In 8dimensional geometry, there are 135 uniform polytopes with A8 symmetry. There is one selfdual regular form, the 8simplex with 9 vertices.
Each can be visualized as symmetric orthographic projections in Coxeter planes of the A8 Coxeter group, and other subgroups.
Graphs
Symmetric orthographic projections of these 135 polytopes can be made in the A8, A7, A6, A5, A4, A3, A2 Coxeter planes. Ak has [k+1] symmetry.
These 135 polytopes are each shown in these 7 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_{8} [9] 
A_{7} [8] 
A_{6} [7] 
A_{5} [6] 
A_{4} [5] 
A_{3} [4] 
A_{2} [3] 

1  t_{0}{3,3,3,3,3,3,3} 8simplex 

2  t_{1}{3,3,3,3,3,3,3} Rectified 8simplex 

3  t_{2}{3,3,3,3,3,3,3} Birectified 8simplex 

4  t_{3}{3,3,3,3,3,3,3} Trirectified 8simplex 

5  t_{0,1}{3,3,3,3,3,3,3} Truncated 8simplex 

6  t_{0,2}{3,3,3,3,3,3,3} Cantellated 8simplex 

7  t_{1,2}{3,3,3,3,3,3,3} Bitruncated 8simplex 

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

9  t_{1,3}{3,3,3,3,3,3,3} Bicantellated 8simplex 

10  t_{2,3}{3,3,3,3,3,3,3} Tritruncated 8simplex 

11  t_{0,4}{3,3,3,3,3,3,3} Stericated 8simplex 

12  t_{1,4}{3,3,3,3,3,3,3} Biruncinated 8simplex 

13  t_{2,4}{3,3,3,3,3,3,3} Tricantellated 8simplex 

14  t_{3,4}{3,3,3,3,3,3,3} Quadritruncated 8simplex 

15  t_{0,5}{3,3,3,3,3,3,3} Pentellated 8simplex 

16  t_{1,5}{3,3,3,3,3,3,3} Bistericated 8simplex 

17  t_{2,5}{3,3,3,3,3,3,3} Triruncinated 8simplex 

18  t_{0,6}{3,3,3,3,3,3,3} Hexicated 8simplex 

19  t_{1,6}{3,3,3,3,3,3,3} Bipentellated 8simplex 

20  t_{0,7}{3,3,3,3,3,3,3} Heptellated 8simplex 

21  t_{0,1,2}{3,3,3,3,3,3,3} Cantitruncated 8simplex 

22  t_{0,1,3}{3,3,3,3,3,3,3} Runcitruncated 8simplex 

23  t_{0,2,3}{3,3,3,3,3,3,3} Runcicantellated 8simplex 

24  t_{1,2,3}{3,3,3,3,3,3,3} Bicantitruncated 8simplex 

25  t_{0,1,4}{3,3,3,3,3,3,3} Steritruncated 8simplex 

26  t_{0,2,4}{3,3,3,3,3,3,3} Stericantellated 8simplex 

27  t_{1,2,4}{3,3,3,3,3,3,3} Biruncitruncated 8simplex 

28  t_{0,3,4}{3,3,3,3,3,3,3} Steriruncinated 8simplex 

29  t_{1,3,4}{3,3,3,3,3,3,3} Biruncicantellated 8simplex 

30  t_{2,3,4}{3,3,3,3,3,3,3} Tricantitruncated 8simplex 

31  t_{0,1,5}{3,3,3,3,3,3,3} Pentitruncated 8simplex 

32  t_{0,2,5}{3,3,3,3,3,3,3} Penticantellated 8simplex 

33  t_{1,2,5}{3,3,3,3,3,3,3} Bisteritruncated 8simplex 

34  t_{0,3,5}{3,3,3,3,3,3,3} Pentiruncinated 8simplex 

