B7 polytope

Orthographic projections in the B7 Coxeter plane

7-cube
7-orthoplex
7-demicube

In 7-dimensional geometry, there are 128 uniform polytopes with B7 symmetry. There are two regular forms, the 7-orthoplex, and 8-cube with 14 and 128 vertices respectively. The 7-demicube is added with half of the symmetry.

They can be visualized as symmetric orthographic projections in Coxeter planes of the B7 Coxeter group, and other subgroups.

Graphs

Symmetric orthographic projections of these 128 polytopes can be made in the B7, B6, B5, B4, B3, B2, A5, A3, Coxeter planes. Ak has [k+1] symmetry, and Bk has [2k] symmetry.

These 128 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-Dynkin diagram
Schläfli symbol
Johnson and Bowers Name
B7
[14]		 B6
[12]		 B5
[10]		 B4
[8]		 B3
[6]		 B2
[4]		 A5
[6]		 A3
[4]
1
{3,3,3,3,3,4}
7-orthoplex
Hecatonicosoctaexon (zee)
2
t1{3,3,3,3,3,4}
Rectified 7-orthoplex
Rectified hecatonicosoctaexon (rez)
3
t2{3,3,3,3,3,4}
Birectified 7-orthoplex
Birectified hecatonicosoctaexon (barz)
4
t3{4,3,3,3,3,3}
Trirectified 7-cube
Hepteractihecatonicosoctaexon (sez)
5
t2{4,3,3,3,3,3}
Birectified 7-cube
Birectified hepteract (bersa)
6
t1{4,3,3,3,3,3}
Rectified 7-cube
Rectified hepteract (rasa)
7
{4,3,3,3,3,3}
7-cube
Hepteract (hept)
128
h{4,3,3,3,3,3}
7-demicube
Hemihepteract
8
t0,1{3,3,3,3,3,4}
Truncated 7-orthoplex
Truncated hecatonicosoctaexon (Taz)
9
t0,2{3,3,3,3,3,4}
Cantellated 7-orthoplex
Small rhombated hecatonicosoctaexon (Sarz)
10
t1,2{3,3,3,3,3,4}
Bitruncated 7-orthoplex
Bitruncated hecatonicosoctaexon (Botaz)
11
t0,3{3,3,3,3,3,4}
Runcinated 7-orthoplex
Small prismated hecatonicosoctaexon (Spaz)
12
t1,3{3,3,3,3,3,4}
Bicantellated 7-orthoplex
Small birhombated hecatonicosoctaexon (Sebraz)
13
t2,3{3,3,3,3,3,4}
Tritruncated 7-orthoplex
Tritruncated hecatonicosoctaexon (Totaz)
14
t0,4{3,3,3,3,3,4}
Stericated 7-orthoplex
Small cellated hecatonicosoctaexon (Scaz)
15
t1,4{3,3,3,3,3,4}
Biruncinated 7-orthoplex
Small biprismated hecatonicosoctaexon (Sibpaz)
16
t2,4{4,3,3,3,3,3}
Tricantellated 7-cube
Small trirhombi-hepteractihecatonicosoctaexon (Strasaz)
17
t2,3{4,3,3,3,3,3}
Tritruncated 7-cube
Tritruncated hepteract (Tatsa)
18
t0,5{3,3,3,3,3,4}
Pentellated 7-orthoplex
Small terated hecatonicosoctaexon (Staz)
19
t1,5{4,3,3,3,3,3}
Bistericated 7-cube
Small bicelli-hepteractihecatonicosoctaexon (Sabcosaz)
20
t1,4{4,3,3,3,3,3}
Biruncinated 7-cube
Small biprismated hepteract (Sibposa)
21
t1,3{4,3,3,3,3,3}
Bicantellated 7-cube
Small birhombated hepteract (Sibrosa)
22
t1,2{4,3,3,3,3,3}
Bitruncated 7-cube
Bitruncated hepteract (Betsa)
23
t0,6{4,3,3,3,3,3}
