Hexicated 7-orthoplexes
| 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
| Hexitruncated 7-orthoplex | |
|---|---|
| Type | 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
- Petitruncated heptacross
Images
| 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] |
Hexicantellated 7-orthoplex
| Hexicantellated 7-orthoplex | |
|---|---|
| Type | 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
- Petirhombated heptacross
Images
| 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] |
Hexicantitruncated 7-orthoplex
| Hexicantitruncated 7-orthoplex | |
|---|---|
| Type | 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
- Petigreatorhombated heptacross
Images
| 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] |
Hexiruncitruncated 7-orthoplex
| Hexiruncitruncated 7-orthoplex | |
|---|---|
| Type | 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
- Petiprismatotruncated heptacross
Images
| 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] |
Hexiruncicantellated 7-orthoplex
| Hexiruncicantellated 7-orthoplex | |
|---|---|
| Type | 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
- Petiprismatorhombated heptacross
Images
| 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] |
Hexisteritruncated 7-orthoplex
| hexisteritruncated 7-orthoplex | |
|---|---|
| Type | 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
- Peticellitruncated heptacross
Images
| 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] |
Hexiruncicantitruncated 7-orthoplex
| Hexiruncicantitruncated 7-orthoplex | |
|---|---|
| Type | 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
- Petigreatoprismated heptacross
Images
| 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] |
Hexistericantitruncated 7-orthoplex
| Hexistericantitruncated 7-orthoplex | |
|---|---|
| Type | 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
- Peticelligreatorhombated heptacross
Images
| 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] |
Hexisteriruncitruncated 7-orthoplex
| Hexisteriruncitruncated 7-orthoplex | |
|---|---|
| Type | 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
- Peticelliprismatotruncated heptacross
Images
| Coxeter 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
| hexipenticantitruncated 7-orthoplex | |
|---|---|
| Type | 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
- Petiterigreatorhombated heptacross
Images
| 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] |
Hexisteriruncicantitruncated 7-orthoplex
| Hexisteriruncicantitruncated 7-orthoplex | |
|---|---|
| Type | 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
- Great petacellated heptacross
Images
| Coxeter 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
| Hexipentiruncicantitruncated 7-orthoplex | |
|---|---|
| Type | 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
- Petiterigreatoprismated heptacross
Images
| Coxeter 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
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
- (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)".
External links
- Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- 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 / E9 / E10 / F4 / G2 | Hn | |||||||
| Regular polygon | Triangle | Square | p-gon | Hexagon | Pentagon | |||||||
| Uniform polyhedron | Tetrahedron | Octahedron • Cube | Demicube | Dodecahedron • Icosahedron | ||||||||
| Uniform 4-polytope | 5-cell | 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 | ||||||||||||
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