D7 polytope

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]
1		80px	80px	80px	80px	80px	80px	80px	= 7-demicube Demihepteract (Hesa)
2		80px	80px	80px	80px	80px	80px	80px	= Cantic 7-cube Truncated demihepteract (Thesa)
3		80px	80px	80px	80px	80px	80px		= Runcic 7-cube Small rhombated demihepteract (Sirhesa)
4		80px	80px	80px	80px	80px	80px		= Steric 7-cube Small prismated demihepteract (Sphosa)
5		80px	80px	80px	80px	80px	80px		= Pentic 7-cube Small cellated demihepteract (Sochesa)
6		80px	80px	80px	80px	80px	80px		= Hexic 7-cube Small terated demihepteract (Suthesa)
7		80px	80px	80px	80px	80px	80px		= Runcicantic 7-cube Great rhombated demihepteract (Girhesa)
8		80px	80px	80px	80px	80px	80px		= Stericantic 7-cube Prismatotruncated demihepteract (Pothesa)
9		80px	80px	80px	80px	80px	80px		= Steriruncic 7-cube Prismatorhomated demihepteract (Prohesa)
10		80px	80px	80px	80px	80px	80px		= Penticantic 7-cube Cellitruncated demihepteract (Cothesa)
11		80px	80px	80px	80px	80px	80px		= Pentiruncic 7-cube Cellirhombated demihepteract (Crohesa)
12		80px	80px	80px	80px	80px	80px		= Pentisteric 7-cube Celliprismated demihepteract (Caphesa)
13		80px	80px	80px	80px	80px	80px		= Hexicantic 7-cube Teritruncated demihepteract (Tuthesa)
14		80px	80px	80px	80px	80px	80px		= Hexiruncic 7-cube Terirhombated demihepteract (Turhesa)
15		80px	80px	80px	80px	80px	80px		= Hexisteric 7-cube Teriprismated demihepteract (Tuphesa)
16		80px	80px	80px	80px	80px	80px		= Hexipentic 7-cube Tericellated demihepteract (Tuchesa)
17		80px	80px	80px	80px	80px	80px		= Steriruncicantic 7-cube Great prismated demihepteract (Gephosa)
18		80px	80px	80px	80px	80px	80px		= Pentiruncicantic 7-cube Celligreatorhombated demihepteract (Cagrohesa)
19		80px	80px	80px	80px	80px	80px		= Pentistericantic 7-cube Celliprismatotruncated demihepteract (Capthesa)
20		80px	80px	80px	80px	80px	80px		= Pentisteriruncic 7-cube Celliprismatorhombated demihepteract (Coprahesa)
21		80px	80px	80px	80px	80px	80px		= Hexiruncicantic 7-cube Terigreatorhombated demihepteract (Tugrohesa)
22		80px	80px	80px	80px	80px	80px		= Hexistericantic 7-cube Teriprismatotruncated demihepteract (Tupthesa)
23		80px	80px	80px	80px	80px	80px		= Hexisteriruncic 7-cube Teriprismatorhombated demihepteract (Tuprohesa)
24		80px	80px	80px	80px	80px	80px		= Hexipenticantic 7-cube Tericellitruncated demihepteract (Tucothesa)
25		80px	80px	80px	80px	80px	80px		= Hexipentiruncic 7-cube Tericellirhombated demihepteract (Tucrohesa)
26		80px	80px	80px	80px	80px	80px		= Hexipentisteric 7-cube Tericelliprismated demihepteract (Tucophesa)
27		80px	80px	80px	80px	80px	80px		= Pentisteriruncicantic 7-cube Great cellated demihepteract (Gochesa)
28		80px	80px	80px	80px	80px	80px		= Hexisteriruncicantic 7-cube Terigreatoprimated demihepteract (Tugphesa)
29		80px	80px	80px	80px	80px	80px		= Hexipentiruncicantic 7-cube Tericelligreatorhombated demihepteract (Tucagrohesa)
30		80px	80px	80px	80px	80px	80px		= Hexipentistericantic 7-cube Tericelliprismatotruncated demihepteract (Tucpathesa)
31		80px	80px	80px	80px	80px	80px		= Hexipentisteriruncic 7-cube Tericellprismatorhombated demihepteract (Tucprohesa)
32		80px	80px	80px	80px	80px	80px		= 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, [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)". https://bendwavy.org/klitzing/dimensions/polypeta.htm.