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 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:
There are no nonconvex regular 7-polytopes.
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 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 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 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 | t1,4{4,3,3,3,3,3} |
Biruncinated 7-cube (Sibposa) | (0,1,1,1,2,2,2)√2 | 67200 | 8960 | ||||||
21 | t1,3{4,3,3,3,3,3} |
Bicantellated 7-cube (Sibrosa) | (0,1,1,2,2,2,2)√2 | 40320 | 6720 | ||||||
22 | t1,2{4,3,3,3,3,3} |
Bitruncated 7-cube (Betsa) | (0,1,2,2,2,2,2)√2 | 9408 | 2688 | ||||||
23 | t0,6{4,3,3,3,3,3} |
Hexicated 7-cube (Supposaz) | (0,0,0,0,0,0,1)√2 + (1,1,1,1,1,1,1) | 5376 | 896 | ||||||
24 | t0,5{4,3,3,3,3,3} |
Pentellated 7-cube (Stesa) | (0,0,0,0,0,1,1)√2 + (1,1,1,1,1,1,1) | 20160 | 2688 | ||||||
25 | t0,4{4,3,3,3,3,3} |
Stericated 7-cube (Scosa) | (0,0,0,0,1,1,1)√2 + (1,1,1,1,1,1,1) | 35840 | 4480 | ||||||
26 | t0,3{4,3,3,3,3,3} |
Runcinated 7-cube (Spesa) | (0,0,0,1,1,1,1)√2 + (1,1,1,1,1,1,1) | 33600 | 4480 | ||||||
27 | t0,2{4,3,3,3,3,3} |
Cantellated 7-cube (Sersa) | (0,0,1,1,1,1,1)√2 + (1,1,1,1,1,1,1) | 16128 | 2688 | ||||||
28 | t0,1{4,3,3,3,3,3} |
Truncated 7-cube (Tasa) | (0,1,1,1,1,1,1)√2 + (1,1,1,1,1,1,1) | 142 | 980 | 2968 | 5040 | 5152 | 3136 | 896 | |
29 | t0,1,2{3,3,3,3,3,4} |
Cantitruncated 7-orthoplex (Garz) | (0,1,2,3,3,3,3)√2 | 8400 | 1680 | ||||||
30 | t0,1,3{3,3,3,3,3,4} |
Runcitruncated 7-orthoplex (Potaz) | (0,1,2,2,3,3,3)√2 | 50400 | 6720 | ||||||
31 | t0,2,3{3,3,3,3,3,4} |
Runcicantellated 7-orthoplex (Parz) | (0,1,1,2,3,3,3)√2 | 33600 | 6720 | ||||||
32 | t1,2,3{3,3,3,3,3,4} |
Bicantitruncated 7-orthoplex (Gebraz) | (0,0,1,2,3,3,3)√2 | 30240 | 6720 | ||||||
33 | t0,1,4{3,3,3,3,3,4} |
Steritruncated 7-orthoplex (Catz) | (0,0,1,1,1,2,3)√2 | 107520 | 13440 | ||||||
34 | t0,2,4{3,3,3,3,3,4} |
Stericantellated 7-orthoplex (Craze) | (0,0,1,1,2,2,3)√2 | 141120 | 20160 | ||||||
35 | t1,2,4{3,3,3,3,3,4} |
Biruncitruncated 7-orthoplex (Baptize) | (0,0,1,1,2,3,3)√2 | 120960 | 20160 | ||||||
36 | t0,3,4{3,3,3,3,3,4} |
