In geometry, 2k1 polytope is a uniform polytope in n dimensions (n = k+4) constructed from the En Coxeter group. The family was named by their Coxeter symbol as 2k1 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence. It can be named by an extended Schläfli symbol {3,3,3k,1}.
Family members
The family starts uniquely as 6-polytopes, but can be extended backwards to include the 5-orthoplex (pentacross) in 5-dimensions, and the 4-simplex (5-cell) in 4-dimensions.
Each polytope is constructed from (n-1)-simplex and 2k-1,1 (n-1)-polytope facets, each has a vertex figure as an (n-1)-demicube, {31,n-2,1}.
The sequence ends with k=6 (n=10), as an infinite hyperbolic tessellation of 9-space.
The complete family of 2k1 polytope polytopes are:
- 5-cell: 201, (5 tetrahedra cells)
- Pentacross: 211, (32 5-cell (201) facets)
- 221, (72 5-simplex and 27 5-orthoplex (211) facets)
- 231, (576 6-simplex and 56 221 facets)
- 241, (17280 7-simplex and 240 231 facets)
- 251, tessellates Euclidean 8-space (∞ 8-simplex and ∞ 241 facets)
- 261, tessellates hyperbolic 9-space (∞ 9-simplex and ∞ 251 facets)