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. A_k has [k+1] symmetry, and B_k 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 in progressive order: red, orange, yellow, green, cyan, blue, purple, magenta, red-violet.

#	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 Hecatonicosaoctaexon (zee)
2									t₁{3,3,3,3,3,4} Rectified 7-orthoplex Rectified hecatonicosaoctaexon (rez)
3									t₂{3,3,3,3,3,4} Birectified 7-orthoplex Birectified hecatonicosaoctaexon (barz)
4									t₃{4,3,3,3,3,3} Trirectified 7-cube Hepteractihecatonicosaoctaexon (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 (hesa)
8									t_0,1{3,3,3,3,3,4} Truncated 7-orthoplex Truncated hecatonicosaoctaexon (Taz)
9									t_0,2{3,3,3,3,3,4} Cantellated 7-orthoplex Small rhombated hecatonicosaoctaexon (Sarz)
10									t_1,2{3,3,3,3,3,4} Bitruncated 7-orthoplex Bitruncated hecatonicosaoctaexon (Botaz)
11									t_0,3{3,3,3,3,3,4} Runcinated 7-orthoplex Small prismated hecatonicosaoctaexon (Spaz)
12									t_1,3{3,3,3,3,3,4} Bicantellated 7-orthoplex Small birhombated hecatonicosaoctaexon (Sebraz)
13									t_2,3{3,3,3,3,3,4} Tritruncated 7-orthoplex Tritruncated hecatonicosaoctaexon (Totaz)
14									t_0,4{3,3,3,3,3,4} Stericated 7-orthoplex Small cellated hecatonicosaoctaexon (Scaz)
15									t_1,4{3,3,3,3,3,4} Biruncinated 7-orthoplex Small biprismated hecatonicosaoctaexon (Sibpaz)
16									t_2,4{4,3,3,3,3,3} Tricantellated 7-cube Small trirhombi-hepteractihecatonicosaoctaexon (Strasaz)
17									t_2,3{4,3,3,3,3,3} Tritruncated 7-cube Tritruncated hepteract (Tatsa)
18									t_0,5{3,3,3,3,3,4} Pentellated 7-orthoplex Small terated hecatonicosaoctaexon (Staz)
19									t_1,5{4,3,3,3,3,3} Bistericated 7-cube Small bicelli-hepteractihecatonicosaoctaexon (Sabcosaz)
20									t_1,4{4,3,3,3,3,3} Biruncinated 7-cube Small biprismated hepteract (Sibposa)
21									t_1,3{4,3,3,3,3,3} Bicantellated 7-cube Small birhombated hepteract (Sibrosa)
22									t_1,2{4,3,3,3,3,3} Bitruncated 7-cube Bitruncated hepteract (Betsa)
23									t_0,6{4,3,3,3,3,3} Hexicated 7-cube Small peti-hepteractihecatonicosaoctaexon (Suposaz)
24									t_0,5{4,3,3,3,3,3} Pentellated 7-cube Small terated hepteract (Stesa)
25									t_0,4{4,3,3,3,3,3} Stericated 7-cube Small cellated hepteract (Scosa)
26									t_0,3{4,3,3,3,3,3} Runcinated 7-cube Small prismated hepteract (Spesa)
27									t_0,2{4,3,3,3,3,3} Cantellated 7-cube Small rhombated hepteract (Sersa)
28									t_0,1{4,3,3,3,3,3} Truncated 7-cube Truncated hepteract (Tasa)
29									t_0,1,2{3,3,3,3,3,4} Cantitruncated 7-orthoplex Great rhombated hecatonicosaoctaexon (Garz)
30									t_0,1,3{3,3,3,3,3,4} Runcitruncated 7-orthoplex Prismatotruncated hecatonicosaoctaexon (Potaz)