Steriruncinated 7-orthoplex (Copaz) | (0,1,1,1,2,3,3)√2 | 67200 | 13440 | ||||||
37 | t1,3,4{3,3,3,3,3,4} |
Biruncicantellated 7-orthoplex (Boparz) | (0,0,1,2,2,3,3)√2 | 100800 | 20160 | ||||||
38 | t2,3,4{4,3,3,3,3,3} |
Tricantitruncated 7-cube (Gotrasaz) | (0,0,0,1,2,3,3)√2 | 53760 | 13440 | ||||||
39 | t0,1,5{3,3,3,3,3,4} |
Pentitruncated 7-orthoplex (Tetaz) | (0,1,1,1,1,2,3)√2 | 87360 | 13440 | ||||||
40 | t0,2,5{3,3,3,3,3,4} |
Penticantellated 7-orthoplex (Teroz) | (0,1,1,1,2,2,3)√2 | 188160 | 26880 | ||||||
41 | t1,2,5{3,3,3,3,3,4} |
Bisteritruncated 7-orthoplex (Boctaz) | (0,1,1,1,2,3,3)√2 | 147840 | 26880 | ||||||
42 | t0,3,5{3,3,3,3,3,4} |
Pentiruncinated 7-orthoplex (Topaz) | (0,1,1,2,2,2,3)√2 | 174720 | 26880 | ||||||
43 | t1,3,5{4,3,3,3,3,3} |
Bistericantellated 7-cube (Bacresaz) | (0,1,1,2,2,3,3)√2 | 241920 | 40320 | ||||||
44 | t1,3,4{4,3,3,3,3,3} |
Biruncicantellated 7-cube (Bopresa) | (0,1,1,2,3,3,3)√2 | 120960 | 26880 | ||||||
45 | t0,4,5{3,3,3,3,3,4} |
Pentistericated 7-orthoplex (Tocaz) | (0,1,2,2,2,2,3)√2 | 67200 | 13440 | ||||||
46 | t1,2,5{4,3,3,3,3,3} |
Bisteritruncated 7-cube (Bactasa) | (0,1,2,2,2,3,3)√2 | 147840 | 26880 | ||||||
47 | t1,2,4{4,3,3,3,3,3} |
Biruncitruncated 7-cube (Biptesa) | (0,1,2,2,3,3,3)√2 | 134400 | 26880 | ||||||
48 | t1,2,3{4,3,3,3,3,3} |
Bicantitruncated 7-cube (Gibrosa) | (0,1,2,3,3,3,3)√2 | 47040 | 13440 | ||||||
49 | t0,1,6{3,3,3,3,3,4} |
Hexitruncated 7-orthoplex (Putaz) | (0,0,0,0,0,1,2)√2 + (1,1,1,1,1,1,1) | 29568 | 5376 | ||||||
50 | t0,2,6{3,3,3,3,3,4} |
Hexicantellated 7-orthoplex (Puraz) | (0,0,0,0,1,1,2)√2 + (1,1,1,1,1,1,1) | 94080 | 13440 | ||||||
51 | t0,4,5{4,3,3,3,3,3} |
Pentistericated 7-cube (Tacosa) | (0,0,0,0,1,2,2)√2 + (1,1,1,1,1,1,1) | 67200 | 13440 | ||||||
52 | t0,3,6{4,3,3,3,3,3} |
Hexiruncinated 7-cube (Pupsez) | (0,0,0,1,1,1,2)√2 + (1,1,1,1,1,1,1) | 134400 | 17920 | ||||||
53 | t0,3,5{4,3,3,3,3,3} |
Pentiruncinated 7-cube (Tapsa) | (0,0,0,1,1,2,2)√2 + (1,1,1,1,1,1,1) | 174720 | 26880 | ||||||
54 | t0,3,4{4,3,3,3,3,3} |
Steriruncinated 7-cube (Capsa) | (0,0,0,1,2,2,2)√2 + (1,1,1,1,1,1,1) | 80640 | 17920 | ||||||
55 | t0,2,6{4,3,3,3,3,3} |
Hexicantellated 7-cube (Purosa) | (0,0,1,1,1,1,2)√2 + (1,1,1,1,1,1,1) | 94080 | 13440 | ||||||
