![]() 7-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Runcinated 7-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Biruncinated 7-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Runcitruncated 7-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Biruncitruncated 7-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Runcicantellated 7-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Biruncicantellated 7-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Runcicantitruncated 7-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Biruncicantitruncated 7-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Orthogonal projections in B7 Coxeter plane |
---|
In seven-dimensional geometry, a runcinated 7-cube is a convex uniform 7-polytope with 3rd order truncations (runcination) of the regular 7-cube.
There are 16 unique runcinations of the 7-cube with permutations of truncations, and cantellations. 8 are more simply constructed from the 7-orthoplex.
These polytopes are among 127 uniform 7-polytopes with B7 symmetry.
Runcinated 7-cube
[edit]Runcinated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,3{4,35} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 33600 |
Vertices | 4480 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
[edit]- Small prismated hepteract (acronym: spesa) (Jonathan Bowers)[1]
Images
[edit]Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | ![]() |
![]() | |
Dihedral symmetry | [6] | [4] |
Biruncinated 7-cube
[edit]Biruncinated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t1,4{4,35} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 67200 |
Vertices | 8960 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
[edit]- Small biprismated hepteract (Acronym sibposa) (Jonathan Bowers)[2]
Images
[edit]Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | ![]() |
![]() | |
Dihedral symmetry | [6] | [4] |
Runcitruncated 7-cube
[edit]Runcitruncated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,1,3{4,35} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 73920 |
Vertices | 13440 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
[edit]- Prismatotruncated hepteract (acronym: petsa) (Jonathan Bowers)[3]
Images
[edit]Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | ![]() |
![]() | |
Dihedral symmetry | [6] | [4] |
Biruncitruncated 7-cube
[edit]Biruncitruncated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t1,2,4{4,35} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 134400 |
Vertices | 26880 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
[edit]- Biprismatotruncated hepteract (acronym: biptesa) (Jonathan Bowers)[4]
Images
[edit]Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | ![]() |
![]() | |
Dihedral symmetry | [6] | [4] |
Runcicantellated 7-cube
[edit]Runcicantellated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,2,3{4,35} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 53760 |
Vertices | 13440 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
[edit]- Prismatorhombated hepteract (acronym: parsa) (Jonathan Bowers)[5]
Images
[edit]Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | ![]() |
![]() | |
Dihedral symmetry | [6] | [4] |
Biruncicantellated 7-cube
[edit]biruncicantellated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t1,3,4{4,35} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 120960 |
Vertices | 26880 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
[edit]- Biprismatorhombated hepteract (acronym: bopresa) (Jonathan Bowers)[6]
Images
[edit]Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | ![]() |
![]() | |
Dihedral symmetry | [6] | [4] |
Runcicantitruncated 7-cube
[edit]Runcicantitruncated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t0,1,2,3{4,35} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 94080 |
Vertices | 26880 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
[edit]- Great prismated hepteract (acronym: gapsa) (Jonathan Bowers)[7]
Images
[edit]Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | ![]() |
![]() | |
Dihedral symmetry | [6] | [4] |
Biruncicantitruncated 7-cube
[edit]biruncicantitruncated 7-cube | |
---|---|
Type | uniform 7-polytope |
Schläfli symbol | t1,2,3,4{4,35} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 188160 |
Vertices | 53760 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
[edit]- Great biprismated hepteract (acronym: gibposa) (Jonathan Bowers)[8]
Images
[edit]Coxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [14] | [12] | [10] |
Coxeter plane | B4 / D5 | B3 / D4 / A2 | B2 / D3 |
Graph | ![]() |
![]() |
![]() |
Dihedral symmetry | [8] | [6] | [4] |
Coxeter plane | A5 | A3 | |
Graph | ![]() |
![]() | |
Dihedral symmetry | [6] | [4] |
Notes
[edit]References
[edit]- 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. (1966)
- Klitzing, Richard. "7D uniform polytopes (polyexa)". x3o3o3x3o3o4o - spo, o3x3o3o3x3o4o - sibpo, x3x3o3x3o3o4o - patto, o3x3x3o3x3o4o - bipto, x3o3x3x3o3o4o - paro, x3x3x3x3o3o4o - gapo, o3x3x3x3x3o3o- gibpo