Convergence of measures: Difference between revisions

Revision as of 19:47, 3 February 2014

Graphs of three regular and related uniform polytopes

File:7-simplex t0.svg 7-simplex	File:7-simplex t1.svg Rectified 7-simplex		File:7-simplex t01.svg Truncated 7-simplex
File:7-simplex t02.svg Cantellated 7-simplex	File:7-simplex t03.svg Runcinated 7-simplex		File:7-simplex t04.svg Stericated 7-simplex
File:7-simplex t05.svg Pentellated 7-simplex		File:7-simplex t06.svg Hexicated 7-simplex
File:7-cube t6.svg 7-orthoplex	File:7-cube t56.svg Truncated 7-orthoplex		File:7-cube t5.svg Rectified 7-orthoplex
File:7-cube t46.svg Cantellated 7-orthoplex	File:7-cube t36.svg Runcinated 7-orthoplex		File:7-cube t26.svg Stericated 7-orthoplex
File:7-cube t16.svg Pentellated 7-orthoplex	File:7-cube t06.svg Hexicated 7-cube		File:7-cube t05.svg Pentellated 7-cube
File:7-cube t04.svg Stericated 7-cube	File:7-cube t02.svg Cantellated 7-cube		File:7-cube t03.svg Runcinated 7-cube
File:7-cube t0.svg 7-cube	File:7-cube t01.svg Truncated 7-cube		File:7-cube t1.svg Rectified 7-cube
File:7-demicube t0 D7.svg 7-demicube	File:7-demicube t01 D7.svg Truncated 7-demicube		File:7-demicube t02 D7.svg Cantellated 7-demicube
File:7-demicube t03 D7.svg Runcinated 7-demicube	File:7-demicube t04 D7.svg Stericated 7-demicube		File:7-demicube t05 D7.svg Pentellated 7-demicube
File:E7 graph.svg 3₂₁	File:Gosset 2 31 polytope.svg 2₃₁		File:Gosset 1 32 petrie.svg 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 which is vertex-transitive, and constructed from uniform 6-polytope facets.

A proposed name for 7-polytopes is polyexon or polyecton.

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:

{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.

Euler characteristic

The Euler characteristic for 7-polytopes that are topological 6-spheres (including all convex 7-polytopes) is two. χ=V-E+F-C+f₄-f₅+f₆=2.

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		Coxeter-Dynkin diagram	Regular and semiregular forms	Uniform count
1	A₇	[3⁶]	Template:CDD	7-simplex - {3⁶}, Template:CDD	71
2	B₇	[4,3⁵]	Template:CDD	7-cube - {4,3⁵}, Template:CDD 7-orthoplex - {3⁵,4}, Template:CDD	127
3	D₇	[3^4,1,1]	Template:CDD	7-demicube, {3^1,4,1}, Template:CDD 7-orthoplex, {3^4,1,1}, Template:CDD	95 (32 unique)
4	E₇	[3^3,2,1]	Template:CDD	3₂₁ - Template:CDD 1₃₂ - Template:CDD 2₃₁ - Template:CDD	127

Prismatic finite Coxeter groups
#	Coxeter group		Coxeter-Dynkin diagram
6+1
1	A₆×A₁	[3⁵]×[ ]	Template:CDD
2	BC₆×A₁	[4,3⁴]×[ ]	Template:CDD
3	D₆×A₁	[3^3,1,1]×[ ]	Template:CDD
4	E₆×A₁	[3^2,2,1]×[ ]	Template:CDD
5+2
1	A₅×I₂(p)	[3,3,3]×[p]	Template:CDD
2	BC₅×I₂(p)	[4,3,3]×[p]	Template:CDD
3	D₅×I₂(p)	[3^2,1,1]×[p]	Template:CDD
5+1+1
1	A₅×A₁²	[3,3,3]×[ ]²	Template:CDD
2	BC₅×A₁²	[4,3,3]×[ ]²	Template:CDD
3	D₅×A₁²	[3^2,1,1]×[ ]²	Template:CDD
4+3
4	A₄×A₃	[3,3,3]×[3,3]	Template:CDD
5	A₄×BC₃	[3,3,3]×[4,3]	Template:CDD
6	A₄×H₃	[3,3,3]×[5,3]	Template:CDD
7	BC₄×A₃	[4,3,3]×[3,3]	Template:CDD
8	BC₄×BC₃	[4,3,3]×[4,3]	Template:CDD
9	BC₄×H₃	[4,3,3]×[5,3]	Template:CDD
10	H₄×A₃	[5,3,3]×[3,3]	Template:CDD
11	H₄×BC₃	[5,3,3]×[4,3]	Template:CDD
12	H₄×H₃	[5,3,3]×[5,3]	Template:CDD
13	F₄×A₃	[3,4,3]×[3,3]	Template:CDD
14	F₄×BC₃	[3,4,3]×[4,3]	Template:CDD
15	F₄×H₃	[3,4,3]×[5,3]	Template:CDD
16	D₄×A₃	[3^1,1,1]×[3,3]	Template:CDD
17	D₄×BC₃	[3^1,1,1]×[4,3]	Template:CDD
18	D₄×H₃	[3^1,1,1]×[5,3]	Template:CDD
4+2+1
5	A₄×I₂(p)×A₁	[3,3,3]×[p]×[ ]	Template:CDD
6	BC₄×I₂(p)×A₁	[4,3,3]×[p]×[ ]	Template:CDD
7	F₄×I₂(p)×A₁	[3,4,3]×[p]×[ ]	Template:CDD
8	H₄×I₂(p)×A₁	[5,3,3]×[p]×[ ]	Template:CDD
9	D₄×I₂(p)×A₁	[3^1,1,1]×[p]×[ ]	Template:CDD
4+1+1+1
5	A₄×A₁³	[3,3,3]×[ ]³	Template:CDD
6	BC₄×A₁³	[4,3,3]×[ ]³	Template:CDD
7	F₄×A₁³	[3,4,3]×[ ]³	Template:CDD
8	H₄×A₁³	[5,3,3]×[ ]³	Template:CDD
9	D₄×A₁³	[3^1,1,1]×[ ]³	Template:CDD
3+3+1
10	A₃×A₃×A₁	[3,3]×[3,3]×[ ]	Template:CDD
11	A₃×BC₃×A₁	[3,3]×[4,3]×[ ]	Template:CDD
12	A₃×H₃×A₁	[3,3]×[5,3]×[ ]	Template:CDD
13	BC₃×BC₃×A₁	[4,3]×[4,3]×[ ]	Template:CDD
14	BC₃×H₃×A₁	[4,3]×[5,3]×[ ]	Template:CDD
15	H₃×A₃×A₁	[5,3]×[5,3]×[ ]	Template:CDD
3+2+2
1	A₃×I₂(p)×I₂(q)	[3,3]×[p]×[q]	Template:CDD
2	BC₃×I₂(p)×I₂(q)	[4,3]×[p]×[q]	Template:CDD
3	H₃×I₂(p)×I₂(q)	[5,3]×[p]×[q]	Template:CDD
3+2+1+1
1	A₃×I₂(p)×A₁²	[3,3]×[p]×[ ]²	Template:CDD
2	BC₃×I₂(p)×A₁²	[4,3]×[p]×[ ]²	Template:CDD
3	H₃×I₂(p)×A₁²	[5,3]×[p]×[ ]²	Template:CDD
3+1+1+1+1
1	A₃×A₁⁴	[3,3]×[ ]⁴	Template:CDD
2	BC₃×A₁⁴	[4,3]×[ ]⁴	Template:CDD
3	H₃×A₁⁴	[5,3]×[ ]⁴	Template:CDD
2+2+2+1
1	I₂(p)×I₂(q)×I₂(r)×A₁	[p]×[q]×[r]×[ ]	Template:CDD
2+2+1+1+1
1	I₂(p)×I₂(q)×A₁³	[p]×[q]×[ ]³	Template:CDD
2+1+1+1+1+1
1	I₂(p)×A₁⁵	[p]×[ ]⁵	Template:CDD
1+1+1+1+1+1+1
1	A₁⁷	[ ]⁷	Template:CDD

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.