56 | t0,2,5{4,3,3,3,3,3} |
Penticantellated 7-cube (Tersa) | (0,0,1,1,1,2,2)√2 + (1,1,1,1,1,1,1) | 188160 | 26880 | ||||||
57 | t0,2,4{4,3,3,3,3,3} |
Stericantellated 7-cube (Carsa) | (0,0,1,1,2,2,2)√2 + (1,1,1,1,1,1,1) | 161280 | 26880 | ||||||
58 | t0,2,3{4,3,3,3,3,3} |
Runcicantellated 7-cube (Parsa) | (0,0,1,2,2,2,2)√2 + (1,1,1,1,1,1,1) | 53760 | 13440 | ||||||
59 | t0,1,6{4,3,3,3,3,3} |
Hexitruncated 7-cube (Putsa) | (0,1,1,1,1,1,2)√2 + (1,1,1,1,1,1,1) | 29568 | 5376 | ||||||
60 | t0,1,5{4,3,3,3,3,3} |
Pentitruncated 7-cube (Tetsa) | (0,1,1,1,1,2,2)√2 + (1,1,1,1,1,1,1) | 87360 | 13440 | ||||||
61 | t0,1,4{4,3,3,3,3,3} |
Steritruncated 7-cube (Catsa) | (0,1,1,1,2,2,2)√2 + (1,1,1,1,1,1,1) | 116480 | 17920 | ||||||
62 | t0,1,3{4,3,3,3,3,3} |
Runcitruncated 7-cube (Petsa) | (0,1,1,2,2,2,2)√2 + (1,1,1,1,1,1,1) | 73920 | 13440 | ||||||
63 | t0,1,2{4,3,3,3,3,3} |
Cantitruncated 7-cube (Gersa) | (0,1,2,2,2,2,2)√2 + (1,1,1,1,1,1,1) | 18816 | 5376 | ||||||
64 | t0,1,2,3{3,3,3,3,3,4} |
Runcicantitruncated 7-orthoplex (Gopaz) | (0,1,2,3,4,4,4)√2 | 60480 | 13440 | ||||||
65 | t0,1,2,4{3,3,3,3,3,4} |
Stericantitruncated 7-orthoplex (Cogarz) | (0,0,1,1,2,3,4)√2 | 241920 | 40320 | ||||||
66 | t0,1,3,4{3,3,3,3,3,4} |
Steriruncitruncated 7-orthoplex (Captaz) | (0,0,1,2,2,3,4)√2 | 181440 | 40320 | ||||||
67 | t0,2,3,4{3,3,3,3,3,4} |
Steriruncicantellated 7-orthoplex (Caparz) | (0,0,1,2,3,3,4)√2 | 181440 | 40320 | ||||||
68 | t1,2,3,4{3,3,3,3,3,4} |
Biruncicantitruncated 7-orthoplex (Gibpaz) | (0,0,1,2,3,4,4)√2 | 161280 | 40320 | ||||||
69 | t0,1,2,5{3,3,3,3,3,4} |
Penticantitruncated 7-orthoplex (Tograz) | (0,1,1,1,2,3,4)√2 | 295680 | 53760 | ||||||
70 | t0,1,3,5{3,3,3,3,3,4} |
Pentiruncitruncated 7-orthoplex (Toptaz) | (0,1,1,2,2,3,4)√2 | 443520 | 80640 | ||||||
71 | t0,2,3,5{3,3,3,3,3,4} |
Pentiruncicantellated 7-orthoplex (Toparz) | (0,1,1,2,3,3,4)√2 | 403200 | 80640 | ||||||
72 | t1,2,3,5{3,3,3,3,3,4} |
Bistericantitruncated 7-orthoplex (Becogarz) | (0,1,1,2,3,4,4)√2 | 362880 | 80640 | ||||||
73 | t0,1,4,5{3,3,3,3,3,4} |
Pentisteritruncated 7-orthoplex (Tacotaz) | (0,1,2,2,2,3,4)√2 | 241920 | 53760 | ||||||
