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.

#	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

• 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, Wiley-Interscience Publication, 1995, 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)".

Notes