31									t_0,2,3{3,3,3,3,3,4} Runcicantellated 7-orthoplex Prismatorhombated hecatonicosaoctaexon (Parz)
32									t_1,2,3{3,3,3,3,3,4} Bicantitruncated 7-orthoplex Great birhombated hecatonicosaoctaexon (Gebraz)
33									t_0,1,4{3,3,3,3,3,4} Steritruncated 7-orthoplex Cellitruncated hecatonicosaoctaexon (Catz)
34									t_0,2,4{3,3,3,3,3,4} Stericantellated 7-orthoplex Cellirhombated hecatonicosaoctaexon (Craze)
35									t_1,2,4{3,3,3,3,3,4} Biruncitruncated 7-orthoplex Biprismatotruncated hecatonicosaoctaexon (Baptize)
36									t_0,3,4{3,3,3,3,3,4} Steriruncinated 7-orthoplex Celliprismated hecatonicosaoctaexon (Copaz)
37									t_1,3,4{3,3,3,3,3,4} Biruncicantellated 7-orthoplex Biprismatorhombated hecatonicosaoctaexon (Boparz)
38									t_2,3,4{4,3,3,3,3,3} Tricantitruncated 7-cube Great trirhombi-hepteractihecatonicosaoctaexon (Gotrasaz)
39									t_0,1,5{3,3,3,3,3,4} Pentitruncated 7-orthoplex Teritruncated hecatonicosaoctaexon (Tetaz)
40									t_0,2,5{3,3,3,3,3,4} Penticantellated 7-orthoplex Terirhombated hecatonicosaoctaexon (Teroz)
41									t_1,2,5{3,3,3,3,3,4} Bisteritruncated 7-orthoplex Bicellitruncated hecatonicosaoctaexon (Boctaz)
42									t_0,3,5{3,3,3,3,3,4} Pentiruncinated 7-orthoplex Teriprismated hecatonicosaoctaexon (Topaz)
43									t_1,3,5{4,3,3,3,3,3} Bistericantellated 7-cube Bicellirhombi-hepteractihecatonicosaoctaexon (Bacresaz)
44									t_1,3,4{4,3,3,3,3,3} Biruncicantellated 7-cube Biprismatorhombated hepteract (Bopresa)
45									t_0,4,5{3,3,3,3,3,4} Pentistericated 7-orthoplex Tericellated hecatonicosaoctaexon (Tocaz)
46									t_1,2,5{4,3,3,3,3,3} Bisteritruncated 7-cube Bicellitruncated hepteract (Bactasa)
47									t_1,2,4{4,3,3,3,3,3} Biruncitruncated 7-cube Biprismatotruncated hepteract (Biptesa)
48									t_1,2,3{4,3,3,3,3,3} Bicantitruncated 7-cube Great birhombated hepteract (Gibrosa)
49									t_0,1,6{3,3,3,3,3,4} Hexitruncated 7-orthoplex Petitruncated hecatonicosaoctaexon (Putaz)
50									t_0,2,6{3,3,3,3,3,4} Hexicantellated 7-orthoplex Petirhombated hecatonicosaoctaexon (Puraz)
51									t_0,4,5{4,3,3,3,3,3} Pentistericated 7-cube Tericellated hepteract (Tacosa)
52									t_0,3,6{4,3,3,3,3,3} Hexiruncinated 7-cube Petiprismato-hepteractihecatonicosaoctaexon (Pupsez)
53									t_0,3,5{4,3,3,3,3,3} Pentiruncinated 7-cube Teriprismated hepteract (Tapsa)
54									t_0,3,4{4,3,3,3,3,3} Steriruncinated 7-cube Celliprismated hepteract (Capsa)
55									t_0,2,6{4,3,3,3,3,3} Hexicantellated 7-cube Petirhombated hepteract (Purosa)
56									t_0,2,5{4,3,3,3,3,3} Penticantellated 7-cube Terirhombated hepteract (Tersa)
57									t_0,2,4{4,3,3,3,3,3} Stericantellated 7-cube Cellirhombated hepteract (Carsa)
58									t_0,2,3{4,3,3,3,3,3} Runcicantellated 7-cube Prismatorhombated hepteract (Parsa)
