B7 polytope
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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[edit]
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 |
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