#	Coxeter-Dynkin diagram	Truncation indices	Johnson name Bowers name (and acronym)	Basepoint	Element counts
#	Coxeter-Dynkin diagram	Truncation indices	Johnson name Bowers name (and acronym)	Basepoint	6	5	4	3	2	1	0
1	Template:CDD	t₀	7-simplex (oca)	(0,0,0,0,0,0,0,1)	8	28	56	70	56	28	8
2	Template:CDD	t₁	Rectified 7-simplex (roc)	(0,0,0,0,0,0,1,1)	16	84	224	350	336	168	28
3	Template:CDD	t₂	Birectified 7-simplex (broc)	(0,0,0,0,0,1,1,1)	16	112	392	770	840	420	56
4	Template:CDD	t₃	Trirectified 7-simplex (he)	(0,0,0,0,1,1,1,1)	16	112	448	980	1120	560	70
5	Template:CDD	t_0,1	Truncated 7-simplex (toc)	(0,0,0,0,0,0,1,2)	16	84	224	350	336	196	56
6	Template:CDD	t_0,2	Cantellated 7-simplex (saro)	(0,0,0,0,0,1,1,2)	44	308	980	1750	1876	1008	168
7	Template:CDD	t_1,2	Bitruncated 7-simplex (bittoc)	(0,0,0,0,0,1,2,2)						588	168
8	Template:CDD	t_0,3	Runcinated 7-simplex (spo)	(0,0,0,0,1,1,1,2)	100	756	2548	4830	4760	2100	280
9	Template:CDD	t_1,3	Bicantellated 7-simplex (sabro)	(0,0,0,0,1,1,2,2)						2520	420
10	Template:CDD	t_2,3	Tritruncated 7-simplex (tattoc)	(0,0,0,0,1,2,2,2)						980	280
11	Template:CDD	t_0,4	Stericated 7-simplex (sco)	(0,0,0,1,1,1,1,2)						2240	280
12	Template:CDD	t_1,4	Biruncinated 7-simplex (sibpo)	(0,0,0,1,1,1,2,2)						4200	560
13	Template:CDD	t_2,4	Tricantellated 7-simplex (stiroh)	(0,0,0,1,1,2,2,2)						3360	560
14	Template:CDD	t_0,5	Pentellated 7-simplex (seto)	(0,0,1,1,1,1,1,2)						1260	168
15	Template:CDD	t_1,5	Bistericated 7-simplex (sabach)	(0,0,1,1,1,1,2,2)						3360	420
16	Template:CDD	t_0,6	Hexicated 7-simplex (suph)	(0,1,1,1,1,1,1,2)						336	56
17	Template:CDD	t_0,1,2	Cantitruncated 7-simplex (garo)	(0,0,0,0,0,1,2,3)						1176	336
18	Template:CDD	t_0,1,3	Runcitruncated 7-simplex (patto)	(0,0,0,0,1,1,2,3)						4620	840
19	Template:CDD	t_0,2,3	Runcicantellated 7-simplex (paro)	(0,0,0,0,1,2,2,3)						3360	840
20	Template:CDD	t_1,2,3	Bicantitruncated 7-simplex (gabro)	(0,0,0,0,1,2,3,3)						2940	840
21	Template:CDD	t_0,1,4	Steritruncated 7-simplex (cato)	(0,0,0,1,1,1,2,3)						7280	1120
22	Template:CDD	t_0,2,4	Stericantellated 7-simplex (caro)	(0,0,0,1,1,2,2,3)						10080	1680
23	Template:CDD	t_1,2,4	Biruncitruncated 7-simplex (bipto)	(0,0,0,1,1,2,3,3)						8400	1680
24	Template:CDD	t_0,3,4	Steriruncinated 7-simplex (cepo)	(0,0,0,1,2,2,2,3)						5040	1120
25	Template:CDD	t_1,3,4	Biruncicantellated 7-simplex (bipro)	(0,0,0,1,2,2,3,3)						7560	1680
26	Template:CDD	t_2,3,4	Tricantitruncated 7-simplex (gatroh)	(0,0,0,1,2,3,3,3)						3920	1120
27	Template:CDD	t_0,1,5	Pentitruncated 7-simplex (teto)	(0,0,1,1,1,1,2,3)						5460	840
28	Template:CDD	t_0,2,5	Penticantellated 7-simplex (tero)	(0,0,1,1,1,2,2,3)						11760	1680
29	Template:CDD	t_1,2,5	Bisteritruncated 7-simplex (bacto)	(0,0,1,1,1,2,3,3)						9240	1680
30	Template:CDD	t_0,3,5	Pentiruncinated 7-simplex (tepo)	(0,0,1,1,2,2,2,3)						10920	1680
31	Template:CDD	t_1,3,5	Bistericantellated 7-simplex (bacroh)	(0,0,1,1,2,2,3,3)						15120	2520
32	Template:CDD	t_0,4,5	Pentistericated 7-simplex (teco)	(0,0,1,2,2,2,2,3)						4200	840
33	Template:CDD	t_0,1,6	Hexitruncated 7-simplex (puto)	(0,1,1,1,1,1,2,3)						1848	336
34	Template:CDD	t_0,2,6	Hexicantellated 7-simplex (puro)	(0,1,1,1,1,2,2,3)						5880	840
35	Template:CDD	t_0,3,6	Hexiruncinated 7-simplex (puph)	(0,1,1,1,2,2,2,3)						8400	1120
36	Template:CDD	t_0,1,2,3	Runcicantitruncated 7-simplex (gapo)	(0,0,0,0,1,2,3,4)						5880	1680
37	Template:CDD	t_0,1,2,4	Stericantitruncated 7-simplex (cagro)	(0,0,0,1,1,2,3,4)						16800	3360
38	Template:CDD	t_0,1,3,4	Steriruncitruncated 7-simplex (capto)	(0,0,0,1,2,2,3,4)						13440	3360
39	Template:CDD	t_0,2,3,4	Steriruncicantellated 7-simplex (capro)	(0,0,0,1,2,3,3,4)						13440	3360
40	Template:CDD	t_1,2,3,4	Biruncicantitruncated 7-simplex (gibpo)	(0,0,0,1,2,3,4,4)						11760	3360
41	Template:CDD	t_0,1,2,5	Penticantitruncated 7-simplex (tegro)	(0,0,1,1,1,2,3,4)						18480	3360
42	Template:CDD	t_0,1,3,5	Pentiruncitruncated 7-simplex (tapto)	(0,0,1,1,2,2,3,4)						27720	5040
43	Template:CDD	t_0,2,3,5	Pentiruncicantellated 7-simplex (tapro)	(0,0,1,1,2,3,3,4)						25200	5040
44	Template:CDD	t_1,2,3,5	Bistericantitruncated 7-simplex (bacogro)	(0,0,1,1,2,3,4,4)						22680	5040
45	Template:CDD	t_0,1,4,5	Pentisteritruncated 7-simplex (tecto)	(0,0,1,2,2,2,3,4)						15120	3360
46	Template:CDD	t_0,2,4,5	Pentistericantellated 7-simplex (tecro)	(0,0,1,2,2,3,3,4)						25200	5040
47	Template:CDD	t_1,2,4,5	Bisteriruncitruncated 7-simplex (bicpath)	(0,0,1,2,2,3,4,4)						20160	5040
48	Template:CDD	t_0,3,4,5	Pentisteriruncinated 7-simplex (tacpo)	(0,0,1,2,3,3,3,4)						15120	3360
49	Template:CDD	t_0,1,2,6	Hexicantitruncated 7-simplex (pugro)	(0,1,1,1,1,2,3,4)						8400	1680
50	Template:CDD	t_0,1,3,6	Hexiruncitruncated 7-simplex (pugato)	(0,1,1,1,2,2,3,4)						20160	3360
51	Template:CDD	t_0,2,3,6	Hexiruncicantellated 7-simplex (pugro)	(0,1,1,1,2,3,3,4)						16800	3360
52	Template:CDD	t_0,1,4,6	Hexisteritruncated 7-simplex (pucto)	(0,1,1,2,2,2,3,4)						20160	3360
53	Template:CDD	t_0,2,4,6	Hexistericantellated 7-simplex (pucroh)	(0,1,1,2,2,3,3,4)						30240	5040
54	Template:CDD	t_0,1,5,6	Hexipentitruncated 7-simplex (putath)	(0,1,2,2,2,2,3,4)						8400	1680
55	Template:CDD	t_0,1,2,3,4	Steriruncicantitruncated 7-simplex (gecco)	(0,0,0,1,2,3,4,5)						23520	6720
56	Template:CDD	t_0,1,2,3,5	Pentiruncicantitruncated 7-simplex (tegapo)	(0,0,1,1,2,3,4,5)						45360	10080
57	Template:CDD	t_0,1,2,4,5	Pentistericantitruncated 7-simplex (tecagro)	(0,0,1,2,2,3,4,5)						40320	10080
58	Template:CDD	t_0,1,3,4,5	Pentisteriruncitruncated 7-simplex (tacpeto)	(0,0,1,2,3,3,4,5)						40320	10080
59	Template:CDD	t_0,2,3,4,5	Pentisteriruncicantellated 7-simplex (tacpro)	(0,0,1,2,3,4,4,5)						40320	10080
60	Template:CDD	t_1,2,3,4,5	Bisteriruncicantitruncated 7-simplex (gabach)	(0,0,1,2,3,4,5,5)						35280	10080
61	Template:CDD	t_0,1,2,3,6	Hexiruncicantitruncated 7-simplex (pugopo)	(0,1,1,1,2,3,4,5)						30240	6720
62	Template:CDD	t_0,1,2,4,6	Hexistericantitruncated 7-simplex (pucagro)	(0,1,1,2,2,3,4,5)						50400	10080
63	Template:CDD	t_0,1,3,4,6	Hexisteriruncitruncated 7-simplex (pucpato)	(0,1,1,2,3,3,4,5)						45360	10080
64	Template:CDD	t_0,2,3,4,6	Hexisteriruncicantellated 7-simplex (pucproh)	(0,1,1,2,3,4,4,5)						45360	10080
65	Template:CDD	t_0,1,2,5,6	Hexipenticantitruncated 7-simplex (putagro)	(0,1,2,2,2,3,4,5)						30240	6720
66	Template:CDD	t_0,1,3,5,6	Hexipentiruncitruncated 7-simplex (putpath)	(0,1,2,2,3,3,4,5)						50400	10080
67	Template:CDD	t_0,1,2,3,4,5	Pentisteriruncicantitruncated 7-simplex (geto)	(0,0,1,2,3,4,5,6)						70560	20160
68	Template:CDD	t_0,1,2,3,4,6	Hexisteriruncicantitruncated 7-simplex (pugaco)	(0,1,1,2,3,4,5,6)						80640	20160
69	Template:CDD	t_0,1,2,3,5,6	Hexipentiruncicantitruncated 7-simplex (putgapo)	(0,1,2,2,3,4,5,6)						80640	20160
70	Template:CDD	t_0,1,2,4,5,6	Hexipentistericantitruncated 7-simplex (putcagroh)	(0,1,2,3,3,4,5,6)						80640	20160
71	Template:CDD	t_{0,1,2,3,4,5,6}	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 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.

