A_{7} polytope
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7simplex 
In 7dimensional geometry, there are 71 uniform polytopes with A_{7} 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 A_{7} Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projections of these 71 polytopes can be made in the A_{7}, A_{6}, A_{5}, A_{4}, A_{3}, A_{2} Coxeter planes. A_{k} 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 
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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 
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22  t_{0,2,4}{3,3,3,3,3,3} Stericantellated 7simplex 
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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 
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50  t_{0,1,3,6}{3,3,3,3,3,3} Hexiruncitruncated 7simplex 
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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 
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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) 380–407, 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)". https://bendwavy.org/klitzing/dimensions/polypeta.htm.
Notes
Original source: https://en.wikipedia.org/wiki/A7 polytope.
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