D_{7} polytope
7demicube 
7orthoplex 
In 7dimensional geometry, there are 95 uniform polytopes with D_{7} symmetry; 32 are unique, and 63 are shared with the B_{7} symmetry. There are two regular forms, the 7orthoplex, and 7demicube with 14 and 64 vertices respectively.
They can be visualized as symmetric orthographic projections in Coxeter planes of the D_{6} Coxeter group, and other subgroups.
Contents
Graphs
Symmetric orthographic projections of these 32 polytopes can be made in the D_{7}, D_{6}, D_{5}, D_{4}, D_{3}, A_{5}, A_{3}, Coxeter planes. A_{k} has [k+1] symmetry, D_{k} has [2(k1)] symmetry. B_{7} 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 


B_{7} [14/2] 
D_{7} [12] 
D_{6} [10] 
D_{5} [8] 
D_{4} [6] 
D_{3} [4] 
A_{5} [6] 
A_{3} [4] 

1 
= 7demicube Demihepteract (Hesa) 

2 
= Cantic 7cube Truncated demihepteract (Thesa) 

3 
= Runcic 7cube Small rhombated demihepteract (Sirhesa) 

4 
= Steric 7cube Small prismated demihepteract (Sphosa) 

5 
= Pentic 7cube Small cellated demihepteract (Sochesa) 

6 
= Hexic 7cube Small terated demihepteract (Suthesa) 

7 
= Runcicantic 7cube Great rhombated demihepteract (Girhesa) 

8 
= Stericantic 7cube Prismatotruncated demihepteract (Pothesa) 

9 
= Steriruncic 7cube Prismatorhomated demihepteract (Prohesa) 

10 
= Penticantic 7cube Cellitruncated demihepteract (Cothesa) 

11 
= Pentiruncic 7cube Cellirhombated demihepteract (Crohesa) 

12 
= Pentisteric 7cube Celliprismated demihepteract (Caphesa) 

13 
= Hexicantic 7cube Teritruncated demihepteract (Tuthesa) 

14 
= Hexiruncic 7cube Terirhombated demihepteract (Turhesa) 

15 
= Hexisteric 7cube Teriprismated demihepteract (Tuphesa) 

16 
= Hexipentic 7cube Tericellated demihepteract (Tuchesa) 

17 
= Steriruncicantic 7cube Great prismated demihepteract (Gephosa) 

18 
= Pentiruncicantic 7cube Celligreatorhombated demihepteract (Cagrohesa) 

19 
= Pentistericantic 7cube Celliprismatotruncated demihepteract (Capthesa) 

20 
= Pentisteriruncic 7cube Celliprismatorhombated demihepteract (Coprahesa) 

21 
= Hexiruncicantic 7cube Terigreatorhombated demihepteract (Tugrohesa) 

22 
= Hexistericantic 7cube Teriprismatotruncated demihepteract (Tupthesa) 

23 
= Hexisteriruncic 7cube Teriprismatorhombated demihepteract (Tuprohesa) 

24 
= Hexipenticantic 7cube Tericellitruncated demihepteract (Tucothesa) 

25 
= Hexipentiruncic 7cube Tericellirhombated demihepteract (Tucrohesa) 

26 
= Hexipentisteric 7cube Tericelliprismated demihepteract (Tucophesa) 

27 
= Pentisteriruncicantic 7cube Great cellated demihepteract (Gochesa) 

28 
= Hexisteriruncicantic 7cube Terigreatoprimated demihepteract (Tugphesa) 

29 
= Hexipentiruncicantic 7cube Tericelligreatorhombated demihepteract (Tucagrohesa) 

30 
= Hexipentistericantic 7cube Tericelliprismatotruncated demihepteract (Tucpathesa) 

31 
= Hexipentisteriruncic 7cube Tericellprismatorhombated demihepteract (Tucprohesa) 

32 
= Hexipentisteriruncicantic 7cube 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, WileyInterscience Publication, 1995, ISBN 9780471010036 ^{[1]}
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10]
 (Paper 23) H.S.M. Coxeter, Regular and SemiRegular Polytopes II, [Math. Zeit. 188 (1985) 559591]
 (Paper 24) H.S.M. Coxeter, Regular and SemiRegular Polytopes III, [Math. Zeit. 200 (1988) 345]
 N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
 Klitzing, Richard. "7D uniform polytopes (polyexa)".
Notes
 ^ Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter