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Uniform 7-polytope

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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.

Regular 7-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:

  1. {3,3,3,3,3,3} - 7-simplex
  2. {4,3,3,3,3,3} - 7-cube
  3. {3,3,3,3,3,4} - 7-orthoplex

There are no nonconvex regular 7-polytopes.

Characteristics

The topology of any given 7-polytope is defined by its Betti numbers and torsion coefficients.

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.

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.

Uniform 7-polytopes by fundamental Coxeter groups

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 71
2 B7 127 + 32
3 D7 95 (0 unique)
4 E7 127
Prismatic finite Coxeter groups
# Coxeter group Coxeter diagram
6+1
1 A6A1 ×
2 BC6A1 ×
3 D6A1 ×
4 E6A1 ×
5+2
1 A5I2(p) ×
2 BC5I2(p) ×
3 D5I2(p) ×
5+1+1
1 A5A1 ×
2 BC5A1 ×
3 D5A1 ×
4+3
1 A4A3 ×
2 A4B3 ×
3 A4H3 ×
4 BC4A3 ×
5 BC4B3 ×
6 BC4H3 ×
7 H4A3 ×
8 H4B3 ×
9 H4H3 ×
10 F4A3 ×
11 F4B3 ×
12 F4H3 ×
13 D4A3 ×
14 D4B3 ×
15 D4H3 ×
4+2+1
1 A4I2(p)A1 ××
2 BC4I2(p)A1 ××
3 F4I2(p)A1 ××
4 H4I2(p)A1 ××
5 D4I2(p)A1 ××
4+1+1+1
1 A4A1 ×
2 BC4A1 ×
3 F4A1 ×
4 H4A1 ×
5 D4A1 ×
3+3+1
1 A3A3A1 ××
2 A3B3A1 ××
3 A3H3A1 ××
4 BC3B3A1 ××
5 BC3H3A1 ××
6 H3A3A1 ××
3+2+2
1 A3I2(p)I2(q) ××
2 BC3I2(p)I2(q) ××
3 H3I2(p)I2(q) ××
3+2+1+1
1 A3I2(p)A1 ××
2 BC3I2(p)A1 ××
3 H3I2(p)A1 ××
3+1+1+1+1
1 A3A1 ×
2 BC3A1 ×
3 H3A1 ×
2+2+2+1
1 I2(p)I2(q)I2(r)A1 ×××
2+2+1+1+1
1 I2(p)I2(q)A1 ××
2+1+1+1+1+1
1 I2(p)A1 ×
1+1+1+1+1+1+1
1 A1

The A7 family

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

The B7 family has symmetry of order 645120 (7 factorial x 2).

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

The D7 family has symmetry of order 322560 (7 factorial x 2).

