Uniform 7-polytope
In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.
A uniform 7-polytope is one whose symmetry group is transitive on vertices and whose facets are uniform 6-polytopes.
Regular 7-polytopes edit
Regular 7-polytopes are represented by the Schläfli symbol {p,q,r,s,t,u} with u {p,q,r,s,t} 6-polytopes facets around each 4-face.
There are exactly three such convex regular 7-polytopes:
- {3,3,3,3,3,3} - 7-simplex
- {4,3,3,3,3,3} - 7-cube
- {3,3,3,3,3,4} - 7-orthoplex
There are no nonconvex regular 7-polytopes.
Characteristics edit
The topology of any given 7-polytope is defined by its Betti numbers and torsion coefficients.[1]
The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.[1]
Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.[1]
Uniform 7-polytopes by fundamental Coxeter groups edit
Uniform 7-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:
# | Coxeter group | Regular and semiregular forms | Uniform count | ||
---|---|---|---|---|---|
1 | A7 | [36] |
| 71 | |
2 | B7 | [4,35] |
| 127 + 32 | |
3 | D7 | [33,1,1] |
| 95 (0 unique) | |
4 | E7 | [33,2,1] | 127 |
Prismatic finite Coxeter groups | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
# | Coxeter group | Coxeter diagram | |||||||||
6+1 | |||||||||||
1 | A6A1 | [35]×[ ] | |||||||||
2 | BC6A1 | [4,34]×[ ] | |||||||||
3 | D6A1 | [33,1,1]×[ ] | |||||||||
4 | E6A1 | [32,2,1]×[ ] | |||||||||
5+2 | |||||||||||
1 | A5I2(p) | [3,3,3]×[p] | |||||||||
2 | BC5I2(p) | [4,3,3]×[p] | |||||||||
3 | D5I2(p) | [32,1,1]×[p] | |||||||||
5+1+1 | |||||||||||
1 | A5A12 | [3,3,3]×[ ]2 | |||||||||
2 | BC5A12 | [4,3,3]×[ ]2 | |||||||||
3 | D5A12 | [32,1,1]×[ ]2 | |||||||||
4+3 | |||||||||||
1 | A4A3 | [3,3,3]×[3,3] | |||||||||
2 | A4B3 | [3,3,3]×[4,3] | |||||||||
3 | A4H3 | [3,3,3]×[5,3] | |||||||||
4 | BC4A3 | [4,3,3]×[3,3] | |||||||||
5 | BC4B3 | [4,3,3]×[4,3] | |||||||||
6 | BC4H3 | [4,3,3]×[5,3] | |||||||||
7 | H4A3 | [5,3,3]×[3,3] | |||||||||
8 | H4B3 | [5,3,3]×[4,3] | |||||||||
9 | H4H3 | [5,3,3]×[5,3] | |||||||||
10 | F4A3 | [3,4,3]×[3,3] | |||||||||
11 | F4B3 | [3,4,3]×[4,3] | |||||||||
12 | F4H3 | [3,4,3]×[5,3] | |||||||||
13 | D4A3 | [31,1,1]×[3,3] | |||||||||
14 | D4B3 | [31,1,1]×[4,3] | |||||||||
15 | D4H3 | [31,1,1]×[5,3] | |||||||||
4+2+1 | |||||||||||
1 | A4I2(p)A1 | [3,3,3]×[p]×[ ] | |||||||||
2 | BC4I2(p)A1 | [4,3,3]×[p]×[ ] | |||||||||
3 | F4I2(p)A1 | [3,4,3]×[p]×[ ] | |||||||||
4 | H4I2(p)A1 | [5,3,3]×[p]×[ ] | |||||||||
5 | D4I2(p)A1 | [31,1,1]×[p]×[ ] | |||||||||
4+1+1+1 | |||||||||||
1 | A4A13 | [3,3,3]×[ ]3 | |||||||||
2 | BC4A13 | [4,3,3]×[ ]3 | |||||||||
3 | F4A13 | [3,4,3]×[ ]3 | |||||||||
4 | H4A13 | [5,3,3]×[ ]3 | |||||||||
5 | D4A13 | [31,1,1]×[ ]3 | |||||||||
3+3+1 | |||||||||||
1 | A3A3A1 | [3,3]×[3,3]×[ ] | |||||||||
2 | A3B3A1 | [3,3]×[4,3]×[ ] | |||||||||
3 | A3H3A1 | [3,3]×[5,3]×[ ] | |||||||||
4 | BC3B3A1 | [4,3]×[4,3]×[ ] | |||||||||
5 | BC3H3A1 | [4,3]×[5,3]×[ ] | |||||||||
6 | H3A3A1 | [5,3]×[5,3]×[ ] | |||||||||
3+2+2 | |||||||||||
1 | A3I2(p)I2(q) | [3,3]×[p]×[q] | |||||||||
2 | BC3I2(p)I2(q) | [4,3]×[p]×[q] | |||||||||
3 | H3I2(p)I2(q) | [5,3]×[p]×[q] | |||||||||
3+2+1+1 | |||||||||||
1 | A3I2(p)A12 | [3,3]×[p]×[ ]2 | |||||||||
2 | BC3I2(p)A12 | [4,3]×[p]×[ ]2 | |||||||||
3 | H3I2(p)A12 | [5,3]×[p]×[ ]2 | |||||||||
3+1+1+1+1 | |||||||||||
1 | A3A14 | [3,3]×[ ]4 | |||||||||
2 | BC3A14 | [4,3]×[ ]4 | |||||||||
3 | H3A14 | [5,3]×[ ]4 | |||||||||
2+2+2+1 | |||||||||||
1 | I2(p)I2(q)I2(r)A1 | [p]×[q]×[r]×[ ] | |||||||||
2+2+1+1+1 | |||||||||||
1 | I2(p)I2(q)A13 | [p]×[q]×[ ]3 | |||||||||
2+1+1+1+1+1 | |||||||||||
1 | I2(p)A15 | [p]×[ ]5 | |||||||||
1+1+1+1+1+1+1 | |||||||||||
1 | A17 | [ ]7 |
The A7 family edit
The A7 family has symmetry of order 40320 (8 factorial).
