In geometry, 1k2 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 1k2 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. It can be named by an extended Schläfli symbol {3,3k,2}.
The family starts uniquely as 6-polytopes, but can be extended backwards to include the 5-demicube (demipenteract) in 5-dimensions, and the 4-simplex (5-cell) in 4-dimensions.
Each polytope is constructed from 1k-1,2 and (n-1)-demicube facets. Each has a vertex figure of a {31,n-2,2} polytope is a birectified n-simplex, t2{3n}.
The sequence ends with k=6 (n=10), as an infinite tessellation of 9-dimensional hyperbolic space.
The complete family of 1k2 polytope polytopes are:
n | 1k2 | Petrie polygon projection |
Name Coxeter-Dynkin diagram |
Facets | Elements | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1k-1,2 | (n-1)-demicube | Vertices | Edges | Faces | Cells | 4-faces | 5-faces | 6-faces | 7-faces | ||||
4 | 102 | ![]() |
120![]() ![]() ![]() ![]() ![]() |
-- | 5 110 ![]() |
5 | 10 | 10![]() |
5![]() |
||||
5 | 112 | ![]() |
121![]() ![]() ![]() ![]() ![]() ![]() ![]() |
16 120 ![]() |
10 111 ![]() |
16 | 80 | 160![]() |
120![]() |
26![]() ![]() |
|||
6 | 122 | ![]() |
122![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
27 112 ![]() |
27 121 ![]() |
72 | 720 | 2160![]() |
2160![]() |
702![]() ![]() |
54![]() |
||
7 | 132 | ![]() |
132![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
56 122 ![]() |
126 131 ![]() |
576 | 10080 | 40320![]() |
50400![]() |
23688![]() ![]() |
4284![]() ![]() |
182![]() ![]() |
|
8 | 142 | ![]() |
142![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
240 132 ![]() |
2160 141 ![]() |
17280 | 483840 | 2419200![]() |
3628800![]() |
2298240![]() ![]() |
725760![]() ![]() |
106080![]() ![]() ![]() |
2400![]() ![]() |
9 | 152 | 152![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() (8-space tessellation) |
∞ 142 ![]() |
∞ 151 ![]() |
∞ | ||||||||
10 | 162 | 162![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() (9-space hyperbolic tessellation) |
∞ 152 |
∞ 161 ![]() |
∞ |