Runcinated 7-simplexes
 7-simplex    |  Runcinated 7-simplex |  Biruncinated 7-simplex |
 Runcitruncated 7-simplex |  Biruncitruncated 7-simplex |  Runcicantellated 7-simplex |
 Biruncicantellated 7-simplex |  Runcicantitruncated 7-simplex |  Biruncicantitruncated 7-simplex |
| Orthogonal projections in A7 Coxeter plane | ||
|---|---|---|
In seven-dimensional geometry, a runcinated 7-simplex is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-simplex.
There are 8 unique runcinations of the 7-simplex with permutations of truncations, and cantellations.
Runcinated 7-simplex
| Runcinated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,3{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 2100 |
| Vertices | 280 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
Alternate names
- Small prismated octaexon (acronym: spo) (Jonathan Bowers)[1]
Coordinates
The vertices of the runcinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,1,2). This construction is based on facets of the runcinated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph |  |  |  |
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |  |  |
| Dihedral symmetry | [5] | [4] | [3] |
Biruncinated 7-simplex
| Biruncinated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t1,4{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 4200 |
| Vertices | 560 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
Alternate names
- Small biprismated octaexon (sibpo) (Jonathan Bowers)[2]
Coordinates
The vertices of the biruncinated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph |  |  |  |
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |  |  |
| Dihedral symmetry | [5] | [4] | [3] |
Runcitruncated 7-simplex
| runcitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,3{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 4620 |
| Vertices | 840 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
Alternate names
- Prismatotruncated octaexon (acronym: patto) (Jonathan Bowers)[3]
Coordinates
The vertices of the runcitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,1,2,3). This construction is based on facets of the runcitruncated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph |  |  |  |
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |  |  |
| Dihedral symmetry | [5] | [4] | [3] |
Biruncitruncated 7-simplex
| Biruncitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t1,2,4{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 8400 |
| Vertices | 1680 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
Alternate names
- Biprismatotruncated octaexon (acronym: bipto) (Jonathan Bowers)[4]
Coordinates
The vertices of the biruncitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,1,2,3,3). This construction is based on facets of the biruncitruncated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph |  |  |  |
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |  |  |
| Dihedral symmetry | [5] | [4] | [3] |
Runcicantellated 7-simplex
| runcicantellated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,2,3{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 3360 |
| Vertices | 840 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
Alternate names
- Prismatorhombated octaexon (acronym: paro) (Jonathan Bowers)[5]
Coordinates
The vertices of the runcicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,2,3). This construction is based on facets of the runcicantellated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph |  |  |  |
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |  |  |
| Dihedral symmetry | [5] | [4] | [3] |
Biruncicantellated 7-simplex
| biruncicantellated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t1,3,4{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | |
| Vertices | |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
Alternate names
- Biprismatorhombated octaexon (acronym: bipro) (Jonathan Bowers)
Coordinates
The vertices of the biruncicantellated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,2,3,3). This construction is based on facets of the biruncicantellated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph |  |  |  |
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |  |  |
| Dihedral symmetry | [5] | [4] | [3] |
Runcicantitruncated 7-simplex
| runcicantitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t0,1,2,3{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 5880 |
| Vertices | 1680 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
Alternate names
- Great prismated octaexon (acronym: gapo) (Jonathan Bowers)[6]
Coordinates
The vertices of the runcicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,0,1,2,3,4). This construction is based on facets of the runcicantitruncated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph |  |  |  |
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |  |  |
| Dihedral symmetry | [5] | [4] | [3] |
Biruncicantitruncated 7-simplex
| biruncicantitruncated 7-simplex | |
|---|---|
| Type | uniform 7-polytope |
| Schläfli symbol | t1,2,3,4{3,3,3,3,3,3} |
| Coxeter-Dynkin diagrams | |
| 6-faces | |
| 5-faces | |
| 4-faces | |
| Cells | |
| Faces | |
| Edges | 11760 |
| Vertices | 3360 |
| Vertex figure | |
| Coxeter group | A7, [36], order 40320 |
| Properties | convex |
Alternate names
- Great biprismated octaexon (acronym: gibpo) (Jonathan Bowers)[7]
Coordinates
The vertices of the biruncicantitruncated 7-simplex can be most simply positioned in 8-space as permutations of (0,0,0,1,2,3,4,4). This construction is based on facets of the biruncicantitruncated 8-orthoplex.
Images
| Ak Coxeter plane | A7 | A6 | A5 |
|---|---|---|---|
| Graph |  |  |  |
| Dihedral symmetry | [8] | [7] | [6] |
| Ak Coxeter plane | A4 | A3 | A2 |
| Graph |  |  |  |
| Dihedral symmetry | [5] | [4] | [3] |
Related polytopes
These polytopes are among 71 uniform 7-polytopes with A7 symmetry.
| A7 polytopes | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
 t0 |  t1 |  t2 |  t3 |  t0,1 |  t0,2 |  t1,2 |  t0,3 | ||||
 t1,3 |  t2,3 |  t0,4 |  t1,4 |  t2,4 |  t0,5 |  t1,5 |  t0,6 | ||||
 t0,1,2 |  t0,1,3 |  t0,2,3 |  t1,2,3 |  t0,1,4 |  t0,2,4 |  t1,2,4 |  t0,3,4 | ||||
 t1,3,4 |  t2,3,4 |  t0,1,5 |  t0,2,5 |  t1,2,5 |  t0,3,5 |  t1,3,5 |  t0,4,5 | ||||
 t0,1,6 |  t0,2,6 |  t0,3,6 |  t0,1,2,3 |  t0,1,2,4 |  t0,1,3,4 |  t0,2,3,4 |  t1,2,3,4 | ||||
 t0,1,2,5 |  t0,1,3,5 |  t0,2,3,5 |  t1,2,3,5 |  t0,1,4,5 |  t0,2,4,5 |  t1,2,4,5 |  t0,3,4,5 | ||||
 t0,1,2,6 |  t0,1,3,6 |  t0,2,3,6 |  t0,1,4,6 |  t0,2,4,6 |  t0,1,5,6 |  t0,1,2,3,4 |  t0,1,2,3,5 | ||||
 t0,1,2,4,5 |  t0,1,3,4,5 |  t0,2,3,4,5 |  t1,2,3,4,5 |  t0,1,2,3,6 |  t0,1,2,4,6 |  t0,1,3,4,6 |  t0,2,3,4,6 | ||||
 t0,1,2,5,6 |  t0,1,3,5,6 |  t0,1,2,3,4,5 |  t0,1,2,3,4,6 |  t0,1,2,3,5,6 |  t0,1,2,4,5,6 |  t0,1,2,3,4,5,6 | |||||
Notes
References
- 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 [1]
- (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. "7D uniform polytopes (polyexa)". x3o3o3x3o3o3o - spo, o3x3o3o3x3o3o - sibpo, x3x3o3x3o3o3o - patto, o3x3x3o3x3o3o - bipto, x3o3x3x3o3o3o - paro, x3x3x3x3o3o3o - gapo, o3x3x3x3x3o3o- gibpo
External links
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
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 | ||||||||||||