#	Coxeter-Dynkin diagram t-notation	Name (BSA)	Base point	Element counts
#	Coxeter-Dynkin diagram t-notation	Name (BSA)	Base point	6	5	4	3	2	1	0
1	Template:CDD 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	Template:CDD 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	Template:CDD 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	Template:CDD 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	Template:CDD 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	Template:CDD 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	Template:CDD 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	Template:CDD t_0,1{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	Template:CDD t_0,2{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	Template:CDD t_1,2{3,3,3,3,3,4}	Bitruncated 7-orthoplex (Botaz)	(0,0,0,0,1,2,2)√2						4200	840
11	Template:CDD t_0,3{3,3,3,3,3,4}	Runcinated 7-orthoplex (Spaz)	(0,0,0,1,1,1,2)√2						23520	2240
12	Template:CDD t_1,3{3,3,3,3,3,4}	Bicantellated 7-orthoplex (Sebraz)	(0,0,0,1,1,2,2)√2						26880	3360
13	Template:CDD t_2,3{3,3,3,3,3,4}	Tritruncated 7-orthoplex (Totaz)	(0,0,0,1,2,2,2)√2						10080	2240
14	Template:CDD t_0,4{3,3,3,3,3,4}	Stericated 7-orthoplex (Scaz)	(0,0,1,1,1,1,2)√2						33600	3360
15	Template:CDD t_1,4{3,3,3,3,3,4}	Biruncinated 7-orthoplex (Sibpaz)	(0,0,1,1,1,2,2)√2						60480	6720
16	Template:CDD t_2,4{4,3,3,3,3,3}	Tricantellated 7-cube (Strasaz)	(0,0,1,1,2,2,2)√2						47040	6720
17	Template:CDD t_2,3{4,3,3,3,3,3}	Tritruncated 7-cube (Tatsa)	(0,0,1,2,2,2,2)√2						13440	3360
18	Template:CDD t_0,5{3,3,3,3,3,4}	Pentellated 7-orthoplex (Staz)	(0,1,1,1,1,1,2)√2						20160	2688
19	Template:CDD t_1,5{4,3,3,3,3,3}	Bistericated 7-cube (Sabcosaz)	(0,1,1,1,1,2,2)√2						53760	6720
20	Template:CDD t_1,4{4,3,3,3,3,3}	Biruncinated 7-cube (Sibposa)	(0,1,1,1,2,2,2)√2						67200	8960
21	Template:CDD t_1,3{4,3,3,3,3,3}	Bicantellated 7-cube (Sibrosa)	(0,1,1,2,2,2,2)√2						40320	6720
22	Template:CDD t_1,2{4,3,3,3,3,3}	Bitruncated 7-cube (Betsa)	(0,1,2,2,2,2,2)√2						9408	2688
23	Template:CDD t_0,6{4,3,3,3,3,3}	Hexicated 7-cube (Suposaz)	(0,0,0,0,0,0,1)√2 + (1,1,1,1,1,1,1)						5376	896
24	Template:CDD t_0,5{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	Template:CDD t_0,4{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	Template:CDD t_0,3{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	Template:CDD t_0,2{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	Template:CDD t_0,1{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	Template:CDD t_0,1,2{3,3,3,3,3,4}	Cantitruncated 7-orthoplex (Garz)	(0,1,2,3,3,3,3)√2						8400	1680
30	Template:CDD t_0,1,3{3,3,3,3,3,4}	Runcitruncated 7-orthoplex (Potaz)	(0,1,2,2,3,3,3)√2						50400	6720
31	Template:CDD t_0,2,3{3,3,3,3,3,4}	Runcicantellated 7-orthoplex (Parz)	(0,1,1,2,3,3,3)√2						33600	6720
32	Template:CDD t_1,2,3{3,3,3,3,3,4}	Bicantitruncated 7-orthoplex (Gebraz)	(0,0,1,2,3,3,3)√2						30240	6720
33	Template:CDD t_0,1,4{3,3,3,3,3,4}	Steritruncated 7-orthoplex (Catz)	(0,0,1,1,1,2,3)√2						107520	13440
34	Template:CDD t_0,2,4{3,3,3,3,3,4}	Stericantellated 7-orthoplex (Craze)	(0,0,1,1,2,2,3)√2						141120	20160
35	Template:CDD t_1,2,4{3,3,3,3,3,4}	Biruncitruncated 7-orthoplex (Baptize)	(0,0,1,1,2,3,3)√2						120960	20160
36	Template:CDD t_0,3,4{3,3,3,3,3,4}	Steriruncinated 7-orthoplex (Copaz)	(0,1,1,1,2,3,3)√2						67200	13440
37	Template:CDD t_1,3,4{3,3,3,3,3,4}	Biruncicantellated 7-orthoplex (Boparz)	(0,0,1,2,2,3,3)√2						100800	20160
38	Template:CDD t_2,3,4{4,3,3,3,3,3}	Tricantitruncated 7-cube (Gotrasaz)	(0,0,0,1,2,3,3)√2						53760	13440
39	Template:CDD t_0,1,5{3,3,3,3,3,4}	Pentitruncated 7-orthoplex (Tetaz)	(0,1,1,1,1,2,3)√2						87360	13440
40	Template:CDD t_0,2,5{3,3,3,3,3,4}	Penticantellated 7-orthoplex (Teroz)	(0,1,1,1,2,2,3)√2						188160	26880
41	Template:CDD t_1,2,5{3,3,3,3,3,4}	Bisteritruncated 7-orthoplex (Boctaz)	(0,1,1,1,2,3,3)√2						147840	26880
42	Template:CDD t_0,3,5{3,3,3,3,3,4}	Pentiruncinated 7-orthoplex (Topaz)	(0,1,1,2,2,2,3)√2						174720	26880
43	Template:CDD t_1,3,5{4,3,3,3,3,3}	Bistericantellated 7-cube (Bacresaz)	(0,1,1,2,2,3,3)√2						241920	40320
44	Template:CDD t_1,3,4{4,3,3,3,3,3}	Biruncicantellated 7-cube (Bopresa)	(0,1,1,2,3,3,3)√2						120960	26880
45	Template:CDD t_0,4,5{3,3,3,3,3,4}	Pentistericated 7-orthoplex (Tocaz)	(0,1,2,2,2,2,3)√2						67200	13440
46	Template:CDD t_1,2,5{4,3,3,3,3,3}	Bisteritruncated 7-cube (Bactasa)	(0,1,2,2,2,3,3)√2						147840	26880
47	Template:CDD t_1,2,4{4,3,3,3,3,3}	Biruncitruncated 7-cube (Biptesa)	(0,1,2,2,3,3,3)√2						134400	26880
48	Template:CDD t_1,2,3{4,3,3,3,3,3}	Bicantitruncated 7-cube (Gibrosa)	(0,1,2,3,3,3,3)√2						47040	13440
49	Template:CDD t_0,1,6{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	Template:CDD t_0,2,6{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	Template:CDD t_0,4,5{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	Template:CDD t_0,3,6{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	Template:CDD t_0,3,5{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	Template:CDD t_0,3,4{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	Template:CDD t_0,2,6{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	Template:CDD t_0,2,5{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	Template:CDD t_0,2,4{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	Template:CDD t_0,2,3{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	Template:CDD t_0,1,6{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	Template:CDD t_0,1,5{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	Template:CDD t_0,1,4{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	Template:CDD t_0,1,3{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	Template:CDD t_0,1,2{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	Template:CDD t_0,1,2,3{3,3,3,3,3,4}	Runcicantitruncated 7-orthoplex (Gopaz)	(0,1,2,3,4,4,4)√2						60480	13440
65	Template:CDD t_0,1,2,4{3,3,3,3,3,4}	Stericantitruncated 7-orthoplex (Cogarz)	(0,0,1,1,2,3,4)√2						241920	40320
66	Template:CDD t_0,1,3,4{3,3,3,3,3,4}	Steriruncitruncated 7-orthoplex (Captaz)	(0,0,1,2,2,3,4)√2						181440	40320
67	Template:CDD t_0,2,3,4{3,3,3,3,3,4}	Steriruncicantellated 7-orthoplex (Caparz)	(0,0,1,2,3,3,4)√2						181440	40320
68	Template:CDD t_1,2,3,4{3,3,3,3,3,4}	Biruncicantitruncated 7-orthoplex (Gibpaz)	(0,0,1,2,3,4,4)√2						161280	40320
69	Template:CDD t_0,1,2,5{3,3,3,3,3,4}	Penticantitruncated 7-orthoplex (Tograz)	(0,1,1,1,2,3,4)√2						295680	53760
70	Template:CDD t_0,1,3,5{3,3,3,3,3,4}	Pentiruncitruncated 7-orthoplex (Toptaz)	(0,1,1,2,2,3,4)√2						443520	80640
71	Template:CDD t_0,2,3,5{3,3,3,3,3,4}	Pentiruncicantellated 7-orthoplex (Toparz)	(0,1,1,2,3,3,4)√2						403200	80640
72	Template:CDD t_1,2,3,5{3,3,3,3,3,4}	Bistericantitruncated 7-orthoplex (Becogarz)	(0,1,1,2,3,4,4)√2						362880	80640
73	Template:CDD t_0,1,4,5{3,3,3,3,3,4}	Pentisteritruncated 7-orthoplex (Tacotaz)	(0,1,2,2,2,3,4)√2						241920	53760
74	Template:CDD t_0,2,4,5{3,3,3,3,3,4}	Pentistericantellated 7-orthoplex (Tocarz)	(0,1,2,2,3,3,4)√2						403200	80640
75	Template:CDD t_1,2,4,5{4,3,3,3,3,3}	Bisteriruncitruncated 7-cube (Bocaptosaz)	(0,1,2,2,3,4,4)√2						322560	80640
76	Template:CDD t_0,3,4,5{3,3,3,3,3,4}	Pentisteriruncinated 7-orthoplex (Tecpaz)	(0,1,2,3,3,3,4)√2						241920	53760
77	Template:CDD t_1,2,3,5{4,3,3,3,3,3}	Bistericantitruncated 7-cube (Becgresa)	(0,1,2,3,3,4,4)√2						362880	80640
78	Template:CDD t_1,2,3,4{4,3,3,3,3,3}	Biruncicantitruncated 7-cube (Gibposa)	(0,1,2,3,4,4,4)√2						188160	53760
79	Template:CDD t_0,1,2,6{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	Template:CDD t_0,1,3,6{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	Template:CDD t_0,2,3,6{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	Template:CDD t_0,3,4,5{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	Template:CDD t_0,1,4,6{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	Template:CDD t_0,2,4,6{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	Template:CDD t_0,2,4,5{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	Template:CDD t_0,2,3,6{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	Template:CDD t_0,2,3,5{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	Template:CDD t_0,2,3,4{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	Template:CDD t_0,1,5,6{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	Template:CDD t_0,1,4,6{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	Template:CDD t_0,1,4,5{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	Template:CDD t_0,1,3,6{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	Template:CDD t_0,1,3,5{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	Template:CDD t_0,1,3,4{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	Template:CDD t_0,1,2,6{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	Template:CDD t_0,1,2,5{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	Template:CDD t_0,1,2,4{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	Template:CDD t_0,1,2,3{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	Template:CDD t_0,1,2,3,4{3,3,3,3,3,4}	Steriruncicantitruncated 7-orthoplex (Gocaz)	(0,0,1,2,3,4,5)√2						322560	80640
100	Template:CDD t_0,1,2,3,5{3,3,3,3,3,4}	Pentiruncicantitruncated 7-orthoplex (Tegopaz)	(0,1,1,2,3,4,5)√2						725760	161280
101	Template:CDD t_0,1,2,4,5{3,3,3,3,3,4}	Pentistericantitruncated 7-orthoplex (Tecagraz)	(0,1,2,2,3,4,5)√2						645120	161280
102	Template:CDD t_0,1,3,4,5{3,3,3,3,3,4}	Pentisteriruncitruncated 7-orthoplex (Tecpotaz)	(0,1,2,3,3,4,5)√2						645120	161280
103	Template:CDD t_0,2,3,4,5{3,3,3,3,3,4}	Pentisteriruncicantellated 7-orthoplex (Tacparez)	(0,1,2,3,4,4,5)√2						645120	161280
104	Template:CDD t_1,2,3,4,5{4,3,3,3,3,3}	Bisteriruncicantitruncated 7-cube (Gabcosaz)	(0,1,2,3,4,5,5)√2						564480	161280
105	Template:CDD t_0,1,2,3,6{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	Template:CDD t_0,1,2,4,6{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	Template:CDD t_0,1,3,4,6{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	Template:CDD t_0,2,3,4,6{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	Template:CDD t_0,2,3,4,5{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	Template:CDD t_0,1,2,5,6{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	Template:CDD t_0,1,3,5,6{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	Template:CDD t_0,1,3,4,6{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	Template:CDD t_0,1,3,4,5{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	Template:CDD t_0,1,2,5,6{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	Template:CDD t_0,1,2,4,6{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	Template:CDD t_0,1,2,4,5{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	Template:CDD t_0,1,2,3,6{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	Template:CDD t_0,1,2,3,5{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	Template:CDD t_0,1,2,3,4{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	Template:CDD t_0,1,2,3,4,5{3,3,3,3,3,4}	Pentisteriruncicantitruncated 7-orthoplex (Gotaz)	(0,1,2,3,4,5,6)√2						1128960	322560
121	Template:CDD t_0,1,2,3,4,6{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	Template:CDD t_0,1,2,3,5,6{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	Template:CDD t_0,1,2,4,5,6{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	Template:CDD t_0,1,2,3,5,6{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	Template:CDD t_0,1,2,3,4,6{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	Template:CDD t_0,1,2,3,4,5{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	Template:CDD t_{0,1,2,3,4,5,6}{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 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.

