Biruncitruncated_6-simplex
In six-dimensional geometry, a runcinated 6-simplex is a convex uniform 6-polytope constructed as a runcination (3rd order truncations) of the regular 6-simplex.
 6-simplex    |
 Runcinated 6-simplex |
 Biruncinated 6-simplex |
 Runcitruncated 6-simplex |
 Biruncitruncated 6-simplex |
 Runcicantellated 6-simplex |
 Runcicantitruncated 6-simplex |
 Biruncicantitruncated 6-simplex | |
| Orthogonal projections in A6 Coxeter plane | ||
|---|---|---|
There are 8 unique runcinations of the 6-simplex with permutations of truncations, and cantellations.
| Runcinated 6-simplex | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | t0,3{3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 70 |
| 4-faces | 455 |
| Cells | 1330 |
| Faces | 1610 |
| Edges | 840 |
| Vertices | 140 |
| Vertex figure | |
| Coxeter group | A6, [35], order 5040 |
| Properties | convex |
Alternate names
- Small prismated heptapeton (Acronym: spil) (Jonathan Bowers)[1]
Coordinates
The vertices of the runcinated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,1,2). This construction is based on facets of the runcinated 7-orthoplex.
Images
| Ak Coxeter plane | A6 | A5 | A4 |
|---|---|---|---|
| Graph | |
 |
 |
| Dihedral symmetry | [7] | [6] | [5] |
| Ak Coxeter plane | A3 | A2 | |
| Graph |  |
 | |
| Dihedral symmetry | [4] | [3] |
| biruncinated 6-simplex | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | t1,4{3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 84 |
| 4-faces | 714 |
| Cells | 2100 |
| Faces | 2520 |
| Edges | 1260 |
| Vertices | 210 |
| Vertex figure | |
| Coxeter group | A6, [[35]], order 10080 |
| Properties | convex |
Alternate names
- Small biprismated tetradecapeton (Acronym: sibpof) (Jonathan Bowers)[2]
Coordinates
The vertices of the biruncinted 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 7-orthoplex.
Images
| Ak Coxeter plane | A6 | A5 | A4 |
|---|---|---|---|
| Graph | |||
| Symmetry | [[7]](*)=[14] | [6] | [[5]](*)=[10] |
| Ak Coxeter plane | A3 | A2 | |
| Graph | |||
| Symmetry | [4] | [[3]](*)=[6] |
- Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.
| Runcitruncated 6-simplex | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | t0,1,3{3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 70 |
| 4-faces | 560 |
| Cells | 1820 |
| Faces | 2800 |
| Edges | 1890 |
| Vertices | 420 |
| Vertex figure | |
| Coxeter group | A6, [35], order 5040 |
| Properties | convex |
Alternate names
- Prismatotruncated heptapeton (Acronym: patal) (Jonathan Bowers)[3]
Coordinates
The vertices of the runcitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,1,2,3). This construction is based on facets of the runcitruncated 7-orthoplex.
Images
| Ak Coxeter plane | A6 | A5 | A4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [7] | [6] | [5] |
| Ak Coxeter plane | A3 | A2 | |
| Graph | |||
| Dihedral symmetry | [4] | [3] |
| biruncitruncated 6-simplex | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | t1,2,4{3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 84 |
| 4-faces | 714 |
| Cells | 2310 |
| Faces | 3570 |
| Edges | 2520 |
| Vertices | 630 |
| Vertex figure | |
| Coxeter group | A6, [35], order 5040 |
| Properties | convex |
Alternate names
- Biprismatorhombated heptapeton (Acronym: bapril) (Jonathan Bowers)[4]
Coordinates
The vertices of the biruncitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,1,2,3,3). This construction is based on facets of the biruncitruncated 7-orthoplex.
Images
| Ak Coxeter plane | A6 | A5 | A4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [7] | [6] | [5] |
| Ak Coxeter plane | A3 | A2 | |
| Graph | |||
| Dihedral symmetry | [4] | [3] |
| Runcicantellated 6-simplex | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | t0,2,3{3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 70 |
| 4-faces | 455 |
| Cells | 1295 |
| Faces | 1960 |
| Edges | 1470 |
| Vertices | 420 |
| Vertex figure | |
| Coxeter group | A6, [35], order 5040 |
| Properties | convex |
Alternate names
- Prismatorhombated heptapeton (Acronym: pril) (Jonathan Bowers)[5]
Coordinates
The vertices of the runcicantellated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,2,3). This construction is based on facets of the runcicantellated 7-orthoplex.
Images
| Ak Coxeter plane | A6 | A5 | A4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [7] | [6] | [5] |
| Ak Coxeter plane | A3 | A2 | |
| Graph | |||
| Dihedral symmetry | [4] | [3] |
| Runcicantitruncated 6-simplex | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | t0,1,2,3{3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 70 |
| 4-faces | 560 |
| Cells | 1820 |
| Faces | 3010 |
| Edges | 2520 |
| Vertices | 840 |
| Vertex figure | |
| Coxeter group | A6, [35], order 5040 |
| Properties | convex |
Alternate names
- Runcicantitruncated heptapeton
- Great prismated heptapeton (Acronym: gapil) (Jonathan Bowers)[6]
Coordinates
The vertices of the runcicantitruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,0,1,2,3,4). This construction is based on facets of the runcicantitruncated 7-orthoplex.
Images
| Ak Coxeter plane | A6 | A5 | A4 |
|---|---|---|---|
| Graph | |||
| Dihedral symmetry | [7] | [6] | [5] |
| Ak Coxeter plane | A3 | A2 | |
| Graph | |||
| Dihedral symmetry | [4] | [3] |
| biruncicantitruncated 6-simplex | |
|---|---|
| Type | uniform 6-polytope |
| Schläfli symbol | t1,2,3,4{3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 5-faces | 84 |
| 4-faces | 714 |
| Cells | 2520 |
| Faces | 4410 |
| Edges | 3780 |
| Vertices | 1260 |
| Vertex figure | |
| Coxeter group | A6, [[35]], order 10080 |
| Properties | convex |
Alternate names
- Biruncicantitruncated heptapeton
- Great biprismated tetradecapeton (Acronym: gibpof) (Jonathan Bowers)[7]
Coordinates
The vertices of the biruncicantittruncated 6-simplex can be most simply positioned in 7-space as permutations of (0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 7-orthoplex.
Images
| Ak Coxeter plane | A6 | A5 | A4 |
|---|---|---|---|
| Graph | |||
| Symmetry | [[7]](*)=[14] | [6] | [[5]](*)=[10] |
| Ak Coxeter plane | A3 | A2 | |
| Graph | |||
| Symmetry | [4] | [[3]](*)=[6] |
- Note: (*) Symmetry doubled for Ak graphs with even k due to symmetrically-ringed Coxeter-Dynkin diagram.
The truncated 6-simplex is one of 35 uniform 6-polytopes based on the [3,3,3,3,3] Coxeter group, all shown here in A6 Coxeter plane orthographic projections.
- H.S.M. Coxeter:
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6
- (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
- Klitzing, Richard. "6D uniform polytopes (polypeta)". x3o3o3x3o3o - spil, o3x3o3o3x3o - sibpof, x3x3o3x3o3o - patal, o3x3x3o3x3o - bapril, x3o3x3x3o3o - pril, x3x3x3x3o3o - gapil, o3x3x3x3x3o - gibpof
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 | ||||||||||||