35  t_{1,3,5}{3,3,3,3,3,3,3} Bistericantellated 8simplex 

36  t_{2,3,5}{3,3,3,3,3,3,3} Triruncitruncated 8simplex 

37  t_{0,4,5}{3,3,3,3,3,3,3} Pentistericated 8simplex 

38  t_{1,4,5}{3,3,3,3,3,3,3} Bisteriruncinated 8simplex 

39  t_{0,1,6}{3,3,3,3,3,3,3} Hexitruncated 8simplex 

40  t_{0,2,6}{3,3,3,3,3,3,3} Hexicantellated 8simplex 

41  t_{1,2,6}{3,3,3,3,3,3,3} Bipentitruncated 8simplex 

42  t_{0,3,6}{3,3,3,3,3,3,3} Hexiruncinated 8simplex 

43  t_{1,3,6}{3,3,3,3,3,3,3} Bipenticantellated 8simplex 

44  t_{0,4,6}{3,3,3,3,3,3,3} Hexistericated 8simplex 

45  t_{0,5,6}{3,3,3,3,3,3,3} Hexipentellated 8simplex 

46  t_{0,1,7}{3,3,3,3,3,3,3} Heptitruncated 8simplex 

47  t_{0,2,7}{3,3,3,3,3,3,3} Hepticantellated 8simplex 

48  t_{0,3,7}{3,3,3,3,3,3,3} Heptiruncinated 8simplex 

49  t_{0,1,2,3}{3,3,3,3,3,3,3} Runcicantitruncated 8simplex 

50  t_{0,1,2,4}{3,3,3,3,3,3,3} Stericantitruncated 8simplex 

51  t_{0,1,3,4}{3,3,3,3,3,3,3} Steriruncitruncated 8simplex 

52  t_{0,2,3,4}{3,3,3,3,3,3,3} Steriruncicantellated 8simplex 

53  t_{1,2,3,4}{3,3,3,3,3,3,3} Biruncicantitruncated 8simplex 

54  t_{0,1,2,5}{3,3,3,3,3,3,3} Penticantitruncated 8simplex 

55  t_{0,1,3,5}{3,3,3,3,3,3,3} Pentiruncitruncated 8simplex 

56  t_{0,2,3,5}{3,3,3,3,3,3,3} Pentiruncicantellated 8simplex 

57  t_{1,2,3,5}{3,3,3,3,3,3,3} Bistericantitruncated 8simplex 

58  t_{0,1,4,5}{3,3,3,3,3,3,3} Pentisteritruncated 8simplex 

59  t_{0,2,4,5}{3,3,3,3,3,3,3} Pentistericantellated 8simplex 

60  t_{1,2,4,5}{3,3,3,3,3,3,3} Bisteriruncitruncated 8simplex 

61  t_{0,3,4,5}{3,3,3,3,3,3,3} Pentisteriruncinated 8simplex 

62  t_{1,3,4,5}{3,3,3,3,3,3,3} Bisteriruncicantellated 8simplex 

63  t_{2,3,4,5}{3,3,3,3,3,3,3} Triruncicantitruncated 8simplex 

64  t_{0,1,2,6}{3,3,3,3,3,3,3} Hexicantitruncated 8simplex 

65  t_{0,1,3,6}{3,3,3,3,3,3,3} Hexiruncitruncated 8simplex 

66  t_{0,2,3,6}{3,3,3,3,3,3,3} Hexiruncicantellated 8simplex 

67  t_{1,2,3,6}{3,3,3,3,3,3,3} Bipenticantitruncated 8simplex 

68  t_{0,1,4,6}{3,3,3,3,3,3,3} Hexisteritruncated 8simplex 

69  t_{0,2,4,6}{3,3,3,3,3,3,3} Hexistericantellated 8simplex 

70  t_{1,2,4,6}{3,3,3,3,3,3,3} Bipentiruncitruncated 8simplex 

71  t_{0,3,4,6}{3,3,3,3,3,3,3} Hexisteriruncinated 8simplex 

72  t_{1,3,4,6}{3,3,3,3,3,3,3} Bipentiruncicantellated 8simplex 