Hexicated 7-cube
Small peti-hepteractihecatonicosoctaexon (Suposaz)
24
t0,5{4,3,3,3,3,3}
Pentellated 7-cube
Small terated hepteract (Stesa)
25
t0,4{4,3,3,3,3,3}
Stericated 7-cube
Small cellated hepteract (Scosa)
26
t0,3{4,3,3,3,3,3}
Runcinated 7-cube
Small prismated hepteract (Spesa)
27
t0,2{4,3,3,3,3,3}
Cantellated 7-cube
Small rhombated hepteract (Sersa)
28
t0,1{4,3,3,3,3,3}
Truncated 7-cube
Truncated hepteract (Tasa)
29
t0,1,2{3,3,3,3,3,4}
Cantitruncated 7-orthoplex
Great rhombated hecatonicosoctaexon (Garz)
30
t0,1,3{3,3,3,3,3,4}
Runcitruncated 7-orthoplex
Prismatotruncated hecatonicosoctaexon (Potaz)
31
t0,2,3{3,3,3,3,3,4}
Runcicantellated 7-orthoplex
Prismatorhombated hecatonicosoctaexon (Parz)
32
t1,2,3{3,3,3,3,3,4}
Bicantitruncated 7-orthoplex
Great birhombated hecatonicosoctaexon (Gebraz)
33
t0,1,4{3,3,3,3,3,4}
Steritruncated 7-orthoplex
Cellitruncated hecatonicosoctaexon (Catz)
34
t0,2,4{3,3,3,3,3,4}
Stericantellated 7-orthoplex
Cellirhombated hecatonicosoctaexon (Craze)
35
t1,2,4{3,3,3,3,3,4}
Biruncitruncated 7-orthoplex
Biprismatotruncated hecatonicosoctaexon (Baptize)
36
t0,3,4{3,3,3,3,3,4}
Steriruncinated 7-orthoplex
Celliprismated hecatonicosoctaexon (Copaz)
37
t1,3,4{3,3,3,3,3,4}
Biruncicantellated 7-orthoplex
Biprismatorhombated hecatonicosoctaexon (Boparz)
38
t2,3,4{4,3,3,3,3,3}
Tricantitruncated 7-cube
Great trirhombi-hepteractihecatonicosoctaexon (Gotrasaz)
39
t0,1,5{3,3,3,3,3,4}
Pentitruncated 7-orthoplex
Teritruncated hecatonicosoctaexon (Tetaz)
40
t0,2,5{3,3,3,3,3,4}
Penticantellated 7-orthoplex
Terirhombated hecatonicosoctaexon (Teroz)
41
t1,2,5{3,3,3,3,3,4}
Bisteritruncated 7-orthoplex
Bicellitruncated hecatonicosoctaexon (Boctaz)
42
t0,3,5{3,3,3,3,3,4}
Pentiruncinated 7-orthoplex
Teriprismated hecatonicosoctaexon (Topaz)
43
t1,3,5{4,3,3,3,3,3}
Bistericantellated 7-cube
Bicellirhombi-hepteractihecatonicosoctaexon (Bacresaz)
44
t1,3,4{4,3,3,3,3,3}
Biruncicantellated 7-cube
Biprismatorhombated hepteract (Bopresa)
45
t0,4,5{3,3,3,3,3,4}
Pentistericated 7-orthoplex
Tericellated hecatonicosoctaexon (Tocaz)
46
t1,2,5{4,3,3,3,3,3}
Bisteritruncated 7-cube
Bicellitruncated hepteract (Bactasa)
47
t1,2,4{4,3,3,3,3,3}
Biruncitruncated 7-cube
Biprismatotruncated hepteract (Biptesa)
48
t1,2,3{4,3,3,3,3,3}
Bicantitruncated 7-cube
Great birhombated hepteract (Gibrosa)
49
t0,1,6{3,3,3,3,3,4}
Hexitruncated 7-orthoplex
Petitruncated hecatonicosoctaexon (Putaz)
50
t0,2,6{3,3,3,3,3,4}
Hexicantellated 7-orthoplex
Petirhombated hecatonicosoctaexon (Puraz)
51
t0,4,5{4,3,3,3,3,3}
Pentistericated 7-cube
Tericellated hepteract (Tacosa)
52
t0,3,6{4,3,3,3,3,3}
Hexiruncinated 7-cube
