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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 in progressive order: red, orange, yellow, green, cyan, blue, purple, magenta, red-violet.

#	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									= Cantic 7-cube Truncated demihepteract (Thesa)
3									= Runcic 7-cube Small rhombated demihepteract (Sirhesa)
4									= Steric 7-cube Small prismated demihepteract (Sphosa)
5									= Pentic 7-cube Small cellated demihepteract (Sochesa)
6									= Hexic 7-cube Small terated demihepteract (Suthesa)
7									= Runcicantic 7-cube Great rhombated demihepteract (Girhesa)
8									= Stericantic 7-cube Prismatotruncated demihepteract (Pothesa)
9									= Steriruncic 7-cube Prismatorhomated demihepteract (Prohesa)
10									= Penticantic 7-cube Cellitruncated demihepteract (Cothesa)
11									= Pentiruncic 7-cube Cellirhombated demihepteract (Crohesa)
12									= Pentisteric 7-cube Celliprismated demihepteract (Caphesa)
13									= Hexicantic 7-cube Teritruncated demihepteract (Tuthesa)
14									= Hexiruncic 7-cube Terirhombated demihepteract (Turhesa)
15									= Hexisteric 7-cube Teriprismated demihepteract (Tuphesa)
16									= Hexipentic 7-cube Tericellated demihepteract (Tuchesa)
17									= Steriruncicantic 7-cube Great prismated demihepteract (Gephosa)
18									= Pentiruncicantic 7-cube Celligreatorhombated demihepteract (Cagrohesa)
19									= Pentistericantic 7-cube Celliprismatotruncated demihepteract (Capthesa)
20									= Pentisteriruncic 7-cube Celliprismatorhombated demihepteract (Coprahesa)
21									= Hexiruncicantic 7-cube Terigreatorhombated demihepteract (Tugrohesa)
22									= Hexistericantic 7-cube Teriprismatotruncated demihepteract (Tupthesa)
23									= Hexisteriruncic 7-cube Teriprismatorhombated demihepteract (Tuprohesa)
24									= Hexipenticantic 7-cube Tericellitruncated demihepteract (Tucothesa)
25									= Hexipentiruncic 7-cube Tericellirhombated demihepteract (Tucrohesa)
26									= Hexipentisteric 7-cube Tericelliprismated demihepteract (Tucophesa)
27									= Pentisteriruncicantic 7-cube Great cellated demihepteract (Gochesa)
28									= Hexisteriruncicantic 7-cube Terigreatoprismated demihepteract (Tugphesa)
29									= Hexipentiruncicantic 7-cube Tericelligreatorhombated demihepteract (Tucagrohesa)
30									= Hexipentistericantic 7-cube Tericelliprismatotruncated demihepteract (Tucpathesa)
31									= Hexipentisteriruncic 7-cube Tericelliprismatorhombated demihepteract (Tucprohesa)
32									= 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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, 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]
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
Klitzing, Richard. "7D uniform polytopes (polyexa) with acronyms".

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	$A n$	$B n$	$I 2 (p)$ / $D n$	$E 6$ / $E 7$ / $E 8$ / $F 4$ / $G 2$	$H n$
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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations

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