73  t_{0,1,5,6}{3,3,3,3,3,3,3} Hexipentitruncated 8simplex 

74  t_{0,2,5,6}{3,3,3,3,3,3,3} Hexipenticantellated 8simplex 

75  t_{1,2,5,6}{3,3,3,3,3,3,3} Bipentisteritruncated 8simplex 

76  t_{0,3,5,6}{3,3,3,3,3,3,3} Hexipentiruncinated 8simplex 

77  t_{0,4,5,6}{3,3,3,3,3,3,3} Hexipentistericated 8simplex 

78  t_{0,1,2,7}{3,3,3,3,3,3,3} Hepticantitruncated 8simplex 

79  t_{0,1,3,7}{3,3,3,3,3,3,3} Heptiruncitruncated 8simplex 

80  t_{0,2,3,7}{3,3,3,3,3,3,3} Heptiruncicantellated 8simplex 

81  t_{0,1,4,7}{3,3,3,3,3,3,3} Heptisteritruncated 8simplex 

82  t_{0,2,4,7}{3,3,3,3,3,3,3} Heptistericantellated 8simplex 

83  t_{0,3,4,7}{3,3,3,3,3,3,3} Heptisteriruncinated 8simplex 

84  t_{0,1,5,7}{3,3,3,3,3,3,3} Heptipentitruncated 8simplex 

85  t_{0,2,5,7}{3,3,3,3,3,3,3} Heptipenticantellated 8simplex 

86  t_{0,1,6,7}{3,3,3,3,3,3,3} Heptihexitruncated 8simplex 

87  t_{0,1,2,3,4}{3,3,3,3,3,3,3} Steriruncicantitruncated 8simplex 

88  t_{0,1,2,3,5}{3,3,3,3,3,3,3} Pentiruncicantitruncated 8simplex 

89  t_{0,1,2,4,5}{3,3,3,3,3,3,3} Pentistericantitruncated 8simplex 

90  t_{0,1,3,4,5}{3,3,3,3,3,3,3} Pentisteriruncitruncated 8simplex 

91  t_{0,2,3,4,5}{3,3,3,3,3,3,3} Pentisteriruncicantellated 8simplex 

92  t_{1,2,3,4,5}{3,3,3,3,3,3,3} Bisteriruncicantitruncated 8simplex 

93  t_{0,1,2,3,6}{3,3,3,3,3,3,3} Hexiruncicantitruncated 8simplex 

94  t_{0,1,2,4,6}{3,3,3,3,3,3,3} Hexistericantitruncated 8simplex 

95  t_{0,1,3,4,6}{3,3,3,3,3,3,3} Hexisteriruncitruncated 8simplex 

96  t_{0,2,3,4,6}{3,3,3,3,3,3,3} Hexisteriruncicantellated 8simplex 

97  t_{1,2,3,4,6}{3,3,3,3,3,3,3} Bipentiruncicantitruncated 8simplex 

98  t_{0,1,2,5,6}{3,3,3,3,3,3,3} Hexipenticantitruncated 8simplex 

99  t_{0,1,3,5,6}{3,3,3,3,3,3,3} Hexipentiruncitruncated 8simplex 

100  t_{0,2,3,5,6}{3,3,3,3,3,3,3} Hexipentiruncicantellated 8simplex 

101  t_{1,2,3,5,6}{3,3,3,3,3,3,3} Bipentistericantitruncated 8simplex 

102  t_{0,1,4,5,6}{3,3,3,3,3,3,3} Hexipentisteritruncated 8simplex 

103  t_{0,2,4,5,6}{3,3,3,3,3,3,3} Hexipentistericantellated 8simplex 

104  t_{0,3,4,5,6}{3,3,3,3,3,3,3} Hexipentisteriruncinated 8simplex 

105  t_{0,1,2,3,7}{3,3,3,3,3,3,3} Heptiruncicantitruncated 8simplex 

106  t_{0,1,2,4,7}{3,3,3,3,3,3,3} Heptistericantitruncated 8simplex 

107  t_{0,1,3,4,7}{3,3,3,3,3,3,3} Heptisteriruncitruncated 8simplex 