Petiprismato-hepteractihecatonicosoctaexon (Pupsez)
53
t0,3,5{4,3,3,3,3,3}
Pentiruncinated 7-cube
Teriprismated hepteract (Tapsa)
54
t0,3,4{4,3,3,3,3,3}
Steriruncinated 7-cube
Celliprismated hepteract (Capsa)
55
t0,2,6{4,3,3,3,3,3}
Hexicantellated 7-cube
Petirhombated hepteract (Purosa)
56
t0,2,5{4,3,3,3,3,3}
Penticantellated 7-cube
Terirhombated hepteract (Tersa)
57
t0,2,4{4,3,3,3,3,3}
Stericantellated 7-cube
Cellirhombated hepteract (Carsa)
58
t0,2,3{4,3,3,3,3,3}
Runcicantellated 7-cube
Prismatorhombated hepteract (Parsa)
59
t0,1,6{4,3,3,3,3,3}
Hexitruncated 7-cube
Petitruncated hepteract (Putsa)
60
t0,1,5{4,3,3,3,3,3}
Pentitruncated 7-cube
Teritruncated hepteract (Tetsa)
61
t0,1,4{4,3,3,3,3,3}
Steritruncated 7-cube
Cellitruncated hepteract (Catsa)
62
t0,1,3{4,3,3,3,3,3}
Runcitruncated 7-cube
Prismatotruncated hepteract (Petsa)
63
t0,1,2{4,3,3,3,3,3}
Cantitruncated 7-cube
Great rhombated hepteract (Gersa)
64
t0,1,2,3{3,3,3,3,3,4}
Runcicantitruncated 7-orthoplex
Great prismated hecatonicosoctaexon (Gopaz)
65
t0,1,2,4{3,3,3,3,3,4}
Stericantitruncated 7-orthoplex
Celligreatorhombated hecatonicosoctaexon (Cogarz)
66
t0,1,3,4{3,3,3,3,3,4}
Steriruncitruncated 7-orthoplex
Celliprismatotruncated hecatonicosoctaexon (Captaz)
67
t0,2,3,4{3,3,3,3,3,4}
Steriruncicantellated 7-orthoplex
Celliprismatorhombated hecatonicosoctaexon (Caparz)
68
t1,2,3,4{3,3,3,3,3,4}
Biruncicantitruncated 7-orthoplex
Great biprismated hecatonicosoctaexon (Gibpaz)
69
t0,1,2,5{3,3,3,3,3,4}
Penticantitruncated 7-orthoplex
Terigreatorhombated hecatonicosoctaexon (Tograz)
70
t0,1,3,5{3,3,3,3,3,4}
Pentiruncitruncated 7-orthoplex
Teriprismatotruncated hecatonicosoctaexon (Toptaz)
71
t0,2,3,5{3,3,3,3,3,4}
Pentiruncicantellated 7-orthoplex
Teriprismatorhombated hecatonicosoctaexon (Toparz)
72
t1,2,3,5{3,3,3,3,3,4}
Bistericantitruncated 7-orthoplex
Bicelligreatorhombated hecatonicosoctaexon (Becogarz)
73
t0,1,4,5{3,3,3,3,3,4}
Pentisteritruncated 7-orthoplex
Tericellitruncated hecatonicosoctaexon (Tacotaz)
74
t0,2,4,5{3,3,3,3,3,4}
Pentistericantellated 7-orthoplex
Tericellirhombated hecatonicosoctaexon (Tocarz)
75
t1,2,4,5{4,3,3,3,3,3}
Bisteriruncitruncated 7-cube
Bicelliprismatotrunki-hepteractihecatonicosoctaexon (Bocaptosaz)
76
t0,3,4,5{3,3,3,3,3,4}
Pentisteriruncinated 7-orthoplex
Tericelliprismated hecatonicosoctaexon (Tecpaz)
77
t1,2,3,5{4,3,3,3,3,3}
Bistericantitruncated 7-cube
Bicelligreatorhombated hepteract (Becgresa)
78
t1,2,3,4{4,3,3,3,3,3}
Biruncicantitruncated 7-cube
Great biprismated hepteract (Gibposa)
79
t0,1,2,6{3,3,3,3,3,4}
Hexicantitruncated 7-orthoplex
Petigreatorhombated hecatonicosoctaexon (Pugarez)
80