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 = steriruncic 7-cube
prismatorhomated demihepteract (prohesa)		 (1,1,1,3,5,5,5) 40320 8960
10 = penticantic 7-cube
cellitruncated demihepteract (cothesa)		 (1,1,3,3,3,5,5) 87360 13440
11 = pentiruncic 7-cube
cellirhombated demihepteract (crohesa)		 (1,1,1,3,3,5,5) 87360 13440
12 = pentisteric 7-cube
celliprismated demihepteract (caphesa)		 (1,1,1,1,3,5,5) 40320 6720
13 = hexicantic 7-cube
tericantic demihepteract (tuthesa)		 (1,1,3,3,3,3,5) 43680 6720
14 = hexiruncic 7-cube
terirhombated demihepteract (turhesa)		 (1,1,1,3,3,3,5) 67200 8960
15 = hexisteric 7-cube
teriprismated demihepteract (tuphesa)		 (1,1,1,1,3,3,5) 53760 6720
16 = hexipentic 7-cube
tericellated demihepteract (tuchesa)		 (1,1,1,1,1,3,5) 21504 2688
17 = steriruncicantic 7-cube
great prismated demihepteract (Gephosa)		 (1,1,3,5,7,7,7) 94080 26880
18 = pentiruncicantic 7-cube
celligreatorhombated demihepteract (cagrohesa)		 (1,1,3,5,5,7,7) 181440 40320
19 = pentistericantic 7-cube
celliprismatotruncated demihepteract (capthesa)		 (1,1,3,3,5,7,7) 181440 40320
20 = pentisteriruncic 7-cube
celliprismatorhombated demihepteract (coprahesa)		 (1,1,1,3,5,7,7) 120960 26880
21 = hexiruncicantic 7-cube
terigreatorhombated demihepteract (tugrohesa)		 (1,1,3,5,5,5,7) 120960 26880
22 = hexistericantic 7-cube
teriprismatotruncated demihepteract (tupthesa)		 (1,1,3,3,5,5,7) 221760 40320
23 = hexisteriruncic 7-cube
teriprismatorhombated demihepteract (tuprohesa)		 (1,1,1,3,5,5,7) 134400 26880
24 = hexipenticantic 7-cube
teriCellitruncated demihepteract (tucothesa)		 (1,1,3,3,3,5,7) 147840 26880
25 = hexipentiruncic 7-cube
tericellirhombated demihepteract (tucrohesa)		 (1,1,1,3,3,5,7) 161280 26880
26 = hexipentisteric 7-cube
tericelliprismated demihepteract (tucophesa)		 (1,1,1,1,3,5,7) 80640 13440
27 = pentisteriruncicantic 7-cube
great cellated demihepteract (gochesa)		 (1,1,3,5,7,9,9) 282240 80640
28 = hexisteriruncicantic 7-cube
terigreatoprimated demihepteract (tugphesa)		 (1,1,3,5,7,7,9) 322560 80640
29 = hexipentiruncicantic 7-cube
tericelligreatorhombated demihepteract (tucagrohesa)		 (1,1,3,5,5,7,9) 322560 80640
30 = hexipentistericantic 7-cube
tericelliprismatotruncated demihepteract (tucpathesa)		 (1,1,3,3,5,7,9) 362880 80640
31 = hexipentisteriruncic 7-cube
tericellprismatorhombated demihepteract (tucprohesa)		 (1,1,1,3,5,7,9) 241920 53760
32 = hexipentisteriruncicantic 7-cube
great terated demihepteract (guthesa)		 (1,1,3,5,7,9,11) 564480 161280

The E7 family

The E7 Coxeter group has order 2,903,040.

There are 127 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.

See also a list of E7 polytopes for symmetric Coxeter plane graphs of these polytopes.