59									t_0,1,6{4,3,3,3,3,3} Hexitruncated 7-cube Petitruncated hepteract (Putsa)
60									t_0,1,5{4,3,3,3,3,3} Pentitruncated 7-cube Teritruncated hepteract (Tetsa)
61									t_0,1,4{4,3,3,3,3,3} Steritruncated 7-cube Cellitruncated hepteract (Catsa)
62									t_0,1,3{4,3,3,3,3,3} Runcitruncated 7-cube Prismatotruncated hepteract (Petsa)
63									t_0,1,2{4,3,3,3,3,3} Cantitruncated 7-cube Great rhombated hepteract (Gersa)
64									t_0,1,2,3{3,3,3,3,3,4} Runcicantitruncated 7-orthoplex Great prismated hecatonicosaoctaexon (Gopaz)
65									t_0,1,2,4{3,3,3,3,3,4} Stericantitruncated 7-orthoplex Celligreatorhombated hecatonicosaoctaexon (Cogarz)
66									t_0,1,3,4{3,3,3,3,3,4} Steriruncitruncated 7-orthoplex Celliprismatotruncated hecatonicosaoctaexon (Captaz)
67									t_0,2,3,4{3,3,3,3,3,4} Steriruncicantellated 7-orthoplex Celliprismatorhombated hecatonicosaoctaexon (Caparz)
68									t_1,2,3,4{3,3,3,3,3,4} Biruncicantitruncated 7-orthoplex Great biprismated hecatonicosaoctaexon (Gibpaz)
69									t_0,1,2,5{3,3,3,3,3,4} Penticantitruncated 7-orthoplex Terigreatorhombated hecatonicosaoctaexon (Tograz)
70									t_0,1,3,5{3,3,3,3,3,4} Pentiruncitruncated 7-orthoplex Teriprismatotruncated hecatonicosaoctaexon (Toptaz)
71									t_0,2,3,5{3,3,3,3,3,4} Pentiruncicantellated 7-orthoplex Teriprismatorhombated hecatonicosaoctaexon (Toparz)
72									t_1,2,3,5{3,3,3,3,3,4} Bistericantitruncated 7-orthoplex Bicelligreatorhombated hecatonicosaoctaexon (Becogarz)
73									t_0,1,4,5{3,3,3,3,3,4} Pentisteritruncated 7-orthoplex Tericellitruncated hecatonicosaoctaexon (Tacotaz)
74									t_0,2,4,5{3,3,3,3,3,4} Pentistericantellated 7-orthoplex Tericellirhombated hecatonicosaoctaexon (Tocarz)
75									t_1,2,4,5{4,3,3,3,3,3} Bisteriruncitruncated 7-cube Bicelliprismatotrunki-hepteractihecatonicosaoctaexon (Bocaptosaz)
76									t_0,3,4,5{3,3,3,3,3,4} Pentisteriruncinated 7-orthoplex Tericelliprismated hecatonicosaoctaexon (Tecpaz)
77									t_1,2,3,5{4,3,3,3,3,3} Bistericantitruncated 7-cube Bicelligreatorhombated hepteract (Becgresa)
78									t_1,2,3,4{4,3,3,3,3,3} Biruncicantitruncated 7-cube Great biprismated hepteract (Gibposa)
79									t_0,1,2,6{3,3,3,3,3,4} Hexicantitruncated 7-orthoplex Petigreatorhombated hecatonicosaoctaexon (Pugarez)
80									t_0,1,3,6{3,3,3,3,3,4} Hexiruncitruncated 7-orthoplex Petiprismatotruncated hecatonicosaoctaexon (Papataz)
81									t_0,2,3,6{3,3,3,3,3,4} Hexiruncicantellated 7-orthoplex Petiprismatorhombated hecatonicosaoctaexon (Puparez)
82									t_0,3,4,5{4,3,3,3,3,3} Pentisteriruncinated 7-cube Tericelliprismated hepteract (Tecpasa)
83									t_0,1,4,6{3,3,3,3,3,4} Hexisteritruncated 7-orthoplex Peticellitruncated hecatonicosaoctaexon (Pucotaz)
