, Eric

edge graphs of polytopes; those that are realizable in this manner are called polytopal graphs. Edge graphs of 3-dimensional polytopes are also called

Regular polytopes: 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 6 regular polytopes in 4 dimensions

6-dimensional geometry, there are 39 uniform polytopes with E6 symmetry. The two simplest forms are the 221 and 122 polytopes, composed of 27 and 72 vertices respectively

families of polytopes based on Cartesian products of lower-dimensional uniform polytopes. Five are formed as the product of a uniform 4-polytope with a regular

facets are uniform 6-polytopes. Regular 7-polytopes are represented by the Schläfli symbol {p,q,r,s,t,u} with u {p,q,r,s,t} 6-polytopes facets around each