List of D7 polytopes

Orthographic projections in the D₇ Coxeter plane
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

Symmetric orthographic projections of these 32 polytopes can be made in the D₇, D₆, D₅, D₄, D₃, A₅, A₃, Coxeter planes. A_k has [k+1] symmetry, D_k 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]	Coxeter diagram Names
1									7-demicube Demihepteract (Hesa)
2									Truncated 7-demicube Truncated demihepteract (Thesa)
3									Cantellated 7-demicube Small rhombated demihepteract (Sirhesa)
4									Runcinated 7-demicube Small prismated demihepteract (Sphosa)
5									Stericated 7-demicube Small cellated demihepteract (Sochesa)
6									Pentellated 7-demicube Small terated demihepteract (Suthesa)
7									Cantitruncated 7-demicube Great rhombated demihepteract (Girhesa)
8									Runcitruncated 7-demicube Prismatotruncated demihepteract (Pothesa)
9									Runcicantellated 7-demicube Prismatorhomated demihepteract (Prohesa)
10									Steritruncated 7-demicube Cellitruncated demihepteract (Cothesa)
11									Stericantellated 7-demicube Cellirhombated demihepteract (Crohesa)
12									Steriruncinated 7-demicube Celliprismated demihepteract (Caphesa)
13									Pentitruncated 7-demicube Teritruncated demihepteract (Tuthesa)
14									Penticantellated 7-demicube Terirhombated demihepteract (Turhesa)
15									Pentiruncinated 7-demicube Teriprismated demihepteract (Tuphesa)
16									Pentistericated 7-demicube Tericellated demihepteract (Tuchesa)
17									Runcicantitruncated 7-demicube Great prismated demihepteract (Gephosa)
18									Stericantitruncated 7-demicube Celligreatorhombated demihepteract (Cagrohesa)
19									Steriruncitruncated 7-demicube Celliprismatotruncated demihepteract (Capthesa)
20									Steriruncicantellated 7-demicube Celliprismatorhombated demihepteract (Coprahesa)
21									Penticantitruncated 7-demicube Terigreatorhombated demihepteract (Tugrohesa)
22									Pentiruncitruncated 7-demicube Teriprismatotruncated demihepteract (Tupthesa)
23									Pentiruncicantellated 7-demicube Teriprismatorhombated demihepteract (Tuprohesa)
24									Pentisteritruncated 7-demicube Tericellitruncated demihepteract (Tucothesa)
25									Pentistericantellated 7-demicube Tericellirhombated demihepteract (Tucrohesa)
26									Pentisteriruncinated 7-demicube Tericelliprismated demihepteract (Tucophesa)
27									Steriruncicantitruncated 7-demicube Great cellated demihepteract (Gochesa)
28									Pentiruncicantitruncated 7-demicube Terigreatoprimated demihepteract (Tugphesa)
29									Pentistericantitruncated 7-demicube Tericelligreatorhombated demihepteract (Tucagrohesa)
30									Pentisteriruncitruncated 7-demicube Tericelliprismatotruncated demihepteract (Tucpathesa)
31									Pentisteriruncicantellated 7-demicube Tericellprismatorhombated demihepteract (Tucprohesa)
32									Pentisteriruncicantitruncated 7-demicube 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 ^[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
Richard Klitzing, 7D, uniform polytopes (polyexa)

Notes

↑ Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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