84									t_0,2,4,6{4,3,3,3,3,3} Hexistericantellated 7-cube Peticellirhombi-hepteractihecatonicosaoctaexon (Pucrosaz)
85									t_0,2,4,5{4,3,3,3,3,3} Pentistericantellated 7-cube Tericellirhombated hepteract (Tecresa)
86									t_0,2,3,6{4,3,3,3,3,3} Hexiruncicantellated 7-cube Petiprismatorhombated hepteract (Pupresa)
87									t_0,2,3,5{4,3,3,3,3,3} Pentiruncicantellated 7-cube Teriprismatorhombated hepteract (Topresa)
88									t_0,2,3,4{4,3,3,3,3,3} Steriruncicantellated 7-cube Celliprismatorhombated hepteract (Copresa)
89									t_0,1,5,6{4,3,3,3,3,3} Hexipentitruncated 7-cube Petiteritrunki-hepteractihecatonicosaoctaexon (Putatosez)
90									t_0,1,4,6{4,3,3,3,3,3} Hexisteritruncated 7-cube Peticellitruncated hepteract (Pacutsa)
91									t_0,1,4,5{4,3,3,3,3,3} Pentisteritruncated 7-cube Tericellitruncated hepteract (Tecatsa)
92									t_0,1,3,6{4,3,3,3,3,3} Hexiruncitruncated 7-cube Petiprismatotruncated hepteract (Pupetsa)
93									t_0,1,3,5{4,3,3,3,3,3} Pentiruncitruncated 7-cube Teriprismatotruncated hepteract (Toptosa)
94									t_0,1,3,4{4,3,3,3,3,3} Steriruncitruncated 7-cube Celliprismatotruncated hepteract (Captesa)
95									t_0,1,2,6{4,3,3,3,3,3} Hexicantitruncated 7-cube Petigreatorhombated hepteract (Pugrosa)
96									t_0,1,2,5{4,3,3,3,3,3} Penticantitruncated 7-cube Terigreatorhombated hepteract (Togresa)
97									t_0,1,2,4{4,3,3,3,3,3} Stericantitruncated 7-cube Celligreatorhombated hepteract (Cogarsa)
98									t_0,1,2,3{4,3,3,3,3,3} Runcicantitruncated 7-cube Great prismated hepteract (Gapsa)
99									t_0,1,2,3,4{3,3,3,3,3,4} Steriruncicantitruncated 7-orthoplex Great cellated hecatonicosaoctaexon (Gocaz)
100									t_0,1,2,3,5{3,3,3,3,3,4} Pentiruncicantitruncated 7-orthoplex Terigreatoprismated hecatonicosaoctaexon (Tegopaz)
101									t_0,1,2,4,5{3,3,3,3,3,4} Pentistericantitruncated 7-orthoplex Tericelligreatorhombated hecatonicosaoctaexon (Tecagraz)
102									t_0,1,3,4,5{3,3,3,3,3,4} Pentisteriruncitruncated 7-orthoplex Tericelliprismatotruncated hecatonicosaoctaexon (Tecpotaz)
103									t_0,2,3,4,5{3,3,3,3,3,4} Pentisteriruncicantellated 7-orthoplex Tericelliprismatorhombated hecatonicosaoctaexon (Tacparez)
104									t_1,2,3,4,5{4,3,3,3,3,3} Bisteriruncicantitruncated 7-cube Great bicelli-hepteractihecatonicosaoctaexon (Gabcosaz)
105									t_0,1,2,3,6{3,3,3,3,3,4} Hexiruncicantitruncated 7-orthoplex Petigreatoprismated hecatonicosaoctaexon (Pugopaz)
106									t_0,1,2,4,6{3,3,3,3,3,4} Hexistericantitruncated 7-orthoplex Peticelligreatorhombated hecatonicosaoctaexon (Pucagraz)
107									t_0,1,3,4,6{3,3,3,3,3,4} Hexisteriruncitruncated 7-orthoplex Peticelliprismatotruncated hecatonicosaoctaexon (Pucpotaz)
108									t_0,2,3,4,6{4,3,3,3,3,3} Hexisteriruncicantellated 7-cube Peticelliprismatorhombi-hepteractihecatonicosaoctaexon (Pucprosaz)