#	Coxeter diagram	Names	Base point (Alternately signed)	Element counts
#	Coxeter diagram	Names	Base point (Alternately signed)	6	5	4	3	2	1	0
1	Template:CDD = Template:CDD	7-demicube Demihepteract (Hesa)	(1,1,1,1,1,1,1)	78	532	1624	2800	2240	672	64
2	Template:CDD = Template:CDD	Truncated 7-demicube Truncated demihepteract (Thesa)	(1,1,3,3,3,3,3)	142	1428	5656	11760	13440	7392	1344
3	Template:CDD = Template:CDD	Cantellated 7-demicube Small rhombated demihepteract (Sirhesa)	(1,1,1,3,3,3,3)						16800	2240
4	Template:CDD = Template:CDD	Runcinated 7-demicube Small prismated demihepteract (Sphosa)	(1,1,1,1,3,3,3)						20160	2240
5	Template:CDD = Template:CDD	Stericated 7-demicube Small cellated demihepteract (Sochesa)	(1,1,1,1,1,3,3)						13440	1344
6	Template:CDD = Template:CDD	Pentellated 7-demicube Small terated demihepteract (Suthesa)	(1,1,1,1,1,1,3)						4704	448
7	Template:CDD = Template:CDD	Cantitruncated 7-demicube Great rhombated demihepteract (Girhesa)	(1,1,3,5,5,5,5)						23520	6720
8	Template:CDD = Template:CDD	Runcitruncated 7-demicube Prismatotruncated demihepteract (Pothesa)	(1,1,3,3,5,5,5)						73920	13440
9	Template:CDD = Template:CDD	Runcicantellated 7-demicube Prismatorhomated demihepteract (Prohesa)	(1,1,1,3,5,5,5)						40320	8960
10	Template:CDD = Template:CDD	Steritruncated 7-demicube Cellitruncated demihepteract (Cothesa)	(1,1,3,3,3,5,5)						87360	13440
11	Template:CDD = Template:CDD	Stericantellated 7-demicube Cellirhombated demihepteract (Crohesa)	(1,1,1,3,3,5,5)						87360	13440
12	Template:CDD = Template:CDD	Steriruncinated 7-demicube Celliprismated demihepteract (Caphesa)	(1,1,1,1,3,5,5)						40320	6720
13	Template:CDD = Template:CDD	Pentitruncated 7-demicube Teritruncated demihepteract (Tuthesa)	(1,1,3,3,3,3,5)						43680	6720
14	Template:CDD = Template:CDD	Penticantellated 7-demicube Terirhombated demihepteract (Turhesa)	(1,1,1,3,3,3,5)						67200	8960
15	Template:CDD = Template:CDD	Pentiruncinated 7-demicube Teriprismated demihepteract (Tuphesa)	(1,1,1,1,3,3,5)						53760	6720
16	Template:CDD = Template:CDD	Pentistericated 7-demicube Tericellated demihepteract (Tuchesa)	(1,1,1,1,1,3,5)						21504	2688
17	Template:CDD = Template:CDD	Runcicantitruncated 7-demicube Great prismated demihepteract (Gephosa)	(1,1,3,5,7,7,7)						94080	26880
18	Template:CDD = Template:CDD	Stericantitruncated 7-demicube Celligreatorhombated demihepteract (Cagrohesa)	(1,1,3,5,5,7,7)						181440	40320
19	Template:CDD = Template:CDD	Steriruncitruncated 7-demicube Celliprismatotruncated demihepteract (Capthesa)	(1,1,3,3,5,7,7)						181440	40320
20	Template:CDD = Template:CDD	Steriruncicantellated 7-demicube Celliprismatorhombated demihepteract (Coprahesa)	(1,1,1,3,5,7,7)						120960	26880
21	Template:CDD = Template:CDD	Penticantitruncated 7-demicube Terigreatorhombated demihepteract (Tugrohesa)	(1,1,3,5,5,5,7)						120960	26880
22	Template:CDD = Template:CDD	Pentiruncitruncated 7-demicube Teriprismatotruncated demihepteract (Tupthesa)	(1,1,3,3,5,5,7)						221760	40320
23	Template:CDD = Template:CDD	Pentiruncicantellated 7-demicube Teriprismatorhombated demihepteract (Tuprohesa)	(1,1,1,3,5,5,7)						134400	26880
24	Template:CDD = Template:CDD	Pentisteritruncated 7-demicube Tericellitruncated demihepteract (Tucothesa)	(1,1,3,3,3,5,7)						147840	26880
25	Template:CDD = Template:CDD	Pentistericantellated 7-demicube Tericellirhombated demihepteract (Tucrohesa)	(1,1,1,3,3,5,7)						161280	26880
26	Template:CDD = Template:CDD	Pentisteriruncinated 7-demicube Tericelliprismated demihepteract (Tucophesa)	(1,1,1,1,3,5,7)						80640	13440
27	Template:CDD = Template:CDD	Steriruncicantitruncated 7-demicube Great cellated demihepteract (Gochesa)	(1,1,3,5,7,9,9)						282240	80640
28	Template:CDD = Template:CDD	Pentiruncicantitruncated 7-demicube Terigreatoprimated demihepteract (Tugphesa)	(1,1,3,5,7,7,9)						322560	80640
29	Template:CDD = Template:CDD	Pentistericantitruncated 7-demicube Tericelligreatorhombated demihepteract (Tucagrohesa)	(1,1,3,5,5,7,9)						322560	80640
30	Template:CDD = Template:CDD	Pentisteriruncitruncated 7-demicube Tericelliprismatotruncated demihepteract (Tucpathesa)	(1,1,3,3,5,7,9)						362880	80640
31	Template:CDD = Template:CDD	Pentisteriruncicantellated 7-demicube Tericellprismatorhombated demihepteract (Tucprohesa)	(1,1,1,3,5,7,9)						241920	53760
32	Template:CDD = Template:CDD	Pentisteriruncicantitruncated 7-demicube 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.

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

Regular and uniform honeycombs

File:Coxeter diagram affine rank7 correspondence.png

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	${\tilde{A}}_{6}$	[3^[7]]	Template:CDD	17
2	${\tilde{C}}_{6}$	[4,3⁴,4]	Template:CDD	71
3	${\tilde{B}}_{6}$	h[4,3⁴,4] [4,3³,3^1,1]	Template:CDD	95 (32 new)
4	${\tilde{D}}_{6}$	q[4,3⁴,4] [3^1,1,3²,3^1,1]	Template:CDD	41 (6 new)
5	${\tilde{E}}_{6}$	[3^2,2,2]	Template:CDD	39

Regular and uniform tessellations include:

${\tilde{A}}_{6}$ , 17 forms
- Uniform 6-simplex honeycomb: {3^[7]} Template:CDD
${\tilde{C}}_{6}$ , [4,3⁴,4], 71 forms
- Regular 6-cube honeycomb, represented by symbols {4,3⁴,4}, Template:CDD
${\tilde{B}}_{6}$ , [3^1,1,3³,4], 95 forms, 64 shared with ${\tilde{C}}_{6}$ , 32 new
- Uniform 6-demicube honeycomb, represented by symbols h{4,3⁴,4} = {3^1,1,3³,4}, Template:CDD = Template:CDD
${\tilde{D}}_{6}$ , [3^1,1,3²,3^1,1], 41 unique ringed permuations, most shared with ${\tilde{B}}_{6}$ and ${\tilde{C}}_{6}$ , and 6 are new. Coxeter calls the first one a quarter 6-cubic honeycomb.
- Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD, Template:CDD
${\tilde{E}}_{6}$ : [3^2,2,2], 39 forms
- Uniform 2₂₂ honeycomb: represented by symbols {3^2,2,2}, Template:CDD

Prismatic groups
#	Coxeter group		Coxeter-Dynkin diagram
1	${\tilde{A}}_{5}$ x ${\tilde{I}}_{1}$	[3^[6],2,∞]	Template:CDD
2	${\tilde{B}}_{5}$ x ${\tilde{I}}_{1}$	[4,3,3^1,1,2,∞]	Template:CDD
3	${\tilde{C}}_{5}$ x ${\tilde{I}}_{1}$	[4,3³,4,2,∞]	Template:CDD
4	${\tilde{D}}_{5}$ x ${\tilde{I}}_{1}$	[3^1,1,3,3^1,1,2,∞]	Template:CDD
5	${\tilde{A}}_{4}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[3^[5],2,∞,2,∞,2,∞]	Template:CDD
6	${\tilde{B}}_{4}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[4,3,3^1,1,2,∞,2,∞]	Template:CDD
7	${\tilde{C}}_{4}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[4,3,3,4,2,∞,2,∞]	Template:CDD
8	${\tilde{D}}_{4}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[3^1,1,1,1,2,∞,2,∞]	Template:CDD
9	${\tilde{F}}_{4}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[3,4,3,3,2,∞,2,∞]	Template:CDD
10	${\tilde{C}}_{3}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[4,3,4,2,∞,2,∞,2,∞]	Template:CDD
11	${\tilde{B}}_{3}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[4,3^1,1,2,∞,2,∞,2,∞]	Template:CDD
12	${\tilde{A}}_{3}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[3^[4],2,∞,2,∞,2,∞]	Template:CDD
13	${\tilde{C}}_{2}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[4,4,2,∞,2,∞,2,∞,2,∞]	Template:CDD
14	${\tilde{H}}_{2}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[6,3,2,∞,2,∞,2,∞,2,∞]	Template:CDD
15	${\tilde{A}}_{2}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[3^[3],2,∞,2,∞,2,∞,2,∞]	Template:CDD
16	${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$ x ${\tilde{I}}_{1}$	[∞,2,∞,2,∞,2,∞,2,∞]	Template:CDD

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 noncompact hyperbolic Coxeter groups of rank 7, each generating uniform honeycombs in 6-space as permutations of rings of the Coxeter diagrams.