E7 uniform polytopes
# Coxeter-Dynkin diagram
Schläfli symbol 		Names Element counts
6 5 4 3 2 1 0
1 231 (laq) 632 4788 16128 20160 10080 2016 126
2 Rectified 231 (rolaq) 758 10332 47880 100800 90720 30240 2016
3 Rectified 132 (rolin) 758 12348 72072 191520 241920 120960 10080
4 132 (lin) 182 4284 23688 50400 40320 10080 576
5 Birectified 321 (branq) 758 12348 68040 161280 161280 60480 4032
6 Rectified 321 (ranq) 758 44352 70560 48384 11592 12096 756
7 321 (naq) 702 6048 12096 10080 4032 756 56
8 Truncated 231 (talq) 758 10332 47880 100800 90720 32256 4032
9 Cantellated 231 (sirlaq) 131040 20160
10 Bitruncated 231 (botlaq) 30240
11 small demified 231 (shilq) 2774 22428 78120 151200 131040 42336 4032
12 demirectified 231 (hirlaq) 12096
13 truncated 132 (tolin) 20160
14 small demiprismated 231 (shiplaq) 20160
15 birectified 132 (berlin) 758 22428 142632 403200 544320 302400 40320
16 tritruncated 321 (totanq) 40320
17 demibirectified 321 (hobranq) 20160
18 small cellated 231 (scalq) 7560
19 small biprismated 231 (sobpalq) 30240
20 small birhombated 321 (sabranq) 60480
21 demirectified 321 (harnaq) 12096
22 bitruncated 321 (botnaq) 12096
23 small terated 321 (stanq) 1512
24 small demicellated 321 (shocanq) 12096
25 small prismated 321 (spanq) 40320
26 small demified 321 (shanq) 4032
27 small rhombated 321 (sranq) 12096
28 Truncated 321 (tanq) 758 11592 48384 70560 44352 12852 1512
29 great rhombated 231 (girlaq) 60480
30 demitruncated 231 (hotlaq) 24192
31 small demirhombated 231 (sherlaq) 60480
32 demibitruncated 231 (hobtalq) 60480
33 demiprismated 231 (hiptalq) 80640
34 demiprismatorhombated 231 (hiprolaq) 120960
35 bitruncated 132 (batlin) 120960
36 small prismated 231 (spalq) 80640
37 small rhombated 132 (sirlin) 120960
38 tritruncated 231 (tatilq) 80640
39 cellitruncated 231 (catalaq) 60480
40 cellirhombated 231 (crilq) 362880
41 biprismatotruncated 231 (biptalq) 181440
42 small prismated 132 (seplin) 60480
43 small biprismated 321 (sabipnaq) 120960
44 small demibirhombated 321 (shobranq) 120960
45 cellidemiprismated 231 (chaplaq) 60480
46 demibiprismatotruncated 321 (hobpotanq) 120960
47 great birhombated 321 (gobranq) 120960
48 demibitruncated 321 (hobtanq) 60480
49 teritruncated 231 (totalq) 24192
50 terirhombated 231 (trilq) 120960
51 demicelliprismated 321 (hicpanq) 120960
52 small teridemified 231 (sethalq) 24192
53 small cellated 321 (scanq) 60480
54 demiprismated 321 (hipnaq) 80640
55 terirhombated 321 (tranq) 60480
56 demicellirhombated 321 (hocranq) 120960
57 prismatorhombated 321 (pranq) 120960
58 small demirhombated 321 (sharnaq) 60480
59 teritruncated 321 (tetanq) 15120
60 demicellitruncated 321 (hictanq) 60480
61 prismatotruncated 321 (potanq) 120960
62 demitruncated 321 (hotnaq) 24192
63 great rhombated 321 (granq) 24192
64 great demified 231 (gahlaq) 120960
65 great demiprismated 231 (gahplaq) 241920
66 prismatotruncated 231 (potlaq) 241920
67 prismatorhombated 231 (prolaq) 241920
68 great rhombated 132 (girlin) 241920
69 celligreatorhombated 231 (cagrilq) 362880
70 cellidemitruncated 231 (chotalq) 241920
71 prismatotruncated 132 (patlin) 362880
72 biprismatorhombated 321 (bipirnaq) 362880
73 tritruncated 132 (tatlin) 241920
74 cellidemiprismatorhombated 231 (chopralq) 362880
75 great demibiprismated 321 (ghobipnaq) 362880
76 celliprismated 231 (caplaq) 241920
77 biprismatotruncated 321 (boptanq) 362880
78 great trirhombated 231 (gatralaq) 241920
79 terigreatorhombated 231 (togrilq) 241920
80 teridemitruncated 231 (thotalq) 120960
81 teridemirhombated 231 (thorlaq) 241920
82 celliprismated 321 (capnaq) 241920
83 teridemiprismatotruncated 231 (thoptalq) 241920
84 teriprismatorhombated 321 (tapronaq) 362880
85 demicelliprismatorhombated 321 (hacpranq) 362880
86 teriprismated 231 (toplaq) 241920
87 cellirhombated 321 (cranq) 362880
88 demiprismatorhombated 321 (hapranq) 241920
89 tericellitruncated 231 (tectalq) 120960
90 teriprismatotruncated 321 (toptanq) 362880
91 demicelliprismatotruncated 321 (hecpotanq) 362880
92 teridemitruncated 321 (thotanq) 120960
93 cellitruncated 321 (catnaq) 241920
94 demiprismatotruncated 321 (hiptanq) 241920
95 terigreatorhombated 321 (tagranq) 120960
96 demicelligreatorhombated 321 (hicgarnq) 241920
97 great prismated 321 (gopanq) 241920
98 great demirhombated 321 (gahranq) 120960
99 great prismated 231 (gopalq) 483840
100 great cellidemified 231 (gechalq) 725760
101 great birhombated 132 (gebrolin) 725760
102 prismatorhombated 132 (prolin) 725760
103 celliprismatorhombated 231 (caprolaq) 725760
104 great biprismated 231 (gobpalq) 725760
105 tericelliprismated 321 (ticpanq) 483840
106 teridemigreatoprismated 231 (thegpalq) 725760
107 teriprismatotruncated 231 (teptalq) 725760
108 teriprismatorhombated 231 (topralq) 725760
109 cellipriemsatorhombated 321 (copranq) 725760
110 tericelligreatorhombated 231 (tecgrolaq) 725760
111 tericellitruncated 321 (tectanq) 483840
112 teridemiprismatotruncated 321 (thoptanq) 725760
113 celliprismatotruncated 321 (coptanq) 725760
114 teridemicelligreatorhombated 321 (thocgranq) 483840
115 terigreatoprismated 321 (tagpanq) 725760
116 great demicellated 321 (gahcnaq) 725760
117 tericelliprismated laq (tecpalq) 483840
118 celligreatorhombated 321 (cogranq) 725760
119 great demified 321 (gahnq) 483840
120 great cellated 231 (gocalq) 1451520
121 terigreatoprismated 231 (tegpalq) 1451520
122 tericelliprismatotruncated 321 (tecpotniq) 1451520
123 tericellidemigreatoprismated 231 (techogaplaq) 1451520
124 tericelligreatorhombated 321 (tacgarnq) 1451520
125 tericelliprismatorhombated 231 (tecprolaq) 1451520
126 great cellated 321 (gocanq) 1451520
127 great terated 321 (gotanq) 2903040

