Uniform 1 k2 polytope

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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}.

Contents

Family members

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:

  1. 5-cell: 102, (5 tetrahedral cells)
  2. 112 polytope, (16 5-cell, and 10 16-cell facets)
  3. 122 polytope, (54 demipenteract facets)
  4. 132 polytope, (56 122 and 126 demihexeract facets)
  5. 142 polytope, (240 132 and 2160 demihepteract facets)
  6. 152 honeycomb, tessellates Euclidean 8-space (∞ 142 and ∞ demiocteract facets)
  7. 162 honeycomb, tessellates hyperbolic 9-space (∞ 152 and ∞ demienneract facets)

Elements

Gosset 1k2 figures
n1k2 Petrie
polygon

projection
Name
Coxeter-Dynkin
diagram
Facets Elements
1k-1,2 (n-1)-demicube Vertices Edges Faces Cells 4-faces5-faces6-faces7-faces
4102 4-simplex t0.svg 120
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01l.png
--5
110
3-simplex t0.svg
51010
2-simplex t0.svg
5
3-simplex t0.svg
    
5112 5-demicube.svg 121
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.png
16
120
4-simplex t0.svg
10
111
4-orthoplex.svg
1680160
2-simplex t0.svg
120
3-simplex t0.svg
26
4-simplex t0.svg 4-orthoplex.svg
   
6122 Up 1 22 t0 E6.svg 122
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
27
112
5-demicube.svg
27
121
5-demicube.svg
727202160
2-simplex t0.svg
2160
3-simplex t0.svg
702
4-simplex t0.svg 4-orthoplex.svg
54
5-demicube.svg
  
7132 Up2 1 32 t0 E7.svg 132
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
56
122
Up 1 22 t0 E6.svg
126
131
6-demicube.svg
5761008040320
2-simplex t0.svg
50400
3-simplex t0.svg
23688
4-simplex t0.svg 4-orthoplex.svg
4284
5-simplex t0.svg 5-demicube.svg
182
Gosset 1 22 polytope.svg 6-demicube.svg
 
8142 Gosset 1 42 polytope petrie.svg 142
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
240
132
Up2 1 32 t0 E7.svg
2160
141
7-demicube.svg
172804838402419200
2-simplex t0.svg
3628800
3-simplex t0.svg
2298240
4-simplex t0.svg 4-orthoplex.svg
725760
5-simplex t0.svg 5-demicube.svg
106080
6-simplex t0.svg 6-demicube.svg Gosset 1 22 polytope.svg
2400
7-demicube.svg 2 41 polytope petrie.svg
9152 152
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
(8-space tessellation)

142
Gosset 1 42 polytope petrie.svg

151
8-demicube.svg
10162 162
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
(9-space hyperbolic tessellation)

152

161
9-demicube.svg

See also

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References

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds
Space Family / /
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152251521
E9 Uniform 9-honeycomb {3[10]}δ10hδ10qδ10
E10Uniform 10-honeycomb{3[11]}δ11hδ11qδ11
En-1Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k22k1k21