${\bar{P}}_{6}$ = [3,3^[6]]: Template:CDD	${\bar{Q}}_{6}$ = [3^1,1,3,3^2,1]: Template:CDD	${\bar{S}}_{6}$ = [4,3,3,3^2,1]: Template:CDD

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}	Template:CDD	Any regular 7-polytope
Rectified	t₁{p,q,r,s,t,u}	Template:CDD	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}	Template:CDD	Birectification reduces cells to their duals.
Truncated	t_0,1{p,q,r,s,t,u}	Template:CDD	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. File:Cube truncation sequence.svg
Bitruncated	t_1,2{p,q,r,s,t,u}	Template:CDD	Bitrunction transforms cells to their dual truncation.
Tritruncated	t_2,3{p,q,r,s,t,u}	Template:CDD	Tritruncation transforms 4-faces to their dual truncation.
Cantellated	t_0,2{p,q,r,s,t,u}	Template:CDD	In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms. File:Cube cantellation sequence.svg
Bicantellated	t_1,3{p,q,r,s,t,u}	Template:CDD	In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms.
Runcinated	t_0,3{p,q,r,s,t,u}	Template:CDD	Runcination reduces cells and creates new cells at the vertices and edges.
Biruncinated	t_1,4{p,q,r,s,t,u}	Template:CDD	Runcination reduces cells and creates new cells at the vertices and edges.
Stericated	t_0,4{p,q,r,s,t,u}	Template:CDD	Sterication reduces 4-faces and creates new 4-faces at the vertices, edges, and faces in the gaps.
Pentellated	t_0,5{p,q,r,s,t,u}	Template:CDD	Pentellation reduces 5-faces and creates new 5-faces at the vertices, edges, faces, and cells in the gaps.
Hexicated	t_0,6{p,q,r,s,t,u}	Template:CDD	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	t_{0,1,2,3,4,5,6}{p,q,r,s,t,u}	Template:CDD	All six operators, truncation, cantellation, runcination, sterication, pentellation, and hexication are applied.

References

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, editied 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, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
- (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
Template:KlitzingPolytopes

