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}.

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}
 ∞

See also

References

External links

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
Fundamental convex regular and uniform honeycombs in dimensions 2–10
Family {\tilde{A}}_{n-1} {\tilde{C}}_{n-1} {\tilde{B}}_{n-1} {\tilde{D}}_{n-1} {\tilde{G}}_2 / {\tilde{F}}_4 / {\tilde{E}}_{n-1}
Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
Uniform 5-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
Uniform 6-honeycomb {3[6]} δ6 hδ6 qδ6
Uniform 7-honeycomb {3[7]} δ7 hδ7 qδ7 222
Uniform 8-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
Uniform 9-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
Uniform n-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
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