D7 polytope
 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
[edit]Symmetric orthographic projections of these 32 polytopes can be made in the D7, D6, D5, D4, D3, A5, A3, Coxeter planes. Ak has [k+1] symmetry, Dk has [2(k-1)] symmetry. B7 is also included although only half of its [14] 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 [14/2] |
D7 [12] |
D6 [10] |
D5 [8] |
D4 [6] |
D3 [4] |
A5 [6] |
A3 [4] | ||
| 1 |  |
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=   7-demicube Demihepteract (Hesa) |
| 2 |  |
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= Cantic 7-cube Truncated demihepteract (Thesa) |
| 3 |  |
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= Runcic 7-cube Small rhombated demihepteract (Sirhesa) |
| 4 |  |
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= Steric 7-cube Small prismated demihepteract (Sphosa) |
| 5 |  |
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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
[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]
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
- Klitzing, Richard. "7D uniform polytopes (polyexa)".