External links

Template:Polytopes

Convergence of measures: Difference between revisions

en>Hairer

Revision as of 19:47, 3 February 2014

Contents

Regular 7-polytopes

Euler characteristic

Uniform 7-polytopes by fundamental Coxeter groups

The A₇ family

The B₇ family

The D₇ family

The E₇ family

Regular and uniform honeycombs

Regular and uniform hyperbolic honeycombs

Notes on the Wythoff construction for the uniform 7-polytopes

References

External links

Navigation menu

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+{| align=right class=wikitable width=300
+|+ Graphs of three [[List of regular polytopes#Convex 4|regular]] and related [[uniform polytope]]s
+| || || || || || || || || || ||
+|-  style="vertical-align:top; text-align:center;"
+|colspan=4|[[File:7-simplex t0.svg|100px]]<br/>[[7-simplex]]
+|colspan=4|[[File:7-simplex t1.svg|100px]]<br/>[[Rectified 7-simplex]]
+|colspan=4|[[File:7-simplex t01.svg|100px]]<br/>[[Truncated 7-simplex]]
+|-  style="vertical-align:top; text-align:center;"
+|colspan=4|[[File:7-simplex t02.svg|100px]]<br/>[[Cantellated 7-simplex]]
+|colspan=4|[[File:7-simplex t03.svg|100px]]<br/>[[Runcinated 7-simplex]]
+|colspan=4|[[File:7-simplex t04.svg|100px]]<br/>[[Stericated 7-simplex]]
+|-  style="vertical-align:top; text-align:center;"
+|colspan=6|[[File:7-simplex t05.svg|150px]]<br/>[[Pentellated 7-simplex]]
+|colspan=6|[[File:7-simplex t06.svg|150px]]<br/>[[Hexicated 7-simplex]]
+|-  style="vertical-align:top; text-align:center;"
+|colspan=4|[[File:7-cube t6.svg|100px]]<br/>[[7-orthoplex]]
+|colspan=4|[[File:7-cube t56.svg|100px]]<br/>[[Truncated 7-orthoplex]]
+|colspan=4|[[File:7-cube t5.svg|100px]]<br/>[[Rectified 7-orthoplex]]
+|-  style="vertical-align:top; text-align:center;"
+|colspan=4|[[File:7-cube t46.svg|100px]]<br/>[[Cantellated 7-orthoplex]]
+|colspan=4|[[File:7-cube t36.svg|100px]]<br/>[[Runcinated 7-orthoplex]]
+|colspan=4|[[File:7-cube t26.svg|100px]]<br/>[[Stericated 7-orthoplex]]
+|-  style="vertical-align:top; text-align:center;"
+|colspan=4|[[File:7-cube t16.svg|100px]]<br/>[[Pentellated 7-orthoplex]]
+|colspan=4|[[File:7-cube t06.svg|100px]]<br/>[[Hexicated 7-cube]]
+|colspan=4|[[File:7-cube t05.svg|100px]]<br/>[[Pentellated 7-cube]]
+|-  style="vertical-align:top; text-align:center;"
+|colspan=4|[[File:7-cube t04.svg|100px]]<br/>[[Stericated 7-cube]]
+|colspan=4|[[File:7-cube t02.svg|100px]]<br/>[[Cantellated 7-cube]]
+|colspan=4|[[File:7-cube t03.svg|100px]]<br/>[[Runcinated 7-cube]]
+|-  style="vertical-align:top; text-align:center;"
+|colspan=4|[[File:7-cube t0.svg|100px]]<br/>[[7-cube]]
+|colspan=4|[[File:7-cube t01.svg|100px]]<br/>[[Truncated 7-cube]]
+|colspan=4|[[File:7-cube t1.svg|100px]]<br/>[[Rectified 7-cube]]
+|-  style="vertical-align:top; text-align:center;"
+|colspan=4|[[File:7-demicube t0 D7.svg|100px]]<br/>[[7-demicube]]
+|colspan=4|[[File:7-demicube t01 D7.svg|100px]]<br/>[[Truncated 7-demicube]]
+|colspan=4|[[File:7-demicube t02 D7.svg|100px]]<br/>[[Cantellated 7-demicube]]
+|-  style="vertical-align:top; text-align:center;"
+|colspan=4|[[File:7-demicube t03 D7.svg|100px]]<br/>[[Runcinated 7-demicube]]
+|colspan=4|[[File:7-demicube t04 D7.svg|100px]]<br/>[[Stericated 7-demicube]]
+|colspan=4|[[File:7-demicube t05 D7.svg|100px]]<br/>[[Pentellated 7-demicube]]
+|-  style="vertical-align:top; text-align:center;"
+|colspan=4|[[File:E7 graph.svg|100px]]<br/>[[3 21 polytope|3<sub>21</sub>]]
+|colspan=4|[[File:Gosset 2 31 polytope.svg|100px]]<br/>[[2 31 polytope|2<sub>31</sub>]]
+|colspan=4|[[File:Gosset 1 32 petrie.svg|100px]]<br/>[[1 32 polytope|1<sub>32</sub>]]
+|}
+In [[seven-dimensional space|seven-dimensional]] [[geometry]], a '''7-polytope''' is a [[polytope]] contained by 6-polytope facets. Each [[5-polytope]] [[Ridge (geometry)|ridge]] being shared by exactly two [[6-polytope]] [[Facet (mathematics)|facets]].
+A '''uniform 7-polytope''' is one which is [[vertex-transitive]], and constructed from uniform [[6-polytope]] facets.
+A proposed name for 7-polytopes is '''polyexon''' or '''polyecton'''.
+== 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 [[Facet (mathematics)|facets]] around each 4-face.
+There are exactly three such [[List of regular polytopes#Convex 4|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.
+== Euler characteristic ==
+The [[Euler characteristic]] for 7-polytopes that are topological [[6-sphere]]s (including all convex 7-polytopes) is two. χ=V-E+F-C+f<sub>4</sub>-f<sub>5</sub>+f<sub>6</sub>=2.
+== 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 diagram]]s:
+{| class="wikitable"
+|-
+!#
+!colspan=2|[[Coxeter group]]
+![[Coxeter-Dynkin diagram|Coxeter-Dynkin<BR>diagram]]
+!Regular and semiregular forms
+!Uniform count
+|-
+|1||A<sub>7</sub>|| [3<sup>6</sup>]||{{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node}}
+|
+* [[7-simplex]] - {3<sup>6</sup>}, {{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}
+|71
+|-
+|2||B<sub>7</sub>||[4,3<sup>5</sup>]||{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node}}
+|
+* [[7-cube]] - {4,3<sup>5</sup>}, {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node}}
+* [[7-orthoplex]] - {3<sup>5</sup>,4}, {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node_1}}
+|127
+|-
+|3||D<sub>7</sub>||[3<sup>4,1,1</sup>]||{{CDD|nodes|split2|node|3|node|3|node|3|node|3|node}}
+|
+* [[7-demicube]], {3<sup>1,4,1</sup>}, {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node}}
+* [[7-orthoplex]], {3<sup>4,1,1</sup>}, {{CDD|nodes|split2|node|3|node|3|node|3|node|3|node_1}}
+|95 (32 unique)
+|-
+|4||[[E7 (mathematics)|E<sub>7</sub>]]||[3<sup>3,2,1</sup>]||{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}
+|
+* '''[[Gosset 3 21 polytope|3<sub>21</sub>]]''' - {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea_1}}
+* '''[[Gosset 1 32 polytope|1<sub>32</sub>]]''' - {{CDD|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea|3a|nodea}}
+* '''[[Gosset 2 31 polytope|2<sub>31</sub>]]''' - {{CDD|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea|3a|nodea}}
+|127
+|}
+{| class="wikitable"
+|+ Prismatic finite Coxeter groups
+|-
+!#
+!colspan=2|[[Coxeter group]]
+![[Coxeter-Dynkin diagram]]
+|-
+!colspan=4|6+1
+|-
+|1 ||A<sub>6</sub>×A<sub>1</sub>|| [3<sup>5</sup>]×[&nbsp;]|| {{CDD|node|3|node|3|node|3|node|3|node|3|node|2|node}}
+|-
+|2 ||BC<sub>6</sub>×A<sub>1</sub>|| [4,3<sup>4</sup>]×[&nbsp;]|| {{CDD|node|4|node|3|node|3|node|3|node|3|node|2|node}}
+|-
+|3 ||D<sub>6</sub>×A<sub>1</sub>|| [3<sup>3,1,1</sup>]×[&nbsp;]|| {{CDD|nodes|split2|node|3|node|3|node|3|node|2|node}}
+|-
+|4 ||E<sub>6</sub>×A<sub>1</sub>|| [3<sup>2,2,1</sup>]×[&nbsp;]|| {{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|2|nodea}}
+|-
+!colspan=4|5+2
+|-
+|1 ||A<sub>5</sub>×I<sub>2</sub>(p)|| [3,3,3]×[p]|| {{CDD|node|3|node|3|node|3|node|3|node|2|node|p|node}}
+|-
+|2 ||BC<sub>5</sub>×I<sub>2</sub>(p)|| [4,3,3]×[p]|| {{CDD|node|4|node|3|node|3|node|3|node|2|node|p|node}}
+|-
+|3 ||D<sub>5</sub>×I<sub>2</sub>(p)|| [3<sup>2,1,1</sup>]×[p]|| {{CDD|nodes|split2|node|3|node|3|node|2|node|p|node}}
+|-
+!colspan=4|5+1+1
+|-
+|1 ||A<sub>5</sub>×A<sub>1</sub><sup>2</sup>|| [3,3,3]×[&nbsp;]<sup>2</sup>|| {{CDD|node|3|node|3|node|3|node|3|node|2|node|2|node}}
+|-
+|2 ||BC<sub>5</sub>×A<sub>1</sub><sup>2</sup>|| [4,3,3]×[&nbsp;]<sup>2</sup>|| {{CDD|node|4|node|3|node|3|node|3|node|2|node|2|node}}
+|-
+|3 ||D<sub>5</sub>×A<sub>1</sub><sup>2</sup>|| [3<sup>2,1,1</sup>]×[&nbsp;]<sup>2</sup>|| {{CDD|nodes|split2|node|3|node|3|node|2|node|2|node}}
+|-
+!colspan=4|4+3
+|-
+|4 ||A<sub>4</sub>×A<sub>3</sub>|| [3,3,3]×[3,3]|| {{CDD|node|3|node|3|node|3|node|2|node|3|node|3|node}}
+|-
+|5 ||A<sub>4</sub>×BC<sub>3</sub>|| [3,3,3]×[4,3]|| {{CDD|node|3|node|3|node|3|node|2|node|4|node|3|node}}
+|-
+|6 ||A<sub>4</sub>×H<sub>3</sub>|| [3,3,3]×[5,3]|| {{CDD|node|3|node|3|node|3|node|2|node|5|node|3|node}}
+|-
+|7 ||BC<sub>4</sub>×A<sub>3</sub>|| [4,3,3]×[3,3]|| {{CDD|node|4|node|3|node|3|node|2|node|3|node|3|node}}
+|-
+|8 ||BC<sub>4</sub>×BC<sub>3</sub>|| [4,3,3]×[4,3]|| {{CDD|node|4|node|3|node|3|node|2|node|4|node|3|node}}
+|-
+|9 ||BC<sub>4</sub>×H<sub>3</sub>|| [4,3,3]×[5,3]|| {{CDD|node|4|node|3|node|3|node|2|node|5|node|3|node}}
+|-
+|10 ||H<sub>4</sub>×A<sub>3</sub>|| [5,3,3]×[3,3]|| {{CDD|node|5|node|3|node|3|node|2|node|3|node|3|node}}
+|-
+|11 ||H<sub>4</sub>×BC<sub>3</sub>|| [5,3,3]×[4,3]|| {{CDD|node|5|node|3|node|3|node|2|node|4|node|3|node}}
+|-
+|12 ||H<sub>4</sub>×H<sub>3</sub>|| [5,3,3]×[5,3]|| {{CDD|node|5|node|3|node|3|node|2|node|5|node|3|node}}
+|-
+|13 ||F<sub>4</sub>×A<sub>3</sub>|| [3,4,3]×[3,3]|| {{CDD|node|3|node|4|node|3|node|2|node|3|node|3|node}}
+|-
+|14 ||F<sub>4</sub>×BC<sub>3</sub>|| [3,4,3]×[4,3]|| {{CDD|node|3|node|4|node|3|node|2|node|4|node|3|node}}
+|-
+|15 ||F<sub>4</sub>×H<sub>3</sub>|| [3,4,3]×[5,3]|| {{CDD|node|3|node|4|node|3|node|2|node|5|node|3|node}}
+|-
+|16 ||D<sub>4</sub>×A<sub>3</sub>|| [3<sup>1,1,1</sup>]×[3,3]|| {{CDD|nodes|split2|node|3|node|2|node|3|node|3|node}}
+|-
+|17 ||D<sub>4</sub>×BC<sub>3</sub>|| [3<sup>1,1,1</sup>]×[4,3]|| {{CDD|nodes|split2|node|3|node|2|node|4|node|3|node}}
+|-
+|18 ||D<sub>4</sub>×H<sub>3</sub>|| [3<sup>1,1,1</sup>]×[5,3]|| {{CDD|nodes|split2|node|3|node|2|node|5|node|3|node}}
+|-
+!colspan=4|4+2+1
+|-
+|5 ||A<sub>4</sub>×I<sub>2</sub>(p)×A<sub>1</sub>|| [3,3,3]×[p]×[&nbsp;]|| {{CDD|node|3|node|3|node|3|node|2|node|p|node|2|node}}
+|-
+|6 ||BC<sub>4</sub>×I<sub>2</sub>(p)×A<sub>1</sub>|| [4,3,3]×[p]×[&nbsp;]|| {{CDD|node|4|node|3|node|3|node|2|node|p|node|2|node}}
+|-