Regular and uniform honeycombs

Coxeter-Dynkin diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence.

There are five fundamental affine Coxeter groups and sixteen prismatic groups that generate regular and uniform tessellations in 6-space:

# Coxeter group Coxeter diagram Forms
1 A ~ 6 {\displaystyle {\tilde {A}}_{6}} 17
2 C ~ 6 {\displaystyle {\tilde {C}}_{6}} 71
3 B ~ 6 {\displaystyle {\tilde {B}}_{6}} h
 95 (32 new)
4 D ~ 6 {\displaystyle {\tilde {D}}_{6}} q
 41 (6 new)
5 E ~ 6 {\displaystyle {\tilde {E}}_{6}} 39

Regular and uniform tessellations include:

  • A ~ 6 {\displaystyle {\tilde {A}}_{6}} , 17 forms
  • C ~ 6 {\displaystyle {\tilde {C}}_{6}} , , 71 forms
  • B ~ 6 {\displaystyle {\tilde {B}}_{6}} , , 95 forms, 64 shared with C ~ 6 {\displaystyle {\tilde {C}}_{6}} , 32 new
  • D ~ 6 {\displaystyle {\tilde {D}}_{6}} , , 41 unique ringed permutations, most shared with B ~ 6 {\displaystyle {\tilde {B}}_{6}} and C ~ 6 {\displaystyle {\tilde {C}}_{6}} , and 6 are new. Coxeter calls the first one a quarter 6-cubic honeycomb.
    • =
    • =
    • =
    • =
    • =
    • =
  • E ~ 6 {\displaystyle {\tilde {E}}_{6}} : , 39 forms
    • Uniform 222 honeycomb: represented by symbols {3,3,3},
    • Uniform t4(222) honeycomb: 4r{3,3,3},
    • Uniform 0222 honeycomb: {3},
    • Uniform t2(0222) honeycomb: 2r{3},
Prismatic groups
# Coxeter group Coxeter-Dynkin diagram
1 A ~ 5 {\displaystyle {\tilde {A}}_{5}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}}
2 B ~ 5 {\displaystyle {\tilde {B}}_{5}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}}
3 C ~ 5 {\displaystyle {\tilde {C}}_{5}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}}
4 D ~ 5 {\displaystyle {\tilde {D}}_{5}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}}
5 A ~ 4 {\displaystyle {\tilde {A}}_{4}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}}
6 B ~ 4 {\displaystyle {\tilde {B}}_{4}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}}
7 C ~ 4 {\displaystyle {\tilde {C}}_{4}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}}
8 D ~ 4 {\displaystyle {\tilde {D}}_{4}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}}
9 F ~ 4 {\displaystyle {\tilde {F}}_{4}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}}
10 C ~ 3 {\displaystyle {\tilde {C}}_{3}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}}
11 B ~ 3 {\displaystyle {\tilde {B}}_{3}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}}
12 A ~ 3 {\displaystyle {\tilde {A}}_{3}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}}
13 C ~ 2 {\displaystyle {\tilde {C}}_{2}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}}
14 H ~ 2 {\displaystyle {\tilde {H}}_{2}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}}
15 A ~ 2 {\displaystyle {\tilde {A}}_{2}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}}
16 I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}} x I ~ 1 {\displaystyle {\tilde {I}}_{1}}

