Orthogonal projections in B4 Coxeter plane | ||||
---|---|---|---|---|
![]() 7-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Hexicated 7-orthoplex Hexicated 7-cube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Hexi-truncated 7-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Hexi-cantellated 7-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Hexicanti-truncated 7-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Hexirunci-truncated 7-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Hexirunci-cantellated 7-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Hexisteri-truncated 7-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Hexiruncicanti-truncated 7-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Hexistericanti-truncated 7-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Hexisterirunci-truncated 7-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Hexipenticanti-truncated 7-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Hexisteriruncicanti-truncated 7-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Hexipentiruncicanti-truncated 7-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
In seven-dimensional geometry, a hexicated 7-orthoplex (also hexicated 7-cube) is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-orthoplex.
There are 32 hexications for the 7-orthoplex, including all permutations of truncations, cantellations, runcinations, sterications, and pentellations. 12 are represented here, while 20 are more easily constructed from the 7-cube.
Hexitruncated 7-orthoplex
editHexitruncated 7-orthoplex | |
---|---|
Typ | Uniform 7-polytope |
Schläfli symbol | t0,1,6{35,4 |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 29568 |
Vertices | 5376 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
edit- Petitruncated heptacross
Images
editCoxeter 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] |
Hexicantellated 7-orthoplex
editHexicantellated 7-orthoplex | |
---|---|
Typ | uniform 7-polytope |
Schläfli symbol | t0,2,6{35,4} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 94080 |
Vertices | 13440 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
edit- Petirhombated heptacross
Images
editCoxeter 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] |
Hexicantitruncated 7-orthoplex
editHexicantitruncated 7-orthoplex | |
---|---|
Typ | uniform 7-polytope |
Schläfli symbol | t0,1,2,6{35,4} |
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- Petigreatorhombated heptacross
Images
editCoxeter 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] |
Hexiruncitruncated 7-orthoplex
editHexiruncitruncated 7-orthoplex | |
---|---|
Typ | uniform 7-polytope |
Schläfli symbol | t0,1,3,6{35,3} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 322560 |
Vertices | 53760 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
edit- Petiprismatotruncated heptacross
Images
editCoxeter 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] |
Hexiruncicantellated 7-orthoplex
editHexiruncicantellated 7-orthoplex | |
---|---|
Typ | uniform 7-polytope |
Schläfli symbol | t0,2,3,6{35,4} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 268800 |
Vertices | 53760 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
In seven-dimensional geometry, a hexiruncicantellated 7-orthoplex is a uniform 7-polytope.
Alternate names
edit- Petiprismatorhombated heptacross
Images
editCoxeter 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] |
Hexisteritruncated 7-orthoplex
edithexisteritruncated 7-orthoplex | |
---|---|
Typ | uniform 7-polytope |
Schläfli symbol | t0,1,4,6{35,4} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 322560 |
Vertices | 53760 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
edit- Peticellitruncated heptacross
Images
editCoxeter 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] |
Hexiruncicantitruncated 7-orthoplex
editHexiruncicantitruncated 7-orthoplex | |
---|---|
Typ | uniform 7-polytope |
Schläfli symbol | t0,1,2,3,6{35,4} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 483840 |
Vertices | 107520 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
edit- Petigreatoprismated heptacross
Images
editCoxeter 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] |
Hexistericantitruncated 7-orthoplex
editHexistericantitruncated 7-orthoplex | |
---|---|
Typ | uniform 7-polytope |
Schläfli symbol | t0,1,2,4,6{35,4} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 806400 |
Vertices | 161280 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
edit- Peticelligreatorhombated heptacross
Images
editCoxeter 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] |
Hexisteriruncitruncated 7-orthoplex
editHexisteriruncitruncated 7-orthoplex | |
---|---|
Typ | uniform 7-polytope |
Schläfli symbol | t0,1,3,4,6{35,4} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 725760 |
Vertices | 161280 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
edit- Peticelliprismatotruncated heptacross
Images
editCoxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | too complex | ||
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] |
Hexipenticantitruncated 7-orthoplex
edithexipenticantitruncated 7-orthoplex | |
---|---|
Typ | uniform 7-polytope |
Schläfli symbol | t0,1,2,5,6{35,4} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 483840 |
Vertices | 107520 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
edit- Petiterigreatorhombated heptacross
Images
editCoxeter 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] |
Hexisteriruncicantitruncated 7-orthoplex
editHexisteriruncicantitruncated 7-orthoplex | |
---|---|
Typ | uniform 7-polytope |
Schläfli symbol | t0,1,2,3,4,6{4,35} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 1290240 |
Vertices | 322560 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
edit- Great petacellated heptacross
Images
editCoxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | too complex | ||
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] |
Hexipentiruncicantitruncated 7-orthoplex
editHexipentiruncicantitruncated 7-orthoplex | |
---|---|
Typ | uniform 7-polytope |
Schläfli symbol | t0,1,2,3,5,6{35,3} |
Coxeter-Dynkin diagrams | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 1290240 |
Vertices | 322560 |
Vertex figure | |
Coxeter groups | B7, [4,35] |
Properties | convex |
Alternate names
edit- Petiterigreatoprismated heptacross
Images
editCoxeter plane | B7 / A6 | B6 / D7 | B5 / D6 / A4 |
---|---|---|---|
Graph | too complex | ||
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
editReferences
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, PhD (1966)
- Klitzing, Richard. "7D uniform polytopes (polyexa)".