 7-demicube    
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 7-orthoplex  
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In 7-dimensional geometry, there are 95 uniform polytopes with D7 symmetry; 32 are unique, and 63 are shared with the B7 symmetry. There are two regular forms, the 7-orthoplex, and 7-demicube with 14 and 64 vertices respectively.
They can be visualized as symmetric orthographic projections in Coxeter planes of the D6 Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projections of these 32 polytopes can be made in the D7, D6, D5, D4, D3, A5, A3, Coxeter planes. Ak has symmetry, Dk has symmetry. B7 is also included although only half of its symmetry exists in these polytopes.
These 32 polytopes are each shown in these 8 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
| # | Coxeter plane graphs | Coxeter diagram Names | |||||||
|---|---|---|---|---|---|---|---|---|---|
| B7 |
D7 |
D6 |
D5 |
D4 |
D3 |
A5 |
A3 | ||
| 1 |  |
|
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=   7-demicube Demihepteract (Hesa) |
| 2 |  |
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 |
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= Cantic 7-cube Truncated demihepteract (Thesa) |
| 3 |  |
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 |
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= Runcic 7-cube Small rhombated demihepteract (Sirhesa) |
| 4 |  |
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 |
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= Steric 7-cube Small prismated demihepteract (Sphosa) |
| 5 |  |
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 |
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= Pentic 7-cube Small cellated demihepteract (Sochesa) |
| 6 |  |
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= Hexic 7-cube Small terated demihepteract (Suthesa) |
| 7 |  |
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= Runcicantic 7-cube Great rhombated demihepteract (Girhesa) |
| 8 |  |
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= Stericantic 7-cube Prismatotruncated demihepteract (Pothesa) |
| 9 |  |
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= Steriruncic 7-cube Prismatorhomated demihepteract (Prohesa) |
| 10 |  |
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= Penticantic 7-cube Cellitruncated demihepteract (Cothesa) |
| 11 |  |
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= Pentiruncic 7-cube Cellirhombated demihepteract (Crohesa) |
| 12 |  |
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= Pentisteric 7-cube Celliprismated demihepteract (Caphesa) |
| 13 |  |
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= Hexicantic 7-cube Teritruncated demihepteract (Tuthesa) |
| 14 |  |
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= Hexiruncic 7-cube Terirhombated demihepteract (Turhesa) |
| 15 |  |
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= Hexisteric 7-cube Teriprismated demihepteract (Tuphesa) |
| 16 |  |
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= Hexipentic 7-cube Tericellated demihepteract (Tuchesa) |
| 17 |  |
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= Steriruncicantic 7-cube Great prismated demihepteract (Gephosa) |
| 18 |  |
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= Pentiruncicantic 7-cube Celligreatorhombated demihepteract (Cagrohesa) |
| 19 |  |
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= Pentistericantic 7-cube Celliprismatotruncated demihepteract (Capthesa) |
| 20 |  |
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= Pentisteriruncic 7-cube Celliprismatorhombated demihepteract (Coprahesa) |
| 21 |  |
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= Hexiruncicantic 7-cube Terigreatorhombated demihepteract (Tugrohesa) |
| 22 |  |
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= Hexistericantic 7-cube Teriprismatotruncated demihepteract (Tupthesa) |
| 23 |  |
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= Hexisteriruncic 7-cube Teriprismatorhombated demihepteract (Tuprohesa) |
| 24 |  |
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= Hexipenticantic 7-cube Tericellitruncated demihepteract (Tucothesa) |
| 25 |  |
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= Hexipentiruncic 7-cube Tericellirhombated demihepteract (Tucrohesa) |
| 26 |  |
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= Hexipentisteric 7-cube Tericelliprismated demihepteract (Tucophesa) |
| 27 |  |
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= Pentisteriruncicantic 7-cube Great cellated demihepteract (Gochesa) |
| 28 |  |
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= Hexisteriruncicantic 7-cube Terigreatoprimated demihepteract (Tugphesa) |
| 29 |  |
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= Hexipentiruncicantic 7-cube Tericelligreatorhombated demihepteract (Tucagrohesa) |
| 30 |  |
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= Hexipentistericantic 7-cube Tericelliprismatotruncated demihepteract (Tucpathesa) |
| 31 |  |
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= Hexipentisteriruncic 7-cube Tericellprismatorhombated demihepteract (Tucprohesa) |
| 32 |  |
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= Hexipentisteriruncicantic 7-cube Great terated demihepteract (Guthesa) |
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,
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II,
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III,
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
- Klitzing, Richard. "7D uniform polytopes (polyexa)".
Notes
| 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 | ||||||||||||