+|7 ||F<sub>4</sub>×I<sub>2</sub>(p)×A<sub>1</sub>|| [3,4,3]×[p]×[&nbsp;]|| {{CDD|node|3|node|4|node|3|node|2|node|p|node|2|node}}
+|-
+|8 ||H<sub>4</sub>×I<sub>2</sub>(p)×A<sub>1</sub>|| [5,3,3]×[p]×[&nbsp;]|| {{CDD|node|5|node|3|node|3|node|2|node|p|node|2|node}}
+|-
+|9 ||D<sub>4</sub>×I<sub>2</sub>(p)×A<sub>1</sub>|| [3<sup>1,1,1</sup>]×[p]×[&nbsp;]|| {{CDD|nodes|split2|node|3|node|2|node|p|node|2|node}}
+|-
+!colspan=4|4+1+1+1
+|-
+|5 ||A<sub>4</sub>×A<sub>1</sub><sup>3</sup>|| [3,3,3]×[&nbsp;]<sup>3</sup>|| {{CDD|node|3|node|3|node|3|node|2|node|2|node|2|node}}
+|-
+|6 ||BC<sub>4</sub>×A<sub>1</sub><sup>3</sup>|| [4,3,3]×[&nbsp;]<sup>3</sup>|| {{CDD|node|4|node|3|node|3|node|2|node|2|node|2|node}}
+|-
+|7 ||F<sub>4</sub>×A<sub>1</sub><sup>3</sup>|| [3,4,3]×[&nbsp;]<sup>3</sup>|| {{CDD|node|3|node|4|node|3|node|2|node|2|node|2|node}}
+|-
+|8 ||H<sub>4</sub>×A<sub>1</sub><sup>3</sup>|| [5,3,3]×[&nbsp;]<sup>3</sup>|| {{CDD|node|5|node|3|node|3|node|2|node|2|node|2|node}}
+|-
+|9 ||D<sub>4</sub>×A<sub>1</sub><sup>3</sup>|| [3<sup>1,1,1</sup>]×[&nbsp;]<sup>3</sup>|| {{CDD|nodes|split2|node|3|node|2|node|2|node|2|node}}
+|-
+!colspan=4|3+3+1
+|-
+|10 ||A<sub>3</sub>×A<sub>3</sub>×A<sub>1</sub>|| [3,3]×[3,3]×[&nbsp;]|| {{CDD|node|3|node|3|node|2|node|3|node|3|node|2|node}}
+|-
+|11 ||A<sub>3</sub>×BC<sub>3</sub>×A<sub>1</sub>|| [3,3]×[4,3]×[&nbsp;]|| {{CDD|node|3|node|3|node|2|node|4|node|3|node|2|node}}
+|-
+|12 ||A<sub>3</sub>×H<sub>3</sub>×A<sub>1</sub>|| [3,3]×[5,3]×[&nbsp;]|| {{CDD|node|3|node|3|node|2|node|5|node|3|node|2|node}}
+|-
+|13 ||BC<sub>3</sub>×BC<sub>3</sub>×A<sub>1</sub>|| [4,3]×[4,3]×[&nbsp;]|| {{CDD|node|4|node|3|node|2|node|4|node|3|node|2|node}}
+|-
+|14 ||BC<sub>3</sub>×H<sub>3</sub>×A<sub>1</sub>|| [4,3]×[5,3]×[&nbsp;]|| {{CDD|node|4|node|3|node|2|node|5|node|3|node|2|node}}
+|-
+|15 ||H<sub>3</sub>×A<sub>3</sub>×A<sub>1</sub>|| [5,3]×[5,3]×[&nbsp;]|| {{CDD|node|5|node|3|node|2|node|5|node|3|node|2|node}}
+|-
+!colspan=4|3+2+2
+|-
+|1 ||A<sub>3</sub>×I<sub>2</sub>(p)×I<sub>2</sub>(q)|| [3,3]×[p]×[q]|| {{CDD|node|3|node|3|node|2|node|p|node|2|node|q|node}}
+|-
+|2 ||BC<sub>3</sub>×I<sub>2</sub>(p)×I<sub>2</sub>(q)|| [4,3]×[p]×[q]|| {{CDD|node|4|node|3|node|2|node|p|node|2|node|q|node}}
+|-
+|3 ||H<sub>3</sub>×I<sub>2</sub>(p)×I<sub>2</sub>(q)|| [5,3]×[p]×[q]|| {{CDD|node|5|node|3|node|2|node|p|node|2|node|q|node}}
+|-
+!colspan=4|3+2+1+1
+|-
+|1 ||A<sub>3</sub>×I<sub>2</sub>(p)×A<sub>1</sub><sup>2</sup>|| [3,3]×[p]×[&nbsp;]<sup>2</sup>|| {{CDD|node|3|node|3|node|2|node|p|node|2|node|2|node}}
+|-
+|2 ||BC<sub>3</sub>×I<sub>2</sub>(p)×A<sub>1</sub><sup>2</sup>|| [4,3]×[p]×[&nbsp;]<sup>2</sup>|| {{CDD|node|4|node|3|node|2|node|p|node|2|node|2|node}}
+|-
+|3 ||H<sub>3</sub>×I<sub>2</sub>(p)×A<sub>1</sub><sup>2</sup>|| [5,3]×[p]×[&nbsp;]<sup>2</sup>|| {{CDD|node|5|node|3|node|2|node|p|node|2|node|2|node}}
+|-
+!colspan=4|3+1+1+1+1
+|-
+|1 ||A<sub>3</sub>×A<sub>1</sub><sup>4</sup>|| [3,3]×[&nbsp;]<sup>4</sup>|| {{CDD|node|3|node|3|node|2|node|2|node|2|node|2|node}}
+|-
+|2 ||BC<sub>3</sub>×A<sub>1</sub><sup>4</sup>|| [4,3]×[&nbsp;]<sup>4</sup>|| {{CDD|node|4|node|3|node|2|node|2|node|2|node|2|node}}
+|-
+|3 ||H<sub>3</sub>×A<sub>1</sub><sup>4</sup>|| [5,3]×[&nbsp;]<sup>4</sup>|| {{CDD|node|5|node|3|node|2|node|2|node|2|node|2|node}}
+|-
+!colspan=4|2+2+2+1
+|-
+|1 ||I<sub>2</sub>(p)×I<sub>2</sub>(q)×I<sub>2</sub>(r)×A<sub>1</sub>|| [p]×[q]×[r]×[&nbsp;]|| {{CDD|node|p|node|2|node|q|node|2|node|r|node|2|node}}
+|-
+!colspan=4|2+2+1+1+1
+|-
+|1 ||I<sub>2</sub>(p)×I<sub>2</sub>(q)×A<sub>1</sub><sup>3</sup>|| [p]×[q]×[&nbsp;]<sup>3</sup>|| {{CDD|node|p|node|2|node|q|node|2|node|2|node|2|node}}
+|-
+!colspan=4|2+1+1+1+1+1
+|-
+|1 ||I<sub>2</sub>(p)×A<sub>1</sub><sup>5</sup>|| [p]×[&nbsp;]<sup>5</sup>|| {{CDD|node|p|node|2|node|2|node|2|node|2|node|2|node}}
+|-
+!colspan=4|1+1+1+1+1+1+1
+|-
+|1 ||A<sub>1</sub><sup>7</sup>|| [&nbsp;]<sup>7</sup>|| {{CDD|node|2|node|2|node|2|node|2|node|2|node|2|node}}
+|}
+== The A<sub>7</sub> family ==
+The A<sub>7</sub> family has symmetry of order 40320 (8 [[factorial]]).
+There are 71 (64+8-1) forms based on all permutations of the [[Coxeter-Dynkin diagram]]s with one or more rings. All 71 are enumerated below. [[Norman Johnson (mathematician)|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.
+{| class="wikitable"
+|-
+!rowspan=2|#
+!rowspan=2|[[Coxeter-Dynkin diagram]]
+!rowspan=2|Truncation<br />indices
+!rowspan=2|[[Norman Johnson (mathematician)|Johnson name]]<br />Bowers name (and acronym)
+!rowspan=2|Basepoint
+!colspan=7|Element counts
+|-
+! 6|| 5|| 4|| 3|| 2|| 1|| 0
+|- style="text-align:center;"
+|1||{{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}||t<sub>0</sub>||[[7-simplex]] (oca)||(0,0,0,0,0,0,0,1)||8||28||56||70||56||28||8
+|- style="text-align:center;"
+|2||{{CDD|node|3|node|3|node|3|node|3|node|3|node_1|3|node}}||t<sub>1</sub>||[[Rectified 7-simplex]] (roc)||(0,0,0,0,0,0,1,1)||16||84||224||350||336||168||28
+|- style="text-align:center;"
+|3||{{CDD|node|3|node|3|node|3|node|3|node_1|3|node|3|node}}||t<sub>2</sub>||[[Birectified 7-simplex]] (broc)||(0,0,0,0,0,1,1,1)||16||112||392||770||840||420||56
+|-  style="text-align:center; background:#e0f0e0;"
+|4||{{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node}}||t<sub>3</sub>||[[Trirectified 7-simplex]] (he)||(0,0,0,0,1,1,1,1)||16||112||448||980||1120||560||70
+|- style="text-align:center;"
+|5||{{CDD|node|3|node|3|node|3|node|3|node|3|node_1|3|node_1}}||t<sub>0,1</sub>||[[Truncated 7-simplex]] (toc)||(0,0,0,0,0,0,1,2)||16||84||224||350||336||196||56
+|- style="text-align:center;"
+|6||{{CDD|node|3|node|3|node|3|node|3|node_1|3|node|3|node_1}}||t<sub>0,2</sub>||[[Cantellated 7-simplex]] (saro)||(0,0,0,0,0,1,1,2)||44||308||980||1750||1876||1008||168
+|- style="text-align:center;"
+|7||{{CDD|node|3|node|3|node|3|node|3|node_1|3|node_1|3|node}}||t<sub>1,2</sub>||[[Bitruncated 7-simplex]] (bittoc)||(0,0,0,0,0,1,2,2)||||||||||||588||168
+|- style="text-align:center;"
+|8||{{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node_1}}||t<sub>0,3</sub>||[[Runcinated 7-simplex]] (spo)||(0,0,0,0,1,1,1,2)||100||756||2548||4830||4760||2100||280
+|- style="text-align:center;"
+|9||{{CDD|node|3|node|3|node|3|node_1|3|node|3|node_1|3|node}}||t<sub>1,3</sub>||[[Bicantellated 7-simplex]] (sabro)||(0,0,0,0,1,1,2,2)||||||||||||2520||420
+|- style="text-align:center;"
+|10||{{CDD|node|3|node|3|node|3|node_1|3|node_1|3|node|3|node}}||t<sub>2,3</sub>||[[Tritruncated 7-simplex]] (tattoc)||(0,0,0,0,1,2,2,2)||||||||||||980||280
+|- style="text-align:center;"
+|11||{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node_1}}||t<sub>0,4</sub>||[[Stericated 7-simplex]] (sco)||(0,0,0,1,1,1,1,2)||||||||||||2240||280
+|- style="text-align:center;"
+|12||{{CDD|node|3|node|3|node_1|3|node|3|node|3|node_1|3|node}}||t<sub>1,4</sub>||[[Biruncinated 7-simplex]] (sibpo)||(0,0,0,1,1,1,2,2)||||||||||||4200||560
+|-  style="text-align:center; background:#e0f0e0;"
+|13||{{CDD|node|3|node|3|node_1|3|node|3|node_1|3|node|3|node}}||t<sub>2,4</sub>||[[Tricantellated 7-simplex]] (stiroh)||(0,0,0,1,1,2,2,2)||||||||||||3360||560
+|- style="text-align:center;"
+|14||{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node_1}}||t<sub>0,5</sub>||[[Pentellated 7-simplex]] (seto)||(0,0,1,1,1,1,1,2)||||||||||||1260||168
+|-  style="text-align:center; background:#e0f0e0;"
+|15||{{CDD|node|3|node_1|3|node|3|node|3|node|3|node_1|3|node}}||t<sub>1,5</sub>||[[Bistericated 7-simplex]] (sabach)||(0,0,1,1,1,1,2,2)||||||||||||3360||420
+|-  style="text-align:center; background:#e0f0e0;"
+|16||{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node_1}}||t<sub>0,6</sub>||[[Hexicated 7-simplex]] (suph)||(0,1,1,1,1,1,1,2)||||||||||||336||56
+|- style="text-align:center;"
+|17||{{CDD|node|3|node|3|node|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2</sub>||[[Cantitruncated 7-simplex]] (garo)||(0,0,0,0,0,1,2,3)||||||||||||1176||336
+|- style="text-align:center;"
+|18||{{CDD|node|3|node|3|node|3|node_1|3|node|3|node_1|3|node_1}}||t<sub>0,1,3</sub>||[[Runcitruncated 7-simplex]] (patto)||(0,0,0,0,1,1,2,3)||||||||||||4620||840
+|- style="text-align:center;"
+|19||{{CDD|node|3|node|3|node|3|node_1|3|node_1|3|node|3|node_1}}||t<sub>0,2,3</sub>||[[Runcicantellated 7-simplex]] (paro)||(0,0,0,0,1,2,2,3)||||||||||||3360||840
+|- style="text-align:center;"
+|20||{{CDD|node|3|node|3|node|3|node_1|3|node_1|3|node_1|3|node}}||t<sub>1,2,3</sub>||[[Bicantitruncated 7-simplex]] (gabro)||(0,0,0,0,1,2,3,3)||||||||||||2940||840
+|- style="text-align:center;"
+|21||{{CDD|node|3|node|3|node_1|3|node|3|node|3|node_1|3|node_1}}||t<sub>0,1,4</sub>||[[Steritruncated 7-simplex]] (cato)||(0,0,0,1,1,1,2,3)||||||||||||7280||1120
+|- style="text-align:center;"
+|22||{{CDD|node|3|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}||t<sub>0,2,4</sub>||[[Stericantellated 7-simplex]] (caro)||(0,0,0,1,1,2,2,3)||||||||||||10080||1680
+|- style="text-align:center;"
+|23||{{CDD|node|3|node|3|node_1|3|node|3|node_1|3|node_1|3|node}}||t<sub>1,2,4</sub>||[[Biruncitruncated 7-simplex]] (bipto)||(0,0,0,1,1,2,3,3)||||||||||||8400||1680
+|- style="text-align:center;"
+|24||{{CDD|node|3|node|3|node_1|3|node_1|3|node|3|node|3|node_1}}||t<sub>0,3,4</sub>||[[Steriruncinated 7-simplex]] (cepo)||(0,0,0,1,2,2,2,3)||||||||||||5040||1120
+|- style="text-align:center;"
+|25||{{CDD|node|3|node|3|node_1|3|node_1|3|node|3|node_1|3|node}}||t<sub>1,3,4</sub>||[[Biruncicantellated 7-simplex]] (bipro)||(0,0,0,1,2,2,3,3)||||||||||||7560||1680
+|-  style="text-align:center; background:#e0f0e0;"