74 | t0,2,4,5{3,3,3,3,3,4} |
Pentistericantellated 7-orthoplex (Tocarz) | (0,1,2,2,3,3,4)√2 | 403200 | 80640 | ||||||
75 | t1,2,4,5{4,3,3,3,3,3} |
Bisteriruncitruncated 7-cube (Bocaptosaz) | (0,1,2,2,3,4,4)√2 | 322560 | 80640 | ||||||
76 | t0,3,4,5{3,3,3,3,3,4} |
Pentisteriruncinated 7-orthoplex (Tecpaz) | (0,1,2,3,3,3,4)√2 | 241920 | 53760 | ||||||
77 | t1,2,3,5{4,3,3,3,3,3} |
Bistericantitruncated 7-cube (Becgresa) | (0,1,2,3,3,4,4)√2 | 362880 | 80640 | ||||||
78 | t1,2,3,4{4,3,3,3,3,3} |
Biruncicantitruncated 7-cube (Gibposa) | (0,1,2,3,4,4,4)√2 | 188160 | 53760 | ||||||
79 | t0,1,2,6{3,3,3,3,3,4} |
Hexicantitruncated 7-orthoplex (Pugarez) | (0,0,0,0,1,2,3)√2 + (1,1,1,1,1,1,1) | 134400 | 26880 | ||||||
80 | t0,1,3,6{3,3,3,3,3,4} |
Hexiruncitruncated 7-orthoplex (Papataz) | (0,0,0,1,1,2,3)√2 + (1,1,1,1,1,1,1) | 322560 | 53760 | ||||||
81 | t0,2,3,6{3,3,3,3,3,4} |
Hexiruncicantellated 7-orthoplex (Puparez) | (0,0,0,1,2,2,3)√2 + (1,1,1,1,1,1,1) | 268800 | 53760 | ||||||
82 | t0,3,4,5{4,3,3,3,3,3} |
Pentisteriruncinated 7-cube (Tecpasa) | (0,0,0,1,2,3,3)√2 + (1,1,1,1,1,1,1) | 241920 | 53760 | ||||||
83 | t0,1,4,6{3,3,3,3,3,4} |
Hexisteritruncated 7-orthoplex (Pucotaz) | (0,0,1,1,1,2,3)√2 + (1,1,1,1,1,1,1) | 322560 | 53760 | ||||||
84 | t0,2,4,6{4,3,3,3,3,3} |
Hexistericantellated 7-cube (Pucrosaz) | (0,0,1,1,2,2,3)√2 + (1,1,1,1,1,1,1) | 483840 | 80640 | ||||||
85 | t0,2,4,5{4,3,3,3,3,3} |
Pentistericantellated 7-cube (Tecresa) | (0,0,1,1,2,3,3)√2 + (1,1,1,1,1,1,1) | 403200 | 80640 | ||||||
86 | t0,2,3,6{4,3,3,3,3,3} |
Hexiruncicantellated 7-cube (Pupresa) | (0,0,1,2,2,2,3)√2 + (1,1,1,1,1,1,1) | 268800 | 53760 | ||||||
87 | t0,2,3,5{4,3,3,3,3,3} |
Pentiruncicantellated 7-cube (Topresa) | (0,0,1,2,2,3,3)√2 + (1,1,1,1,1,1,1) | 403200 | 80640 | ||||||
88 | t0,2,3,4{4,3,3,3,3,3} |
Steriruncicantellated 7-cube (Copresa) | (0,0,1,2,3,3,3)√2 + (1,1,1,1,1,1,1) | 215040 | 53760 | ||||||
89 | t0,1,5,6{4,3,3,3,3,3} |
Hexipentitruncated 7-cube (Putatosez) | (0,1,1,1,1,2,3)√2 + (1,1,1,1,1,1,1) | 134400 | 26880 | ||||||
90 | t0,1,4,6{4,3,3,3,3,3} |
Hexisteritruncated 7-cube (Pacutsa) | (0,1,1,1,2,2,3)√2 + (1,1,1,1,1,1,1) | 322560 | 53760 | ||||||