108  t_{0,2,3,4,7}{3,3,3,3,3,3,3} Heptisteriruncicantellated 8simplex 

109  t_{0,1,2,5,7}{3,3,3,3,3,3,3} Heptipenticantitruncated 8simplex 

110  t_{0,1,3,5,7}{3,3,3,3,3,3,3} Heptipentiruncitruncated 8simplex 

111  t_{0,2,3,5,7}{3,3,3,3,3,3,3} Heptipentiruncicantellated 8simplex 

112  t_{0,1,4,5,7}{3,3,3,3,3,3,3} Heptipentisteritruncated 8simplex 

113  t_{0,1,2,6,7}{3,3,3,3,3,3,3} Heptihexicantitruncated 8simplex 

114  t_{0,1,3,6,7}{3,3,3,3,3,3,3} Heptihexiruncitruncated 8simplex 

115  t_{0,1,2,3,4,5}{3,3,3,3,3,3,3} Pentisteriruncicantitruncated 8simplex 

116  t_{0,1,2,3,4,6}{3,3,3,3,3,3,3} Hexisteriruncicantitruncated 8simplex 

117  t_{0,1,2,3,5,6}{3,3,3,3,3,3,3} Hexipentiruncicantitruncated 8simplex 

118  t_{0,1,2,4,5,6}{3,3,3,3,3,3,3} Hexipentistericantitruncated 8simplex 

119  t_{0,1,3,4,5,6}{3,3,3,3,3,3,3} Hexipentisteriruncitruncated 8simplex 

120  t_{0,2,3,4,5,6}{3,3,3,3,3,3,3} Hexipentisteriruncicantellated 8simplex 

121  t_{1,2,3,4,5,6}{3,3,3,3,3,3,3} Bipentisteriruncicantitruncated 8simplex 

122  t_{0,1,2,3,4,7}{3,3,3,3,3,3,3} Heptisteriruncicantitruncated 8simplex 

123  t_{0,1,2,3,5,7}{3,3,3,3,3,3,3} Heptipentiruncicantitruncated 8simplex 

124  t_{0,1,2,4,5,7}{3,3,3,3,3,3,3} Heptipentistericantitruncated 8simplex 

125  t_{0,1,3,4,5,7}{3,3,3,3,3,3,3} Heptipentisteriruncitruncated 8simplex 

126  t_{0,2,3,4,5,7}{3,3,3,3,3,3,3} Heptipentisteriruncicantellated 8simplex 

127  t_{0,1,2,3,6,7}{3,3,3,3,3,3,3} Heptihexiruncicantitruncated 8simplex 

128  t_{0,1,2,4,6,7}{3,3,3,3,3,3,3} Heptihexistericantitruncated 8simplex 

129  t_{0,1,3,4,6,7}{3,3,3,3,3,3,3} Heptihexisteriruncitruncated 8simplex 

130  t_{0,1,2,5,6,7}{3,3,3,3,3,3,3} Heptihexipenticantitruncated 8simplex 

131  t_{0,1,2,3,4,5,6}{3,3,3,3,3,3,3} Hexipentisteriruncicantitruncated 8simplex 

132  t_{0,1,2,3,4,5,7}{3,3,3,3,3,3,3} Heptipentisteriruncicantitruncated 8simplex 

133  t_{0,1,2,3,4,6,7}{3,3,3,3,3,3,3} Heptihexisteriruncicantitruncated 8simplex 

134  t_{0,1,2,3,5,6,7}{3,3,3,3,3,3,3} Heptihexipentiruncicantitruncated 8simplex 

135  t_{0,1,2,3,4,5,6,7}{3,3,3,3,3,3,3} Omnitruncated 8simplex 
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
External links
Klitzing, Richard. "8D uniform polytopes (polyzetta)".
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 
Hellenica World  Scientific Library
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