![]() 7-demicube ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() 7-orthoplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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)".