In 7dimensional geometry, there are 71 uniform polytopes with A7 symmetry. There is one selfdual regular form, the 7simplex with 8 vertices.
Each can be visualized as symmetric orthographic projections in Coxeter planes of the A7 Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projections of these 71 polytopes can be made in the A7, A6, A5, A4, A3, A2 Coxeter planes. Ak has [k+1] symmetry. For even k and symmetrically ringeddiagrams, symmetry doubles to [2(k+1)].
These 71 polytopes are each shown in these 6 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.
#  CoxeterDynkin diagram Schläfli symbol Johnson name 
A_{k} orthogonal projection graphs  

A_{7} [8] 
A_{6} [7] 
A_{5} [6] 
A_{4} [5] 
A_{3} [4] 
A_{2} [3] 

1  t_{0}{3,3,3,3,3,3} 7simplex 

2  t_{1}{3,3,3,3,3,3} Rectified 7simplex 

3  t_{2}{3,3,3,3,3,3} Birectified 7simplex 

4  t_{3}{3,3,3,3,3,3} Trirectified 7simplex 

5  t_{0,1}{3,3,3,3,3,3} Truncated 7simplex 

6  t_{0,2}{3,3,3,3,3,3} Cantellated 7simplex 

7  t_{1,2}{3,3,3,3,3,3} Bitruncated 7simplex 

8  t_{0,3}{3,3,3,3,3,3} Runcinated 7simplex 

9  t_{1,3}{3,3,3,3,3,3} Bicantellated 7simplex 

10  t_{2,3}{3,3,3,3,3,3} Tritruncated 7simplex 

11  t_{0,4}{3,3,3,3,3,3} Stericated 7simplex 

12  t_{1,4}{3,3,3,3,3,3} Biruncinated 7simplex 

13  t_{2,4}{3,3,3,3,3,3} Tricantellated 7simplex 

14  t_{0,5}{3,3,3,3,3,3} Pentellated 7simplex 

15  t_{1,5}{3,3,3,3,3,3} Bistericated 7simplex 

16  t_{0,6}{3,3,3,3,3,3} Hexicated 7simplex 

17  t_{0,1,2}{3,3,3,3,3,3} Cantitruncated 7simplex 

18  t_{0,1,3}{3,3,3,3,3,3} Runcitruncated 7simplex 

19  t_{0,2,3}{3,3,3,3,3,3} Runcicantellated 7simplex 

20  t_{1,2,3}{3,3,3,3,3,3} Bicantitruncated 7simplex 

21  t_{0,1,4}{3,3,3,3,3,3} Steritruncated 7simplex 

22  t_{0,2,4}{3,3,3,3,3,3} Stericantellated 7simplex 

23  t_{1,2,4}{3,3,3,3,3,3} Biruncitruncated 7simplex 

24  t_{0,3,4}{3,3,3,3,3,3} Steriruncinated 7simplex 

25  t_{1,3,4}{3,3,3,3,3,3} Biruncicantellated 7simplex 

26  t_{2,3,4}{3,3,3,3,3,3} Tricantitruncated 7simplex 

27  t_{0,1,5}{3,3,3,3,3,3} Pentitruncated 7simplex 

28  t_{0,2,5}{3,3,3,3,3,3} Penticantellated 7simplex 

29  t_{1,2,5}{3,3,3,3,3,3} Bisteritruncated 7simplex 

30  t_{0,3,5}{3,3,3,3,3,3} Pentiruncinated 7simplex 

31  t_{1,3,5}{3,3,3,3,3,3} Bistericantellated 7simplex 

32  t_{0,4,5}{3,3,3,3,3,3} Pentistericated 7simplex 

33  t_{0,1,6}{3,3,3,3,3,3} Hexitruncated 7simplex 

34  t_{0,2,6}{3,3,3,3,3,3} Hexicantellated 7simplex 

35  t_{0,3,6}{3,3,3,3,3,3} Hexiruncinated 7simplex 

36  t_{0,1,2,3}{3,3,3,3,3,3} Runcicantitruncated 7simplex 

37  t_{0,1,2,4}{3,3,3,3,3,3} Stericantitruncated 7simplex 

38  t_{0,1,3,4}{3,3,3,3,3,3} Steriruncitruncated 7simplex 

39  t_{0,2,3,4}{3,3,3,3,3,3} Steriruncicantellated 7simplex 

40  t_{1,2,3,4}{3,3,3,3,3,3} Biruncicantitruncated 7simplex 

41  t_{0,1,2,5}{3,3,3,3,3,3} Penticantitruncated 7simplex 

