Ingeometry, 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 
|
∞ | ||||||||
|
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 polychoron | Pentachoron | 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 2–9
| ||||||
|---|---|---|---|---|---|---|
| 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 | 133 • 331 |
| E8 | Uniform 8-honeycomb | {3[9]} | δ9 | hδ9 | qδ9 | 152 • 251 • 521 |
| E9 | Uniform 9-honeycomb | {3[10]} | δ10 | hδ10 | qδ10 | |
| E10 | Uniform 10-honeycomb | {3[11]} | δ11 | hδ11 | qδ11 | |
| En-1 | Uniform (n-1)-honeycomb | {3[n]} | δn | hδn | qδn | 1k2 • 2k1 • k21 |