Regular and uniform hyperbolic honeycombs

There are no compact hyperbolic Coxeter groups of rank 7, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 3 paracompact hyperbolic Coxeter groups of rank 7, each generating uniform honeycombs in 6-space as permutations of rings of the Coxeter diagrams.

P ¯ 6 {\displaystyle {\bar {P}}_{6}} = :
 Q ¯ 6 {\displaystyle {\bar {Q}}_{6}} = :
 S ¯ 6 {\displaystyle {\bar {S}}_{6}} = :

Notes on the Wythoff construction for the uniform 7-polytopes

The reflective 7-dimensional uniform polytopes are constructed through a Wythoff construction process, and represented by a Coxeter-Dynkin diagram, where each node represents a mirror. An active mirror is represented by a ringed node. Each combination of active mirrors generates a unique uniform polytope. Uniform polytopes are named in relation to the regular polytopes in each family. Some families have two regular constructors and thus may be named in two equally valid ways.

Here are the primary operators available for constructing and naming the uniform 7-polytopes.

The prismatic forms and bifurcating graphs can use the same truncation indexing notation, but require an explicit numbering system on the nodes for clarity.

Operation Extended
Schläfli symbol 		Coxeter-
Dynkin
diagram 		Description
Parent t0{p,q,r,s,t,u} Any regular 7-polytope
Rectified t1{p,q,r,s,t,u} The edges are fully truncated into single points. The 7-polytope now has the combined faces of the parent and dual.
Birectified t2{p,q,r,s,t,u} Birectification reduces cells to their duals.
Truncated t0,1{p,q,r,s,t,u} Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated 7-polytope. The 7-polytope has its original faces doubled in sides, and contains the faces of the dual.
Bitruncated t1,2{p,q,r,s,t,u} Bitrunction transforms cells to their dual truncation.
Tritruncated t2,3{p,q,r,s,t,u} Tritruncation transforms 4-faces to their dual truncation.
Cantellated t0,2{p,q,r,s,t,u} In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is halfway between both the parent and dual forms.
Bicantellated t1,3{p,q,r,s,t,u} In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is halfway between both the parent and dual forms.
Runcinated t0,3{p,q,r,s,t,u} Runcination reduces cells and creates new cells at the vertices and edges.
Biruncinated t1,4{p,q,r,s,t,u} Runcination reduces cells and creates new cells at the vertices and edges.
Stericated t0,4{p,q,r,s,t,u} Sterication reduces 4-faces and creates new 4-faces at the vertices, edges, and faces in the gaps.
Pentellated t0,5{p,q,r,s,t,u} Pentellation reduces 5-faces and creates new 5-faces at the vertices, edges, faces, and cells in the gaps.
Hexicated t0,6{p,q,r,s,t,u} Hexication reduces 6-faces and creates new 6-faces at the vertices, edges, faces, cells, and 4-faces in the gaps. (expansion operation for 7-polytopes)
Omnitruncated t0,1,2,3,4,5,6{p,q,r,s,t,u} All six operators, truncation, cantellation, runcination, sterication, pentellation, and hexication are applied.

References

  1. ^ Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.
  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  • A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
  • H.S.M. Coxeter:
    • H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
    • 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 http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I,
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II,
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III,
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
  • Klitzing, Richard. "7D uniform polytopes (polyexa)".

External links

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
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 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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