91 | t0,1,4,5{4,3,3,3,3,3} |
Pentisteritruncated 7-cube (Tecatsa) | (0,1,1,1,2,3,3)√2 + (1,1,1,1,1,1,1) | 241920 | 53760 | ||||||
92 | t0,1,3,6{4,3,3,3,3,3} |
Hexiruncitruncated 7-cube (Pupetsa) | (0,1,1,2,2,2,3)√2 + (1,1,1,1,1,1,1) | 322560 | 53760 | ||||||
93 | t0,1,3,5{4,3,3,3,3,3} |
Pentiruncitruncated 7-cube (Toptosa) | (0,1,1,2,2,3,3)√2 + (1,1,1,1,1,1,1) | 443520 | 80640 | ||||||
94 | t0,1,3,4{4,3,3,3,3,3} |
Steriruncitruncated 7-cube (Captesa) | (0,1,1,2,3,3,3)√2 + (1,1,1,1,1,1,1) | 215040 | 53760 | ||||||
95 | t0,1,2,6{4,3,3,3,3,3} |
Hexicantitruncated 7-cube (Pugrosa) | (0,1,2,2,2,2,3)√2 + (1,1,1,1,1,1,1) | 134400 | 26880 | ||||||
96 | t0,1,2,5{4,3,3,3,3,3} |
Penticantitruncated 7-cube (Togresa) | (0,1,2,2,2,3,3)√2 + (1,1,1,1,1,1,1) | 295680 | 53760 | ||||||
97 | t0,1,2,4{4,3,3,3,3,3} |
Stericantitruncated 7-cube (Cogarsa) | (0,1,2,2,3,3,3)√2 + (1,1,1,1,1,1,1) | 268800 | 53760 | ||||||
98 | t0,1,2,3{4,3,3,3,3,3} |
Runcicantitruncated 7-cube (Gapsa) | (0,1,2,3,3,3,3)√2 + (1,1,1,1,1,1,1) | 94080 | 26880 | ||||||
99 | t0,1,2,3,4{3,3,3,3,3,4} |
Steriruncicantitruncated 7-orthoplex (Gocaz) | (0,0,1,2,3,4,5)√2 | 322560 | 80640 | ||||||
100 | t0,1,2,3,5{3,3,3,3,3,4} |
Pentiruncicantitruncated 7-orthoplex (Tegopaz) | (0,1,1,2,3,4,5)√2 | 725760 | 161280 | ||||||
101 | t0,1,2,4,5{3,3,3,3,3,4} |
Pentistericantitruncated 7-orthoplex (Tecagraz) | (0,1,2,2,3,4,5)√2 | 645120 | 161280 | ||||||
102 | t0,1,3,4,5{3,3,3,3,3,4} |
Pentisteriruncitruncated 7-orthoplex (Tecpotaz) | (0,1,2,3,3,4,5)√2 | 645120 | 161280 | ||||||
103 | t0,2,3,4,5{3,3,3,3,3,4} |
Pentisteriruncicantellated 7-orthoplex (Tacparez) | (0,1,2,3,4,4,5)√2 | 645120 | 161280 | ||||||
104 | t1,2,3,4,5{4,3,3,3,3,3} |
Bisteriruncicantitruncated 7-cube (Gabcosaz) | (0,1,2,3,4,5,5)√2 | 564480 | 161280 | ||||||
105 | t0,1,2,3,6{3,3,3,3,3,4} |
Hexiruncicantitruncated 7-orthoplex (Pugopaz) | (0,0,0,1,2,3,4)√2 + (1,1,1,1,1,1,1) | 483840 | 107520 | ||||||
106 | t0,1,2,4,6{3,3,3,3,3,4} |
Hexistericantitruncated 7-orthoplex (Pucagraz) | (0,0,1,1,2,3,4)√2 + (1,1,1,1,1,1,1) | 806400 | 161280 | ||||||
107 | t0,1,3,4,6{3,3,3,3,3,4} |
Hexisteriruncitruncated 7-orthoplex (Pucpotaz) | (0,0,1,2,2,3,4)√2 + (1,1,1,1,1,1,1) | 725760 | 161280 | ||||||
