Concept
Uniform 1 k2 polytope
Related concepts (7)
Uniform 2 k1 polytope
DISPLAYTITLE:Uniform 2 k1 polytope 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}. 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.
1 42 polytope
DISPLAYTITLE:1 42 polytope In 8-dimensional geometry, the 142 is a uniform 8-polytope, constructed within the symmetry of the E8 group. Its Coxeter symbol is 142, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequences. The rectified 142 is constructed by points at the mid-edges of the 142 and is the same as the birectified 241, and the quadrirectified 421.
Gosset–Elte figures
In geometry, the Gosset–Elte figures, named by Coxeter after Thorold Gosset and E. L. Elte, are a group of uniform polytopes which are not regular, generated by a Wythoff construction with mirrors all related by order-2 and order-3 dihedral angles. They can be seen as one-end-ringed Coxeter–Dynkin diagrams. The Coxeter symbol for these figures has the form ki,j, where each letter represents a length of order-3 branches on a Coxeter–Dynkin diagram with a single ring on the end node of a k length sequence of branches.
1 32 polytope
DISPLAYTITLE:1 32 polytope In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group. Its Coxeter symbol is 132, describing its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences. The rectified 132 is constructed by points at the mid-edges of the 132. These polytopes are part of a family of 127 (27-1) convex uniform polytopes in 7-dimensions, made of uniform polytope facets and vertex figures, defined by all permutations of rings in this Coxeter-Dynkin diagram: .
1 52 honeycomb
DISPLAYTITLE:1 52 honeycomb In geometry, the 152 honeycomb is a uniform tessellation of 8-dimensional Euclidean space. It contains 142 and 151 facets, in a birectified 8-simplex vertex figure. It is the final figure in the 1k2 polytope family. It is created by a Wythoff construction upon a set of 9 hyperplane mirrors in 8-dimensional space. The facet information can be extracted from its Coxeter-Dynkin diagram. Removing the node on the end of the 2-length branch leaves the 8-demicube, 151.
E9 honeycomb
DISPLAYTITLE:E9 honeycomb In geometry, an E9 honeycomb is a tessellation of uniform polytopes in hyperbolic 9-dimensional space. , also (E10) is a paracompact hyperbolic group, so either facets or vertex figures will not be bounded. E10 is last of the series of Coxeter groups with a bifurcated Coxeter-Dynkin diagram of lengths 6,2,1. There are 1023 unique E10 honeycombs by all combinations of its Coxeter-Dynkin diagram.
Uniform k 21 polytope
DISPLAYTITLE:Uniform k 21 polytope In geometry, a uniform k21 polytope is a polytope in k + 4 dimensions constructed from the En Coxeter group, and having only regular polytope facets. The family was named by their Coxeter symbol k21 by its bifurcating Coxeter–Dynkin diagram, with a single ring on the end of the k-node sequence. Thorold Gosset discovered this family as a part of his 1900 enumeration of the regular and semiregular polytopes, and so they are sometimes called Gosset's semiregular figures.