There are 71 (64+8-1) forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. All 71 are enumerated below. Norman Johnson's truncation names are given. Bowers names and acronym are also given for cross-referencing.
See also a list of A7 polytopes for symmetric Coxeter plane graphs of these polytopes.
A7 uniform polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
# | Coxeter-Dynkin diagram | Truncation indices | Johnson name Bowers name (and acronym) | Basepoint | Element counts | ||||||
6 | 5 | 4 | 3 | 2 | 1 | 0 | |||||
1 | t0 | 7-simplex (oca) | (0,0,0,0,0,0,0,1) | 8 | 28 | 56 | 70 | 56 | 28 | 8 | |
2 | t1 | Rectified 7-simplex (roc) | (0,0,0,0,0,0,1,1) | 16 | 84 | 224 | 350 | 336 | 168 | 28 | |
3 | t2 | Birectified 7-simplex (broc) | (0,0,0,0,0,1,1,1) | 16 | 112 | 392 | 770 | 840 | 420 | 56 | |
4 | t3 | Trirectified 7-simplex (he) | (0,0,0,0,1,1,1,1) | 16 | 112 | 448 | 980 | 1120 | 560 | 70 | |
5 | t0,1 | Truncated 7-simplex (toc) | (0,0,0,0,0,0,1,2) | 16 | 84 | 224 | 350 | 336 | 196 | 56 | |
6 | t0,2 | Cantellated 7-simplex (saro) | (0,0,0,0,0,1,1,2) | 44 | 308 | 980 | 1750 | 1876 | 1008 | 168 | |
7 | t1,2 | Bitruncated 7-simplex (bittoc) | (0,0,0,0,0,1,2,2) | 588 | 168 | ||||||
8 | t0,3 | Runcinated 7-simplex (spo) | (0,0,0,0,1,1,1,2) | 100 | 756 | 2548 | 4830 | 4760 | 2100 | 280 | |
9 | t1,3 | Bicantellated 7-simplex (sabro) | (0,0,0,0,1,1,2,2) | 2520 | 420 | ||||||
10 | t2,3 | Tritruncated 7-simplex (tattoc) | (0,0,0,0,1,2,2,2) | 980 | 280 | ||||||
11 | t0,4 | Stericated 7-simplex (sco) | (0,0,0,1,1,1,1,2) | 2240 | 280 | ||||||
12 | t1,4 | Biruncinated 7-simplex (sibpo) | (0,0,0,1,1,1,2,2) | 4200 | 560 | ||||||
13 | t2,4 | Tricantellated 7-simplex (stiroh) | (0,0,0,1,1,2,2,2) | 3360 | 560 | ||||||
14 | t0,5 | Pentellated 7-simplex (seto) | (0,0,1,1,1,1,1,2) | 1260 | 168 | ||||||
15 | t1,5 | Bistericated 7-simplex (sabach) | (0,0,1,1,1,1,2,2) | 3360 | 420 | ||||||
16 | t0,6 | Hexicated 7-simplex (suph) | (0,1,1,1,1,1,1,2) | 336 | 56 | ||||||
17 | t0,1,2 | Cantitruncated 7-simplex (garo) | (0,0,0,0,0,1,2,3) | 1176 | 336 | ||||||
18 | t0,1,3 | Runcitruncated 7-simplex (patto) | (0,0,0,0,1,1,2,3) | 4620 | 840 | ||||||
19 | t0,2,3 | Runcicantellated 7-simplex (paro) | (0,0,0,0,1,2,2,3) | 3360 | 840 | ||||||
20 | t1,2,3 | Bicantitruncated 7-simplex (gabro) | (0,0,0,0,1,2,3,3) | 2940 | 840 | ||||||
21 | t0,1,4 | Steritruncated 7-simplex (cato) | (0,0,0,1,1,1,2,3) | 7280 | 1120 | ||||||
22 | t0,2,4 | Stericantellated 7-simplex (caro) | (0,0,0,1,1,2,2,3) | 10080 | 1680 | ||||||
23 | t1,2,4 | Biruncitruncated 7-simplex (bipto) | (0,0,0,1,1,2,3,3) | 8400 | 1680 | ||||||
24 | t0,3,4 | Steriruncinated 7-simplex (cepo) | (0,0,0,1,2,2,2,3) | 5040 | 1120 | ||||||
25 | t1,3,4 | Biruncicantellated 7-simplex (bipro) | (0,0,0,1,2,2,3,3) | 7560 | 1680 | ||||||
26 | t2,3,4 | Tricantitruncated 7-simplex (gatroh) | (0,0,0,1,2,3,3,3) | 3920 | 1120 | ||||||
27 | t0,1,5 | Pentitruncated 7-simplex (teto) | (0,0,1,1,1,1,2,3) | 5460 | 840 | ||||||
28 | t0,2,5 | Penticantellated 7-simplex (tero) | (0,0,1,1,1,2,2,3) | 11760 | 1680 | ||||||