108 | t0,2,3,4,6{4,3,3,3,3,3} |
Hexisteriruncicantellated 7-cube (Pucprosaz) | (0,0,1,2,3,3,4)√2 + (1,1,1,1,1,1,1) | 725760 | 161280 | ||||||
109 | t0,2,3,4,5{4,3,3,3,3,3} |
Pentisteriruncicantellated 7-cube (Tocpresa) | (0,0,1,2,3,4,4)√2 + (1,1,1,1,1,1,1) | 645120 | 161280 | ||||||
110 | t0,1,2,5,6{3,3,3,3,3,4} |
Hexipenticantitruncated 7-orthoplex (Putegraz) | (0,1,1,1,2,3,4)√2 + (1,1,1,1,1,1,1) | 483840 | 107520 | ||||||
111 | t0,1,3,5,6{4,3,3,3,3,3} |
Hexipentiruncitruncated 7-cube (Putpetsaz) | (0,1,1,2,2,3,4)√2 + (1,1,1,1,1,1,1) | 806400 | 161280 | ||||||
112 | t0,1,3,4,6{4,3,3,3,3,3} |
Hexisteriruncitruncated 7-cube (Pucpetsa) | (0,1,1,2,3,3,4)√2 + (1,1,1,1,1,1,1) | 725760 | 161280 | ||||||
113 | t0,1,3,4,5{4,3,3,3,3,3} |
Pentisteriruncitruncated 7-cube (Tecpetsa) | (0,1,1,2,3,4,4)√2 + (1,1,1,1,1,1,1) | 645120 | 161280 | ||||||
114 | t0,1,2,5,6{4,3,3,3,3,3} |
Hexipenticantitruncated 7-cube (Putgresa) | (0,1,2,2,2,3,4)√2 + (1,1,1,1,1,1,1) | 483840 | 107520 | ||||||
115 | t0,1,2,4,6{4,3,3,3,3,3} |
Hexistericantitruncated 7-cube (Pucagrosa) | (0,1,2,2,3,3,4)√2 + (1,1,1,1,1,1,1) | 806400 | 161280 | ||||||
116 | t0,1,2,4,5{4,3,3,3,3,3} |
Pentistericantitruncated 7-cube (Tecgresa) | (0,1,2,2,3,4,4)√2 + (1,1,1,1,1,1,1) | 645120 | 161280 | ||||||
117 | t0,1,2,3,6{4,3,3,3,3,3} |
Hexiruncicantitruncated 7-cube (Pugopsa) | (0,1,2,3,3,3,4)√2 + (1,1,1,1,1,1,1) | 483840 | 107520 | ||||||
118 | t0,1,2,3,5{4,3,3,3,3,3} |
Pentiruncicantitruncated 7-cube (Togapsa) | (0,1,2,3,3,4,4)√2 + (1,1,1,1,1,1,1) | 725760 | 161280 | ||||||
119 | t0,1,2,3,4{4,3,3,3,3,3} |
Steriruncicantitruncated 7-cube (Gacosa) | (0,1,2,3,4,4,4)√2 + (1,1,1,1,1,1,1) | 376320 | 107520 | ||||||
120 | t0,1,2,3,4,5{3,3,3,3,3,4} |
Pentisteriruncicantitruncated 7-orthoplex (Gotaz) | (0,1,2,3,4,5,6)√2 | 1128960 | 322560 | ||||||
121 | t0,1,2,3,4,6{3,3,3,3,3,4} |
Hexisteriruncicantitruncated 7-orthoplex (Pugacaz) | (0,0,1,2,3,4,5)√2 + (1,1,1,1,1,1,1) | 1290240 | 322560 | ||||||
122 | t0,1,2,3,5,6{3,3,3,3,3,4} |
Hexipentiruncicantitruncated 7-orthoplex (Putgapaz) | (0,1,1,2,3,4,5)√2 + (1,1,1,1,1,1,1) | 1290240 | 322560 | ||||||
123 | t0,1,2,4,5,6{4,3,3,3,3,3} |