109									t_0,2,3,4,5{4,3,3,3,3,3} Pentisteriruncicantellated 7-cube Tericelliprismatorhombated hepteract (Tocpresa)
110									t_0,1,2,5,6{3,3,3,3,3,4} Hexipenticantitruncated 7-orthoplex Petiterigreatorhombated hecatonicosaoctaexon (Putegraz)
111									t_0,1,3,5,6{4,3,3,3,3,3} Hexipentiruncitruncated 7-cube Petiteriprismatotrunki-hepteractihecatonicosaoctaexon (Putpetsaz)
112									t_0,1,3,4,6{4,3,3,3,3,3} Hexisteriruncitruncated 7-cube Peticelliprismatotruncated hepteract (Pucpetsa)
113									t_0,1,3,4,5{4,3,3,3,3,3} Pentisteriruncitruncated 7-cube Tericelliprismatotruncated hepteract (Tecpetsa)
114									t_0,1,2,5,6{4,3,3,3,3,3} Hexipenticantitruncated 7-cube Petiterigreatorhombated hepteract (Putgresa)
115									t_0,1,2,4,6{4,3,3,3,3,3} Hexistericantitruncated 7-cube Peticelligreatorhombated hepteract (Pucagrosa)
116									t_0,1,2,4,5{4,3,3,3,3,3} Pentistericantitruncated 7-cube Tericelligreatorhombated hepteract (Tecgresa)
117									t_0,1,2,3,6{4,3,3,3,3,3} Hexiruncicantitruncated 7-cube Petigreatoprismated hepteract (Pugopsa)
118									t_0,1,2,3,5{4,3,3,3,3,3} Pentiruncicantitruncated 7-cube Terigreatoprismated hepteract (Togapsa)
119									t_0,1,2,3,4{4,3,3,3,3,3} Steriruncicantitruncated 7-cube Great cellated hepteract (Gacosa)
120									t_0,1,2,3,4,5{3,3,3,3,3,4} Pentisteriruncicantitruncated 7-orthoplex Great terated hecatonicosaoctaexon (Gotaz)
121									t_0,1,2,3,4,6{3,3,3,3,3,4} Hexisteriruncicantitruncated 7-orthoplex Petigreatocellated hecatonicosaoctaexon (Pugacaz)
122									t_0,1,2,3,5,6{3,3,3,3,3,4} Hexipentiruncicantitruncated 7-orthoplex Petiterigreatoprismated hecatonicosaoctaexon (Putgapaz)
123									t_0,1,2,4,5,6{4,3,3,3,3,3} Hexipentistericantitruncated 7-cube Petitericelligreatorhombi-hepteractihecatonicosaoctaexon (Putcagrasaz)
124									t_0,1,2,3,5,6{4,3,3,3,3,3} Hexipentiruncicantitruncated 7-cube Petiterigreatoprismated hepteract (Putgapsa)
125									t_0,1,2,3,4,6{4,3,3,3,3,3} Hexisteriruncicantitruncated 7-cube Petigreatocellated hepteract (Pugacasa)
126									t_0,1,2,3,4,5{4,3,3,3,3,3} Pentisteriruncicantitruncated 7-cube Great terated hepteract (Gotesa)
127									t_{0,1,2,3,4,5,6}{4,3,3,3,3,3} Omnitruncated 7-cube Great peti-hepteractihecatonicosaoctaexon (Guposaz)

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 Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, ISBN 978-0-471-01003-6
- (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 Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
Klitzing, Richard. "7D uniform polytopes (polyexa) with acronyms".

v t e 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	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations

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Graphs

References