t0,1,3,6{3,3,3,3,3,4}
Hexiruncitruncated 7-orthoplex
Petiprismatotruncated hecatonicosoctaexon (Papataz)
81
t0,2,3,6{3,3,3,3,3,4}
Hexiruncicantellated 7-orthoplex
Petiprismatorhombated hecatonicosoctaexon (Puparez)
82
t0,3,4,5{4,3,3,3,3,3}
Pentisteriruncinated 7-cube
Tericelliprismated hepteract (Tecpasa)
83
t0,1,4,6{3,3,3,3,3,4}
Hexisteritruncated 7-orthoplex
Peticellitruncated hecatonicosoctaexon (Pucotaz)
84
t0,2,4,6{4,3,3,3,3,3}
Hexistericantellated 7-cube
Peticellirhombi-hepteractihecatonicosoctaexon (Pucrosaz)
85
t0,2,4,5{4,3,3,3,3,3}
Pentistericantellated 7-cube
Tericellirhombated hepteract (Tecresa)
86
t0,2,3,6{4,3,3,3,3,3}
Hexiruncicantellated 7-cube
Petiprismatorhombated hepteract (Pupresa)
87
t0,2,3,5{4,3,3,3,3,3}
Pentiruncicantellated 7-cube
Teriprismatorhombated hepteract (Topresa)
88
t0,2,3,4{4,3,3,3,3,3}
Steriruncicantellated 7-cube
Celliprismatorhombated hepteract (Copresa)
89
t0,1,5,6{4,3,3,3,3,3}
Hexipentitruncated 7-cube
Petiteritrunki-hepteractihecatonicosoctaexon (Putatosez)
90
t0,1,4,6{4,3,3,3,3,3}
Hexisteritruncated 7-cube
Peticellitruncated hepteract (Pacutsa)
91
t0,1,4,5{4,3,3,3,3,3}
Pentisteritruncated 7-cube
Tericellitruncated hepteract (Tecatsa)
92
t0,1,3,6{4,3,3,3,3,3}
Hexiruncitruncated 7-cube
Petiprismatotruncated hepteract (Pupetsa)
93
t0,1,3,5{4,3,3,3,3,3}
Pentiruncitruncated 7-cube
Teriprismatotruncated hepteract (Toptosa)
94
t0,1,3,4{4,3,3,3,3,3}
Steriruncitruncated 7-cube
Celliprismatotruncated hepteract (Captesa)
95
t0,1,2,6{4,3,3,3,3,3}
Hexicantitruncated 7-cube
Petigreatorhombated hepteract (Pugrosa)
96
t0,1,2,5{4,3,3,3,3,3}
Penticantitruncated 7-cube
Terigreatorhombated hepteract (Togresa)
97
t0,1,2,4{4,3,3,3,3,3}
Stericantitruncated 7-cube
Celligreatorhombated hepteract (Cogarsa)
98
t0,1,2,3{4,3,3,3,3,3}
Runcicantitruncated 7-cube
Great prismated hepteract (Gapsa)
99
t0,1,2,3,4{3,3,3,3,3,4}
Steriruncicantitruncated 7-orthoplex
Great cellated hecatonicosoctaexon (Gocaz)
100
t0,1,2,3,5{3,3,3,3,3,4}
Pentiruncicantitruncated 7-orthoplex
Terigreatoprismated hecatonicosoctaexon (Tegopaz)
101
t0,1,2,4,5{3,3,3,3,3,4}
Pentistericantitruncated 7-orthoplex
Tericelligreatorhombated hecatonicosoctaexon (Tecagraz)
102
t0,1,3,4,5{3,3,3,3,3,4}
Pentisteriruncitruncated 7-orthoplex
Tericelliprismatotruncated hecatonicosoctaexon (Tecpotaz)
103
t0,2,3,4,5{3,3,3,3,3,4}
Pentisteriruncicantellated 7-orthoplex
Tericelliprismatorhombated hecatonicosoctaexon (Tacparez)
104
t1,2,3,4,5{4,3,3,3,3,3}
Bisteriruncicantitruncated 7-cube
Great bicelli-hepteractihecatonicosoctaexon (Gabcosaz)
105
t0,1,2,3,6{3,3,3,3,3,4}
Hexiruncicantitruncated 7-orthoplex