29 | t1,2,5 | Bisteritruncated 7-simplex (bacto) | (0,0,1,1,1,2,3,3) | 9240 | 1680 | ||||||
30 | t0,3,5 | Pentiruncinated 7-simplex (tepo) | (0,0,1,1,2,2,2,3) | 10920 | 1680 | ||||||
31 | t1,3,5 | Bistericantellated 7-simplex (bacroh) | (0,0,1,1,2,2,3,3) | 15120 | 2520 | ||||||
32 | t0,4,5 | Pentistericated 7-simplex (teco) | (0,0,1,2,2,2,2,3) | 4200 | 840 | ||||||
33 | t0,1,6 | Hexitruncated 7-simplex (puto) | (0,1,1,1,1,1,2,3) | 1848 | 336 | ||||||
34 | t0,2,6 | Hexicantellated 7-simplex (puro) | (0,1,1,1,1,2,2,3) | 5880 | 840 | ||||||
35 | t0,3,6 | Hexiruncinated 7-simplex (puph) | (0,1,1,1,2,2,2,3) | 8400 | 1120 | ||||||
36 | t0,1,2,3 | Runcicantitruncated 7-simplex (gapo) | (0,0,0,0,1,2,3,4) | 5880 | 1680 | ||||||
37 | t0,1,2,4 | Stericantitruncated 7-simplex (cagro) | (0,0,0,1,1,2,3,4) | 16800 | 3360 | ||||||
38 | t0,1,3,4 | Steriruncitruncated 7-simplex (capto) | (0,0,0,1,2,2,3,4) | 13440 | 3360 | ||||||
39 | t0,2,3,4 | Steriruncicantellated 7-simplex (capro) | (0,0,0,1,2,3,3,4) | 13440 | 3360 | ||||||
40 | t1,2,3,4 | Biruncicantitruncated 7-simplex (gibpo) | (0,0,0,1,2,3,4,4) | 11760 | 3360 | ||||||
41 | t0,1,2,5 | Penticantitruncated 7-simplex (tegro) | (0,0,1,1,1,2,3,4) | 18480 | 3360 | ||||||
42 | t0,1,3,5 | Pentiruncitruncated 7-simplex (tapto) | (0,0,1,1,2,2,3,4) | 27720 | 5040 | ||||||
43 | t0,2,3,5 | Pentiruncicantellated 7-simplex (tapro) | (0,0,1,1,2,3,3,4) | 25200 | 5040 | ||||||
44 | t1,2,3,5 | Bistericantitruncated 7-simplex (bacogro) | (0,0,1,1,2,3,4,4) | 22680 | 5040 | ||||||
45 | t0,1,4,5 | Pentisteritruncated 7-simplex (tecto) | (0,0,1,2,2,2,3,4) | 15120 | 3360 | ||||||
46 | t0,2,4,5 | Pentistericantellated 7-simplex (tecro) | (0,0,1,2,2,3,3,4) | 25200 | 5040 | ||||||
47 | t1,2,4,5 | Bisteriruncitruncated 7-simplex (bicpath) | (0,0,1,2,2,3,4,4) | 20160 | 5040 | ||||||
48 | t0,3,4,5 | Pentisteriruncinated 7-simplex (tacpo) | (0,0,1,2,3,3,3,4) | 15120 | 3360 | ||||||
49 | t0,1,2,6 | Hexicantitruncated 7-simplex (pugro) | (0,1,1,1,1,2,3,4) | 8400 | 1680 | ||||||
50 | t0,1,3,6 | Hexiruncitruncated 7-simplex (pugato) | (0,1,1,1,2,2,3,4) | 20160 | 3360 | ||||||
51 | t0,2,3,6 | Hexiruncicantellated 7-simplex (pugro) | (0,1,1,1,2,3,3,4) | 16800 | 3360 | ||||||
52 | t0,1,4,6 | Hexisteritruncated 7-simplex (pucto) | (0,1,1,2,2,2,3,4) | 20160 | 3360 | ||||||
53 | t0,2,4,6 | Hexistericantellated 7-simplex (pucroh) | (0,1,1,2,2,3,3,4) | 30240 | 5040 | ||||||
54 | t0,1,5,6 | Hexipentitruncated 7-simplex (putath) | (0,1,2,2,2,2,3,4) | 8400 | 1680 | ||||||
55 | t0,1,2,3,4 | Steriruncicantitruncated 7-simplex (gecco) | (0,0,0,1,2,3,4,5) | 23520 | 6720 | ||||||
56 | t0,1,2,3,5 | Pentiruncicantitruncated 7-simplex (tegapo) | (0,0,1,1,2,3,4,5) | 45360 | 10080 | ||||||
57 | t0,1,2,4,5 | Pentistericantitruncated 7-simplex (tecagro) | (0,0,1,2,2,3,4,5) | 40320 | 10080 | ||||||
58 | t0,1,3,4,5 | Pentisteriruncitruncated 7-simplex (tacpeto) | (0,0,1,2,3,3,4,5) | 40320 | 10080 | ||||||
59 | t0,2,3,4,5 | Pentisteriruncicantellated 7-simplex (tacpro) | (0,0,1,2,3,4,4,5) | 40320 | 10080 | ||||||
60 | t1,2,3,4,5 | Bisteriruncicantitruncated 7-simplex (gabach) | (0,0,1,2,3,4,5,5) | 35280 | 10080 | ||||||
