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Uniform 2 k1 polytope

## Summary

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:

1. 5-cell: 201, (5 tetrahedra cells)
2. Pentacross: 211, (32 5-cell (201) facets)
3. 221, (72 5-simplex and 27 5-orthoplex (211) facets)
4. 231, (576 6-simplex and 56 221 facets)
5. 241, (17280 7-simplex and 240 231 facets)
6. 251, tessellates Euclidean 8-space (∞ 8-simplex and ∞ 241 facets)
7. 261, tessellates hyperbolic 9-space (∞ 9-simplex and ∞ 251 facets)

## Elements

Gosset 2k1 figures
n 2k1 Petrie
polygon

projection
Name
Coxeter-Dynkin
diagram
Facets Elements
2k-1,1 polytope (n-1)-simplex Vertices Edges Faces Cells 4-faces 5-faces 6-faces 7-faces
4 201   5-cell

{32,0,1}
-- 5
{33}

5 10 10

5
5 211   pentacross

{32,1,1}
16
{32,0,1}

16
{34}

10 40 80

80

32

6 221   2 21 polytope

{32,2,1}
27
{32,1,1}

72
{35}

27 216 720

1080

648

99

7 231   2 31 polytope

{32,3,1}
56
{32,2,1}

576
{36}

126 2016 10080

20160

16128

4788

632

8 241   2 41 polytope

{32,4,1}
240
{32,3,1}

17280
{37}

2160 69120 483840

1209600

1209600

544320

144960

17520

9 251 2 51 honeycomb

(8-space tessellation)
{32,5,1}

{32,4,1}

{38}

10 261 2 61 honeycomb

(9-space tessellation)
{32,6,1}

{32,5,1}

{39}