Petigreatoprismated hecatonicosoctaexon (Pugopaz)
106
t0,1,2,4,6{3,3,3,3,3,4}
Hexistericantitruncated 7-orthoplex
Peticelligreatorhombated hecatonicosoctaexon (Pucagraz)
107
t0,1,3,4,6{3,3,3,3,3,4}
Hexisteriruncitruncated 7-orthoplex
Peticelliprismatotruncated hecatonicosoctaexon (Pucpotaz)
108
t0,2,3,4,6{4,3,3,3,3,3}
Hexisteriruncicantellated 7-cube
Peticelliprismatorhombi-hepteractihecatonicosoctaexon (Pucprosaz)
109
t0,2,3,4,5{4,3,3,3,3,3}
Pentisteriruncicantellated 7-cube
Tericelliprismatorhombated hepteract (Tocpresa)
110
t0,1,2,5,6{3,3,3,3,3,4}
Hexipenticantitruncated 7-orthoplex
Petiterigreatorhombated hecatonicosoctaexon (Putegraz)
111
t0,1,3,5,6{4,3,3,3,3,3}
Hexipentiruncitruncated 7-cube
Petiteriprismatotrunki-hepteractihecatonicosoctaexon (Putpetsaz)
112
t0,1,3,4,6{4,3,3,3,3,3}
Hexisteriruncitruncated 7-cube
Peticelliprismatotruncated hepteract (Pucpetsa)
113
t0,1,3,4,5{4,3,3,3,3,3}
Pentisteriruncitruncated 7-cube
Tericelliprismatotruncated hepteract (Tecpetsa)
114
t0,1,2,5,6{4,3,3,3,3,3}
Hexipenticantitruncated 7-cube
Petiterigreatorhombated hepteract (Putgresa)
115
t0,1,2,4,6{4,3,3,3,3,3}
Hexistericantitruncated 7-cube
Peticelligreatorhombated hepteract (Pucagrosa)
116
t0,1,2,4,5{4,3,3,3,3,3}
Pentistericantitruncated 7-cube
Tericelligreatorhombated hepteract (Tecgresa)
117
t0,1,2,3,6{4,3,3,3,3,3}
Hexiruncicantitruncated 7-cube
Petigreatoprismated hepteract (Pugopsa)
118
t0,1,2,3,5{4,3,3,3,3,3}
Pentiruncicantitruncated 7-cube
Terigreatoprismated hepteract (Togapsa)
119
t0,1,2,3,4{4,3,3,3,3,3}
Steriruncicantitruncated 7-cube
Great cellated hepteract (Gacosa)
120
t0,1,2,3,4,5{3,3,3,3,3,4}
Pentisteriruncicantitruncated 7-orthoplex
Great terated hecatonicosoctaexon (Gotaz)
121
t0,1,2,3,4,6{3,3,3,3,3,4}
Hexisteriruncicantitruncated 7-orthoplex
Petigreatocellated hecatonicosoctaexon (Pugacaz)
122
t0,1,2,3,5,6{3,3,3,3,3,4}
Hexipentiruncicantitruncated 7-orthoplex
Petiterigreatoprismated hecatonicosoctaexon (Putgapaz)
123
t0,1,2,4,5,6{4,3,3,3,3,3}
Hexipentistericantitruncated 7-cube
Petitericelligreatorhombi-hepteractihecatonicosoctaexon (Putcagrasaz)
124
t0,1,2,3,5,6{4,3,3,3,3,3}
Hexipentiruncicantitruncated 7-cube
Petiterigreatoprismated hepteract (Putgapsa)
125
t0,1,2,3,4,6{4,3,3,3,3,3}
Hexisteriruncicantitruncated 7-cube
Petigreatocellated hepteract (Pugacasa)
126
t0,1,2,3,4,5{4,3,3,3,3,3}
Pentisteriruncicantitruncated 7-cube
Great terated hepteract (Gotesa)
127
t0,1,2,3,4,5,6{4,3,3,3,3,3}
Omnitruncated 7-cube
Great peti-hepteractihecatonicosoctaexon (Guposaz)

References

Notes
  1. ↑ Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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