42  t_{0,1,3,5}{3,3,3,3,3,3} Pentiruncitruncated 7simplex 

43  t_{0,2,3,5}{3,3,3,3,3,3} Pentiruncicantellated 7simplex 

44  t_{1,2,3,5}{3,3,3,3,3,3} Bistericantitruncated 7simplex 

45  t_{0,1,4,5}{3,3,3,3,3,3} Pentisteritruncated 7simplex 

46  t_{0,2,4,5}{3,3,3,3,3,3} Pentistericantellated 7simplex 

47  t_{1,2,4,5}{3,3,3,3,3,3} Bisteriruncitruncated 7simplex 

48  t_{0,3,4,5}{3,3,3,3,3,3} Pentisteriruncinated 7simplex 

49  t_{0,1,2,6}{3,3,3,3,3,3} Hexicantitruncated 7simplex 

50  t_{0,1,3,6}{3,3,3,3,3,3} Hexiruncitruncated 7simplex 

51  t_{0,2,3,6}{3,3,3,3,3,3} Hexiruncicantellated 7simplex 

52  t_{0,1,4,6}{3,3,3,3,3,3} Hexisteritruncated 7simplex 

53  t_{0,2,4,6}{3,3,3,3,3,3} Hexistericantellated 7simplex 

54  t_{0,1,5,6}{3,3,3,3,3,3} Hexipentitruncated 7simplex 

55  t_{0,1,2,3,4}{3,3,3,3,3,3} Steriruncicantitruncated 7simplex 

56  t_{0,1,2,3,5}{3,3,3,3,3,3} Pentiruncicantitruncated 7simplex 

57  t_{0,1,2,4,5}{3,3,3,3,3,3} Pentistericantitruncated 7simplex 

58  t_{0,1,3,4,5}{3,3,3,3,3,3} Pentisteriruncitruncated 7simplex 

59  t_{0,2,3,4,5}{3,3,3,3,3,3} Pentisteriruncicantellated 7simplex 

60  t_{1,2,3,4,5}{3,3,3,3,3,3} Bisteriruncicantitruncated 7simplex 

61  t_{0,1,2,3,6}{3,3,3,3,3,3} Hexiruncicantitruncated 7simplex 

62  t_{0,1,2,4,6}{3,3,3,3,3,3} Hexistericantitruncated 7simplex 

63  t_{0,1,3,4,6}{3,3,3,3,3,3} Hexisteriruncitruncated 7simplex 

64  t_{0,2,3,4,6}{3,3,3,3,3,3} Hexisteriruncicantellated 7simplex 

65  t_{0,1,2,5,6}{3,3,3,3,3,3} Hexipenticantitruncated 7simplex 

66  t_{0,1,3,5,6}{3,3,3,3,3,3} Hexipentiruncitruncated 7simplex 

67  t_{0,1,2,3,4,5}{3,3,3,3,3,3} Pentisteriruncicantitruncated 7simplex 

68  t_{0,1,2,3,4,6}{3,3,3,3,3,3} Hexisteriruncicantitruncated 7simplex 

69  t_{0,1,2,3,5,6}{3,3,3,3,3,3} Hexipentiruncicantitruncated 7simplex 

70  t_{0,1,2,4,5,6}{3,3,3,3,3,3} Hexipentistericantitruncated 7simplex 

71  t_{0,1,2,3,4,5,6}{3,3,3,3,3,3} Omnitruncated 7simplex 
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
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  pgon  Hexagon  Pentagon  
Uniform polyhedron  Tetrahedron  Octahedron • Cube  Demicube  Dodecahedron • Icosahedron  
Uniform 4polytope  5cell  16cell • Tesseract  Demitesseract  24cell  120cell • 600cell  
Uniform 5polytope  5simplex  5orthoplex • 5cube  5demicube  
Uniform 6polytope  6simplex  6orthoplex • 6cube  6demicube  1_{22} • 2_{21}  
Uniform 7polytope  7simplex  7orthoplex • 7cube  7demicube  1_{32} • 2_{31} • 3_{21}  
Uniform 8polytope  8simplex  8orthoplex • 8cube  8demicube  1_{42} • 2_{41} • 4_{21}  
Uniform 9polytope  9simplex  9orthoplex • 9cube  9demicube  
Uniform 10polytope  10simplex  10orthoplex • 10cube  10demicube  
Uniform npolytope  nsimplex  northoplex • ncube  ndemicube  1_{k2} • 2_{k1} • k_{21}  npentagonal polytope  
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds 
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