| Orthogonal projections in A7 Coxeter plane |
|---|
| 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 |
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)
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
Orthographic projections| 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)
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
Orthographic projections| 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)
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
Orthographic projections| 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)
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
Orthographic projections| 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)
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
Orthographic projections| 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
Orthographic projections| 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)
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
Orthographic projections| 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)
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
Orthographic projections| 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
- 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 Ivić 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. . x3o3o3x3o3o3o - spo, o3x3o3o3x3o3o - sibpo, x3x3o3x3o3o3o - patto, o3x3x3o3x3o3o - bipto, x3o3x3x3o3o3o - paro, x3x3x3x3o3o3o - gapo, o3x3x3x3x3o3o - gibpo
External links