61 | t0,1,2,3,6 | Hexiruncicantitruncated 7-simplex (pugopo) | (0,1,1,1,2,3,4,5) | 30240 | 6720 | ||||||
62 | t0,1,2,4,6 | Hexistericantitruncated 7-simplex (pucagro) | (0,1,1,2,2,3,4,5) | 50400 | 10080 | ||||||
63 | t0,1,3,4,6 | Hexisteriruncitruncated 7-simplex (pucpato) | (0,1,1,2,3,3,4,5) | 45360 | 10080 | ||||||
64 | t0,2,3,4,6 | Hexisteriruncicantellated 7-simplex (pucproh) | (0,1,1,2,3,4,4,5) | 45360 | 10080 | ||||||
65 | t0,1,2,5,6 | Hexipenticantitruncated 7-simplex (putagro) | (0,1,2,2,2,3,4,5) | 30240 | 6720 | ||||||
66 | t0,1,3,5,6 | Hexipentiruncitruncated 7-simplex (putpath) | (0,1,2,2,3,3,4,5) | 50400 | 10080 | ||||||
67 | t0,1,2,3,4,5 | Pentisteriruncicantitruncated 7-simplex (geto) | (0,0,1,2,3,4,5,6) | 70560 | 20160 | ||||||
68 | t0,1,2,3,4,6 | Hexisteriruncicantitruncated 7-simplex (pugaco) | (0,1,1,2,3,4,5,6) | 80640 | 20160 | ||||||
69 | t0,1,2,3,5,6 | Hexipentiruncicantitruncated 7-simplex (putgapo) | (0,1,2,2,3,4,5,6) | 80640 | 20160 | ||||||
70 | t0,1,2,4,5,6 | Hexipentistericantitruncated 7-simplex (putcagroh) | (0,1,2,3,3,4,5,6) | 80640 | 20160 | ||||||
71 | t0,1,2,3,4,5,6 | Omnitruncated 7-simplex (guph) | (0,1,2,3,4,5,6,7) | 141120 | 40320 |
The B7 family edit
The B7 family has symmetry of order 645120 (7 factorial x 27).
There are 127 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Johnson and Bowers names.
See also a list of B7 polytopes for symmetric Coxeter plane graphs of these polytopes.
B7 uniform polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
# | Coxeter-Dynkin diagram t-notation | Name (BSA) | Base point | Element counts | |||||||
6 | 5 | 4 | 3 | 2 | 1 | 0 | |||||
1 | t0{3,3,3,3,3,4} | 7-orthoplex (zee) | (0,0,0,0,0,0,1)√2 | 128 | 448 | 672 | 560 | 280 | 84 | 14 | |
2 | t1{3,3,3,3,3,4} | Rectified 7-orthoplex (rez) | (0,0,0,0,0,1,1)√2 | 142 | 1344 | 3360 | 3920 | 2520 | 840 | 84 | |
3 | t2{3,3,3,3,3,4} | Birectified 7-orthoplex (barz) | (0,0,0,0,1,1,1)√2 | 142 | 1428 | 6048 | 10640 | 8960 | 3360 | 280 | |
4 | t3{4,3,3,3,3,3} | Trirectified 7-cube (sez) | (0,0,0,1,1,1,1)√2 | 142 | 1428 | 6328 | 14560 | 15680 | 6720 | 560 | |
5 | t2{4,3,3,3,3,3} | Birectified 7-cube (bersa) | (0,0,1,1,1,1,1)√2 | 142 | 1428 | 5656 | 11760 | 13440 | 6720 | 672 | |
6 | t1{4,3,3,3,3,3} | Rectified 7-cube (rasa) | (0,1,1,1,1,1,1)√2 | 142 | 980 | 2968 | 5040 | 5152 | 2688 | 448 | |
7 | t0{4,3,3,3,3,3} | 7-cube (hept) | (0,0,0,0,0,0,0)√2 + (1,1,1,1,1,1,1) | 14 | 84 | 280 | 560 | 672 | 448 | 128 | |
8 | t0,1{3,3,3,3,3,4} | Truncated 7-orthoplex (Taz) | (0,0,0,0,0,1,2)√2 | 142 | 1344 | 3360 | 4760 | 2520 | 924 | 168 | |
9 | t0,2{3,3,3,3,3,4} | Cantellated 7-orthoplex (Sarz) | (0,0,0,0,1,1,2)√2 | 226 | 4200 | 15456 | 24080 | 19320 | 7560 | 840 | |
10 | t1,2{3,3,3,3,3,4} | Bitruncated 7-orthoplex (Botaz) | (0,0,0,0,1,2,2)√2 | 4200 | 840 | ||||||
11 | t0,3{3,3,3,3,3,4} | Runcinated 7-orthoplex (Spaz) | (0,0,0,1,1,1,2)√2 | 23520 | 2240 | ||||||
12 | t1,3{3,3,3,3,3,4} | Bicantellated 7-orthoplex (Sebraz) | (0,0,0,1,1,2,2)√2 | 26880 | 3360 | ||||||
13 | t2,3{3,3,3,3,3,4} | Tritruncated 7-orthoplex (Totaz) | (0,0,0,1,2,2,2)√2 | 10080 | 2240 | ||||||
14 | t0,4{3,3,3,3,3,4} | Stericated 7-orthoplex (Scaz) | (0,0,1,1,1,1,2)√2 | 33600 | 3360 | ||||||
15 | t1,4{3,3,3,3,3,4} | Biruncinated 7-orthoplex (Sibpaz) | (0,0,1,1,1,2,2)√2 | 60480 | 6720 | ||||||
16 | t2,4{4,3,3,3,3,3} | Tricantellated 7-cube (Strasaz) | (0,0,1,1,2,2,2)√2 | 47040 | 6720 | ||||||
17 | t2,3{4,3,3,3,3,3} | Tritruncated 7-cube (Tatsa) | (0,0,1,2,2,2,2)√2 | 13440 | 3360 | ||||||
18 | t0,5{3,3,3,3,3,4} | Pentellated 7-orthoplex (Staz) | (0,1,1,1,1,1,2)√2 | 20160 | 2688 | ||||||
19 | t1,5{4,3,3,3,3,3} | Bistericated 7-cube (Sabcosaz) | (0,1,1,1,1,2,2)√2 | 53760 | 6720 | ||||||
20 | uniform, polytope, polytope, graphs, three, regular, related, uniform, polytopes, simplex, rectified, simplex, truncated, simplex, cantellated, simplex, runcinated, simplex, stericated, simplex, pentellated, simplex, hexicated, simplex, orthoplex, truncated, o. Polytope Graphs of three regular and related uniform polytopes 7 simplex Rectified 7 simplex Truncated 7 simplex Cantellated 7 simplex Runcinated 7 simplex Stericated 7 simplex Pentellated 7 simplex Hexicated 7 simplex 7 orthoplex Truncated 7 orthoplex Rectified 7 orthoplex Cantellated 7 orthoplex Runcinated 7 orthoplex Stericated 7 orthoplex Pentellated 7 orthoplex Hexicated 7 cube Pentellated 7 cube Stericated 7 cube Cantellated 7 cube Runcinated 7 cube 7 cube Truncated 7 cube Rectified 7 cube 7 demicube Cantic 7 cube Runcic 7 cube Steric 7 cube Pentic 7 cube Hexic 7 cube 321 231 132 In seven dimensional geometry a 7 polytope is a polytope contained by 6 polytope facets Each 5 polytope ridge being shared by exactly two 6 polytope facets A uniform 7 polytope is one whose symmetry group is transitive on vertices and whose facets are uniform 6 polytopes Contents 1 Regular 7 polytopes 2 Characteristics 3 Uniform 7 polytopes by fundamental Coxeter groups 4 The A7 family 5 The B7 family 6 The D7 family 7 The E7 family 8 Regular and uniform honeycombs 8 1 Regular and uniform hyperbolic honeycombs 9 Notes on the Wythoff construction for the uniform 7 polytopes 10 References 11 External links Regular 7 polytopes edit Regular 7 polytopes are represented by the Schlafli symbol p q r s t u with u p q r s t 6 polytopes facets around each 4 face There are exactly three such convex regular 7 polytopes 3 3 3 3 3 3 7 simplex 4 3 3 3 3 3 7 cube 3 3 3 3 3 4 7 orthoplex There are no nonconvex regular 7 polytopes Characteristics edit The topology of any given 7 polytope is defined by its Betti numbers and torsion coefficients 91 1 93 The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions whatever their underlying topology This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers 91 1 93 Similarly the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes and this led to the use of torsion coefficients 91 1 93 Uniform 7 polytopes by fundamental Coxeter groups edit Uniform 7 polytopes with reflective symmetry can be generated by these four Coxeter groups represented by permutations of rings of the Coxeter Dynkin diagrams Coxeter group Regular and semiregular forms Uniform count 1 A7 36 7 simplex 36 71 2 B7 4 35 7 cube 4 35 7 orthoplex 35 4 7 demicube h 4 35 127 32 3 D7 33 1 1 7 demicube 3 34 1 7 orthoplex 34 31 1 95 0 unique 4 E7 33 2 1 321 132 231 127 Prismatic finite Coxeter groups Coxeter group Coxeter diagram 6 1 1 A6A1 35 160 2 BC6A1 4 34 160 3 D6A1 33 1 1 160 4 E6A1 32 2 1 160 5 2 1 A5I2 p 3 3 3 p 2 BC5I2 p 4 3 3 p 3 D5I2 p 32 1 1 p 5 1 1 1 A5A12 3 3 3 160 2 2 BC5A12 4 3 3 160 2 3 D5A12 32 1 1 160 2 4 3 1 A4A3 3 3 3 3 3 2 A4B3 3 3 3 4 3 3 A4H3 3 3 3 5 3 4 BC4A3 4 3 3 3 3 5 BC4B3 4 3 3 4 3 6 BC4H3 4 3 3 5 3 7 H4A3 5 3 3 3 3 8 H4B3 5 3 3 4 3 9 H4H3 5 3 3 5 3 10 F4A3 3 4 3 3 3 11 F4B3 3 4 3 4 3 12 F4H3 3 4 3 5 3 13 D4A3 31 1 1 3 3 14 D4B3 31 1 1 4 3 15 D4H3 31 1 1 5 3 4 2 1 1 A4I2 p A1 3 3 3 p 160 2 BC4I2 p A1 4 3 3 p 160 3 F4I2 p A1 3 4 3 p 160 4 H4I2 p A1 5 3 3 p 160 5 D4I2 p A1 31 1 1 p 160 4 1 1 1 1 A4A13 3 3 3 160 3 2 BC4A13 4 3 3 160 3 3 F4A13 3 4 3 160 3 4 H4A13 5 3 3 160 3 5 D4A13 31 1 1 160 3 3 3 1 1 A3A3A1 3 3 3 3 160 2 A3B3A1 3 3 4 3 160 3 A3H3A1 3 3 5 3 160 4 BC3B3A1 4 3 4 3 160 5 BC3H3A1 4 3 5 3 160 6 H3A3A1 5 3 5 3 160 3 2 2 1 A3I2 p I2 q 3 3 p q 2 BC3I2 p I2 q 4 3 p q 3 H3I2 p I2 q 5 3 p q 3 2 1 1 1 A3I2 p A12 3 3 p 160 2 2 BC3I2 p A12 4 3 p 160 2 3 H3I2 p A12 5 3 p 160 2 3 1 1 1 1 1 A3A14 3 3 160 4 2 BC3A14 4 3 160 4 3 H3A14 5 3 160 4 2 2 2 1 1 I2 p I2 q I2 r A1 p q r 160 2 2 1 1 1 1 I2 p I2 q A13 p q 160 3 2 1 1 1 1 1 1 I2 p A15 p 160 5 1 1 1 1 1 1 1 1 A17 160 7 The A7 family edit The A7 family has symmetry of order 40320 8 factorial There are 71 64 8 1 forms based on all permutations of the Coxeter Dynkin diagrams with one or more rings All 71 are enumerated below Norman Johnson s truncation names are given Bowers names and acronym are also given for cross referencing See also a list of A7 polytopes for symmetric Coxeter plane graphs of these polytopes A7 uniform polytopes Coxeter Dynkin diagram Truncationindices Johnson nameBowers name and acronym Basepoint Element counts 6 5 4 3 2 1 0 1 t0 7 simplex oca 0 0 0 0 0 0 0 1 8 28 56 70 56 28 8 2 t1 Rectified 7 simplex roc 0 0 0 0 0 0 1 1 16 84 224 350 336 168 28 3 t2 Birectified 7 simplex broc 0 0 0 0 0 1 1 1 16 112 392 770 840 420 56 4 t3 Trirectified 7 simplex he 0 0 0 0 1 1 1 1 16 112 448 980 1120 560 70 5 t0 1 Truncated 7 simplex toc 0 0 0 0 0 0 1 2 16 84 224 350 336 196 56 6 t0 2 Cantellated 7 simplex saro 0 0 0 0 0 1 1 2 44 308 980 1750 1876 1008 168 7 t1 2 Bitruncated 7 simplex bittoc 0 0 0 0 0 1 2 2 588 168 8 t0 3 Runcinated 7 simplex spo 0 0 0 0 1 1 1 2 100 756 2548 4830 4760 2100 280 9 t1 3 Bicantellated 7 simplex sabro 0 0 0 0 1 1 2 2 2520 420 10 t2 3 Tritruncated 7 simplex tattoc 0 0 0 0 1 2 2 2 980 280 11 t0 4 Stericated 7 simplex sco 0 0 0 1 1 1 1 2 2240 280 12 t1 4 Biruncinated 7 simplex sibpo 0 0 0 1 1 1 2 2 4200 560 13 t2 4 Tricantellated 7 simplex stiroh 0 0 0 1 1 2 2 2 3360 560 14 t0 5 Pentellated 7 simplex seto 0 0 1 1 1 1 1 2 1260 168 15 t1 5 Bistericated 7 simplex sabach 0 0 1 1 1 1 2 2 3360 420 16 t0 6 Hexicated 7 simplex suph 0 1 1 1 1 1 1 2 336 56 17 t0 1 2 Cantitruncated 7 simplex garo 0 0 0 0 0 1 2 3 1176 