Hexipentistericantitruncated 7-cube (Putcagrasaz) | (0,1,2,2,3,4,5)√2 + (1,1,1,1,1,1,1) | 1290240 | 322560 | ||||||
124 | t0,1,2,3,5,6{4,3,3,3,3,3} |
Hexipentiruncicantitruncated 7-cube (Putgapsa) | (0,1,2,3,3,4,5)√2 + (1,1,1,1,1,1,1) | 1290240 | 322560 | ||||||
125 | t0,1,2,3,4,6{4,3,3,3,3,3} |
Hexisteriruncicantitruncated 7-cube (Pugacasa) | (0,1,2,3,4,4,5)√2 + (1,1,1,1,1,1,1) | 1290240 | 322560 | ||||||
126 | t0,1,2,3,4,5{4,3,3,3,3,3} |
Pentisteriruncicantitruncated 7-cube (Gotesa) | (0,1,2,3,4,5,5)√2 + (1,1,1,1,1,1,1) | 1128960 | 322560 | ||||||
127 | t0,1,2,3,4,5,6{4,3,3,3,3,3} |
Omnitruncated 7-cube (Guposaz) | (0,1,2,3,4,5,6)√2 + (1,1,1,1,1,1,1) | 2257920 | 645120 |
The D7 family has symmetry of order 322560 (7 factorial x 26).
This family has 3×32−1=95 Wythoffian uniform polytopes, generated by marking one or more nodes of the D7 Coxeter-Dynkin diagram. Of these, 63 (2×32−1) are repeated from the B7 family and 32 are unique to this family, listed below. Bowers names and acronym are given for cross-referencing.
See also list of D7 polytopes for Coxeter plane graphs of these polytopes.
D7 uniform polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
# | Coxeter diagram | Names | Base point (Alternately signed) |
Element counts | |||||||
6 | 5 | 4 | 3 | 2 | 1 | 0 | |||||
1 | = | 7-cube demihepteract (hesa) |
(1,1,1,1,1,1,1) | 78 | 532 | 1624 | 2800 | 2240 | 672 | 64 | |
2 | = | cantic 7-cube truncated demihepteract (thesa) |
(1,1,3,3,3,3,3) | 142 | 1428 | 5656 | 11760 | 13440 | 7392 | 1344 | |
3 | = | runcic 7-cube small rhombated demihepteract (sirhesa) |
(1,1,1,3,3,3,3) | 16800 | 2240 | ||||||
4 | = | steric 7-cube small prismated demihepteract (sphosa) |
(1,1,1,1,3,3,3) | 20160 | 2240 | ||||||
5 | = | pentic 7-cube small cellated demihepteract (sochesa) |
(1,1,1,1,1,3,3) | 13440 | 1344 | ||||||
6 | = | hexic 7-cube small terated demihepteract (suthesa) |
(1,1,1,1,1,1,3) | 4704 | 448 | ||||||
7 | = | runcicantic 7-cube great rhombated demihepteract (Girhesa) |
(1,1,3,5,5,5,5) | 23520 | 6720 | ||||||
8 | = | stericantic 7-cube prismatotruncated demihepteract (pothesa) |
(1,1,3,3,5,5,5) | 73920 | 13440 | ||||||
9 | = |