336 18 t0 1 3 Runcitruncated 7 simplex patto 0 0 0 0 1 1 2 3 4620 840 19 t0 2 3 Runcicantellated 7 simplex paro 0 0 0 0 1 2 2 3 3360 840 20 t1 2 3 Bicantitruncated 7 simplex gabro 0 0 0 0 1 2 3 3 2940 840 21 t0 1 4 Steritruncated 7 simplex cato 0 0 0 1 1 1 2 3 7280 1120 22 t0 2 4 Stericantellated 7 simplex caro 0 0 0 1 1 2 2 3 10080 1680 23 t1 2 4 Biruncitruncated 7 simplex bipto 0 0 0 1 1 2 3 3 8400 1680 24 t0 3 4 Steriruncinated 7 simplex cepo 0 0 0 1 2 2 2 3 5040 1120 25 t1 3 4 Biruncicantellated 7 simplex bipro 0 0 0 1 2 2 3 3 7560 1680 26 t2 3 4 Tricantitruncated 7 simplex gatroh 0 0 0 1 2 3 3 3 3920 1120 27 t0 1 5 Pentitruncated 7 simplex teto 0 0 1 1 1 1 2 3 5460 840 28 t0 2 5 Penticantellated 7 simplex tero 0 0 1 1 1 2 2 3 11760 1680 29 t1 2 5 Bisteritruncated 7 simplex bacto 0 0 1 1 1 2 3 3 9240 1680 30 t0 3 5 Pentiruncinated 7 simplex tepo 0 0 1 1 2 2 2 3 10920 1680 31 t1 3 5 Bistericantellated 7 simplex bacroh 0 0 1 1 2 2 3 3 15120 2520 32 t0 4 5 Pentistericated 7 simplex teco 0 0 1 2 2 2 2 3 4200 840 33 t0 1 6 Hexitruncated 7 simplex puto 0 1 1 1 1 1 2 3 1848 336 34 t0 2 6 Hexicantellated 7 simplex puro 0 1 1 1 1 2 2 3 5880 840 35 t0 3 6 Hexiruncinated 7 simplex puph 0 1 1 1 2 2 2 3 8400 1120 36 t0 1 2 3 Runcicantitruncated 7 simplex gapo 0 0 0 0 1 2 3 4 5880 1680 37 t0 1 2 4 Stericantitruncated 7 simplex cagro 0 0 0 1 1 2 3 4 16800 3360 38 t0 1 3 4 Steriruncitruncated 7 simplex capto 0 0 0 1 2 2 3 4 13440 3360 39 t0 2 3 4 Steriruncicantellated 7 simplex capro 0 0 0 1 2 3 3 4 13440 3360 40 t1 2 3 4 Biruncicantitruncated 7 simplex gibpo 0 0 0 1 2 3 4 4 11760 3360 41 t0 1 2 5 Penticantitruncated 7 simplex tegro 0 0 1 1 1 2 3 4 18480 3360 42 t0 1 3 5 Pentiruncitruncated 7 simplex tapto 0 0 1 1 2 2 3 4 27720 5040 43 t0 2 3 5 Pentiruncicantellated 7 simplex tapro 0 0 1 1 2 3 3 4 25200 5040 44 t1 2 3 5 Bistericantitruncated 7 simplex bacogro 0 0 1 1 2 3 4 4 22680 5040 45 t0 1 4 5 Pentisteritruncated 7 simplex tecto 0 0 1 2 2 2 3 4 15120 3360 46 t0 2 4 5 Pentistericantellated 7 simplex tecro 0 0 1 2 2 3 3 4 25200 5040 47 t1 2 4 5 Bisteriruncitruncated 7 simplex bicpath 0 0 1 2 2 3 4 4 20160 5040 48 t0 3 4 5 Pentisteriruncinated 7 simplex tacpo 0 0 1 2 3 3 3 4 15120 3360 49 t0 1 2 6 Hexicantitruncated 7 simplex pugro 0 1 1 1 1 2 3 4 8400 1680 50 t0 1 3 6 Hexiruncitruncated 7 simplex pugato 0 1 1 1 2 2 3 4 20160 3360 51 t0 2 3 6 Hexiruncicantellated 7 simplex pugro 0 1 1 1 2 3 3 4 16800 3360 52 t0 1 4 6 Hexisteritruncated 7 simplex pucto 0 1 1 2 2 2 3 4 20160 3360 53 t0 2 4 6 Hexistericantellated 7 simplex pucroh 0 1 1 2 2 3 3 4 30240 5040 54 t0 1 5 6 Hexipentitruncated 7 simplex putath 0 1 2 2 2 2 3 4 8400 1680 55 t0 1 2 3 4 Steriruncicantitruncated 7 simplex gecco 0 0 0 1 2 3 4 5 23520 6720 56 t0 1 2 3 5 Pentiruncicantitruncated 7 simplex tegapo 0 0 1 1 2 3 4 5 45360 10080 57 t0 1 2 4 5 Pentistericantitruncated 7 simplex tecagro 0 0 1 2 2 3 4 5 40320 10080 58 t0 1 3 4 5 Pentisteriruncitruncated 7 simplex tacpeto 0 0 1 2 3 3 4 5 40320 10080 59 t0 2 3 4 5 Pentisteriruncicantellated 7 simplex tacpro 0 0 1 2 3 4 4 5 40320 10080 60 t1 2 3 4 5 Bisteriruncicantitruncated 7 simplex gabach 0 0 1 2 3 4 5 5 35280 10080 61 t0 1 2 3 6 Hexiruncicantitruncated 7 simplex pugopo 0 1 1 1 2 3 4 5 30240 6720 62 t0 1 2 4 6 Hexistericantitruncated 7 simplex pucagro 0 1 1 2 2 3 4 5 50400 10080 63 t0 1 3 4 6 Hexisteriruncitruncated 7 simplex pucpato 0 1 1 2 3 3 4 5 45360 10080 64 t0 2 3 4 6 Hexisteriruncicantellated 7 simplex pucproh 0 1 1 2 3 4 4 5 45360 10080 65 t0 1 2 5 6 Hexipenticantitruncated 7 simplex putagro 0 1 2 2 2 3 4 5 30240 6720 66 t0 1 3 5 6 Hexipentiruncitruncated 7 simplex putpath 0 1 2 2 3 3 4 5 50400 10080 67 t0 1 2 3 4 5 Pentisteriruncicantitruncated 7 simplex geto 0 0 1 2 3 4 5 6 70560 20160 68 t0 1 2 3 4 6 Hexisteriruncicantitruncated 7 simplex pugaco 0 1 1 2 3 4 5 6 80640 20160 69 t0 1 2 3 5 6 Hexipentiruncicantitruncated 7 simplex putgapo 0 1 2 2 3 4 5 6 80640 20160 70 t0 1 2 4 5 6 Hexipentistericantitruncated 7 simplex putcagroh 0 1 2 3 3 4 5 6 80640 20160 71 t0 1 2 3 4 5 6 Omnitruncated 7 simplex guph 0 1 2 3 4 5 6 7 141120 40320 The B7 family edit The B7 family has symmetry of order 645120 7 factorial x 27 There are 127 forms based on all permutations of the Coxeter Dynkin diagrams with one or more rings Johnson and Bowers names See also a list of B7 polytopes for symmetric Coxeter plane graphs of these polytopes B7 uniform polytopes Coxeter Dynkin diagramt notation Name BSA Base point Element counts 6 5 4 3 2 1 0 1 t0 3 3 3 3 3 4 7 orthoplex zee 0 0 0 0 0 0 1 2 128 448 672 560 280 84 14 2 t1 3 3 3 3 3 4 Rectified 7 orthoplex rez 0 0 0 0 0 1 1 2 142 1344 3360 3920 2520 840 84 3 t2 3 3 3 3 3 4 Birectified 7 orthoplex barz 0 0 0 0 1 1 1 2 142 1428 6048 10640 8960 3360 280 4 t3 4 3 3 3 3 3 Trirectified 7 cube sez 0 0 0 1 1 1 1 2 142 1428 6328 14560 15680 6720 560 5 t2 4 3 3 3 3 3 Birectified 7 cube bersa 0 0 1 1 1 1 1 2 142 1428 5656 11760 13440 6720 672 6 t1 4 3 3 3 3 3 Rectified 7 cube rasa 0 1 1 1 1 1 1 2 142 980 2968 5040 5152 2688 448 7 t0 4 3 3 3 3 3 7 cube hept 0 0 0 0 0 0 0 2 1 1 1 1 1 1 1 14 84 280 560 672 448 128 8 t0 1 3 3 3 3 3 4 Truncated 7 orthoplex Taz 0 0 0 0 0 1 2 2 142 1344 3360 4760 2520 924 168 9 t0 2 3 3 3 3 3 4 Cantellated 7 orthoplex Sarz 0 0 0 0 1 1 2 2 226 4200 15456 24080 19320 7560 840 10 t1 2 3 3 3 3 3 4 Bitruncated 7 orthoplex Botaz 0 0 0 0 1 2 2 2 4200 840 11 t0 3 3 3 3 3 3 4 Runcinated 7 orthoplex Spaz 0 0 0 1 1 1 2 2 23520 2240 12 t1 3 3 3 3 3 3 4 Bicantellated 7 orthoplex Sebraz 0 0 0 1 1 2 2 2 26880 3360 13 t2 3 3 3 3 3 3 4 Tritruncated 7 orthoplex Totaz 0 0 0 1 2 2 2 2 10080 2240 14 t0 4 3 3 3 3 3 4 Stericated 7 orthoplex Scaz 0 0 1 1 1 1 2 2 33600 3360 15 t1 4 3 3 3 3 3 4 Biruncinated 7 orthoplex Sibpaz 0 0 1 1 1 2 2 2 60480 6720 16 t2 4 4 3 3 3 3 3 Tricantellated 7 cube Strasaz 0 0 1 1 2 2 2 2 47040 6720 17 t2 3 4 3 3 3 3 3 Tritruncated 7 cube Tatsa 0 0 1 2 2 2 2 2 13440 3360 18 t0 5 3 3 3 3 3 4 Pentellated 7 orthoplex Staz 0 1 1 1 1 1 2 2 20160 2688 19 t1 5 4 3 3 3 3 3 Bistericated 7 cube Sabcosaz 0 1 1 1 1 2 2 2 53760 6720 20 img dat, wikipedia, wiki, book, books, library, article, read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games. |