+|26||{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}||t<sub>2,3,4</sub>||[[Tricantitruncated 7-simplex]] (gatroh)||(0,0,0,1,2,3,3,3)||||||||||||3920||1120
+|- style="text-align:center;"
+|27||{{CDD|node|3|node_1|3|node|3|node|3|node|3|node_1|3|node_1}}||t<sub>0,1,5</sub>||[[Pentitruncated 7-simplex]] (teto)||(0,0,1,1,1,1,2,3)||||||||||||5460||840
+|- style="text-align:center;"
+|28||{{CDD|node|3|node_1|3|node|3|node|3|node_1|3|node|3|node_1}}||t<sub>0,2,5</sub>||[[Penticantellated 7-simplex]] (tero)||(0,0,1,1,1,2,2,3)||||||||||||11760||1680
+|- style="text-align:center;"
+|29||{{CDD|node|3|node_1|3|node|3|node|3|node_1|3|node_1|3|node}}||t<sub>1,2,5</sub>||[[Bisteritruncated 7-simplex]] (bacto)||(0,0,1,1,1,2,3,3)||||||||||||9240||1680
+|- style="text-align:center;"
+|30||{{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node|3|node_1}}||t<sub>0,3,5</sub>||[[Pentiruncinated 7-simplex]] (tepo)||(0,0,1,1,2,2,2,3)||||||||||||10920||1680
+|-  style="text-align:center; background:#e0f0e0;"
+|31||{{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node_1|3|node}}||t<sub>1,3,5</sub>||[[Bistericantellated 7-simplex]] (bacroh)||(0,0,1,1,2,2,3,3)||||||||||||15120||2520
+|- style="text-align:center;"
+|32||{{CDD|node|3|node_1|3|node_1|3|node|3|node|3|node|3|node_1}}||t<sub>0,4,5</sub>||[[Pentistericated 7-simplex]] (teco)||(0,0,1,2,2,2,2,3)||||||||||||4200||840
+|- style="text-align:center;"
+|33||{{CDD|node_1|3|node|3|node|3|node|3|node|3|node_1|3|node_1}}||t<sub>0,1,6</sub>||[[Hexitruncated 7-simplex]] (puto)||(0,1,1,1,1,1,2,3)||||||||||||1848||336
+|- style="text-align:center;"
+|34||{{CDD|node_1|3|node|3|node|3|node|3|node_1|3|node|3|node_1}}||t<sub>0,2,6</sub>||[[Hexicantellated 7-simplex]] (puro)||(0,1,1,1,1,2,2,3)||||||||||||5880||840
+|-  style="text-align:center; background:#e0f0e0;"
+|35||{{CDD|node_1|3|node|3|node|3|node_1|3|node|3|node|3|node_1}}||t<sub>0,3,6</sub>||[[Hexiruncinated 7-simplex]] (puph)||(0,1,1,1,2,2,2,3)||||||||||||8400||1120
+|- style="text-align:center;"
+|36||{{CDD|node|3|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3</sub>||[[Runcicantitruncated 7-simplex]] (gapo)||(0,0,0,0,1,2,3,4)||||||||||||5880||1680
+|- style="text-align:center;"
+|37||{{CDD|node|3|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,4</sub>||[[Stericantitruncated 7-simplex]] (cagro)||(0,0,0,1,1,2,3,4)||||||||||||16800||3360
+|- style="text-align:center;"
+|38||{{CDD|node|3|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}||t<sub>0,1,3,4</sub>||[[Steriruncitruncated 7-simplex]] (capto)||(0,0,0,1,2,2,3,4)||||||||||||13440||3360
+|- style="text-align:center;"
+|39||{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}||t<sub>0,2,3,4</sub>||[[Steriruncicantellated 7-simplex]] (capro)||(0,0,0,1,2,3,3,4)||||||||||||13440||3360
+|- style="text-align:center;"
+|40||{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}||t<sub>1,2,3,4</sub>||[[Biruncicantitruncated 7-simplex]] (gibpo)||(0,0,0,1,2,3,4,4)||||||||||||11760||3360
+|- style="text-align:center;"
+|41||{{CDD|node|3|node_1|3|node|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,5</sub>||[[Penticantitruncated 7-simplex]] (tegro)||(0,0,1,1,1,2,3,4)||||||||||||18480||3360
+|- style="text-align:center;"
+|42||{{CDD|node|3|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1}}||t<sub>0,1,3,5</sub>||[[Pentiruncitruncated 7-simplex]] (tapto)||(0,0,1,1,2,2,3,4)||||||||||||27720||5040
+|- style="text-align:center;"
+|43||{{CDD|node|3|node_1|3|node|3|node_1|3|node_1|3|node|3|node_1}}||t<sub>0,2,3,5</sub>||[[Pentiruncicantellated 7-simplex]] (tapro)||(0,0,1,1,2,3,3,4)||||||||||||25200||5040
+|- style="text-align:center;"
+|44||{{CDD|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node}}||t<sub>1,2,3,5</sub>||[[Bistericantitruncated 7-simplex]] (bacogro)||(0,0,1,1,2,3,4,4)||||||||||||22680||5040
+|- style="text-align:center;"
+|45||{{CDD|node|3|node_1|3|node_1|3|node|3|node|3|node_1|3|node_1}}||t<sub>0,1,4,5</sub>||[[Pentisteritruncated 7-simplex]] (tecto)||(0,0,1,2,2,2,3,4)||||||||||||15120||3360
+|- style="text-align:center;"
+|46||{{CDD|node|3|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1}}||t<sub>0,2,4,5</sub>||[[Pentistericantellated 7-simplex]] (tecro)||(0,0,1,2,2,3,3,4)||||||||||||25200||5040
+|-  style="text-align:center; background:#e0f0e0;"
+|47||{{CDD|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node}}||t<sub>1,2,4,5</sub>||[[Bisteriruncitruncated 7-simplex]] (bicpath)||(0,0,1,2,2,3,4,4)||||||||||||20160||5040
+|- style="text-align:center;"
+|48||{{CDD|node|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1}}||t<sub>0,3,4,5</sub>||[[Pentisteriruncinated 7-simplex]] (tacpo)||(0,0,1,2,3,3,3,4)||||||||||||15120||3360
+|- style="text-align:center;"
+|49||{{CDD|node_1|3|node|3|node|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,6</sub>||[[Hexicantitruncated 7-simplex]] (pugro)||(0,1,1,1,1,2,3,4)||||||||||||8400||1680
+|- style="text-align:center;"
+|50||{{CDD|node_1|3|node|3|node|3|node_1|3|node|3|node_1|3|node_1}}||t<sub>0,1,3,6</sub>||[[Hexiruncitruncated 7-simplex]] (pugato)||(0,1,1,1,2,2,3,4)||||||||||||20160||3360
+|- style="text-align:center;"
+|51||{{CDD|node_1|3|node|3|node|3|node_1|3|node_1|3|node|3|node_1}}||t<sub>0,2,3,6</sub>||[[Hexiruncicantellated 7-simplex]] (pugro)||(0,1,1,1,2,3,3,4)||||||||||||16800||3360
+|- style="text-align:center;"
+|52||{{CDD|node_1|3|node|3|node_1|3|node|3|node|3|node_1|3|node_1}}||t<sub>0,1,4,6</sub>||[[Hexisteritruncated 7-simplex]] (pucto)||(0,1,1,2,2,2,3,4)||||||||||||20160||3360
+|-  style="text-align:center; background:#e0f0e0;"
+|53||{{CDD|node_1|3|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}||t<sub>0,2,4,6</sub>||[[Hexistericantellated 7-simplex]] (pucroh)||(0,1,1,2,2,3,3,4)||||||||||||30240||5040
+|-  style="text-align:center; background:#e0f0e0;"
+|54||{{CDD|node_1|3|node_1|3|node|3|node|3|node|3|node_1|3|node_1}}||t<sub>0,1,5,6</sub>||[[Hexipentitruncated 7-simplex]] (putath)||(0,1,2,2,2,2,3,4)||||||||||||8400||1680
+|- style="text-align:center;"
+|55||{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3,4</sub>||[[Steriruncicantitruncated 7-simplex]] (gecco)||(0,0,0,1,2,3,4,5)||||||||||||23520||6720
+|- style="text-align:center;"
+|56||{{CDD|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3,5</sub>||[[Pentiruncicantitruncated 7-simplex]] (tegapo)||(0,0,1,1,2,3,4,5)||||||||||||45360||10080
+|- style="text-align:center;"
+|57||{{CDD|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,4,5</sub>||[[Pentistericantitruncated 7-simplex]] (tecagro)||(0,0,1,2,2,3,4,5)||||||||||||40320||10080
+|- style="text-align:center;"
+|58||{{CDD|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}||t<sub>0,1,3,4,5</sub>||[[Pentisteriruncitruncated 7-simplex]] (tacpeto)||(0,0,1,2,3,3,4,5)||||||||||||40320||10080
+|- style="text-align:center;"
+|59||{{CDD|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}||t<sub>0,2,3,4,5</sub>||[[Pentisteriruncicantellated 7-simplex]] (tacpro)||(0,0,1,2,3,4,4,5)||||||||||||40320||10080
+|-  style="text-align:center; background:#e0f0e0;"
+|60||{{CDD|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}||t<sub>1,2,3,4,5</sub>||[[Bisteriruncicantitruncated 7-simplex]] (gabach)||(0,0,1,2,3,4,5,5)||||||||||||35280||10080
+|- style="text-align:center;"
+|61||{{CDD|node_1|3|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3,6</sub>||[[Hexiruncicantitruncated 7-simplex]] (pugopo)||(0,1,1,1,2,3,4,5)||||||||||||30240||6720
+|- style="text-align:center;"
+|62||{{CDD|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,4,6</sub>||[[Hexistericantitruncated 7-simplex]] (pucagro)||(0,1,1,2,2,3,4,5)||||||||||||50400||10080
+|- style="text-align:center;"
+|63||{{CDD|node_1|3|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}||t<sub>0,1,3,4,6</sub>||[[Hexisteriruncitruncated 7-simplex]] (pucpato)||(0,1,1,2,3,3,4,5)||||||||||||45360||10080
+|-  style="text-align:center; background:#e0f0e0;"
+|64||{{CDD|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}||t<sub>0,2,3,4,6</sub>||[[Hexisteriruncicantellated 7-simplex]] (pucproh)||(0,1,1,2,3,4,4,5)||||||||||||45360||10080
+|- style="text-align:center;"
+|65||{{CDD|node_1|3|node_1|3|node|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,5,6</sub>||[[Hexipenticantitruncated 7-simplex]] (putagro)||(0,1,2,2,2,3,4,5)||||||||||||30240||6720
+|-  style="text-align:center; background:#e0f0e0;"
+|66||{{CDD|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1}}||t<sub>0,1,3,5,6</sub>||[[Hexipentiruncitruncated 7-simplex]] (putpath)||(0,1,2,2,3,3,4,5)||||||||||||50400||10080
+|- style="text-align:center;"
+|67||{{CDD|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3,4,5</sub>||[[Pentisteriruncicantitruncated 7-simplex]] (geto)||(0,0,1,2,3,4,5,6)||||||||||||70560||20160
+|- style="text-align:center;"
+|68||{{CDD|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3,4,6</sub>||[[Hexisteriruncicantitruncated 7-simplex]] (pugaco)||(0,1,1,2,3,4,5,6)||||||||||||80640||20160
+|- style="text-align:center;"
+|69||{{CDD|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3,5,6</sub>||[[Hexipentiruncicantitruncated 7-simplex]] (putgapo)||(0,1,2,2,3,4,5,6)||||||||||||80640||20160
+|-  style="text-align:center; background:#e0f0e0;"
+|70||{{CDD|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,4,5,6</sub>||[[Hexipentistericantitruncated 7-simplex]] (putcagroh)||(0,1,2,3,3,4,5,6)||||||||||||80640||20160
+|-  style="text-align:center; background:#e0f0e0;"
+|71||{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}||t<sub>0,1,2,3,4,5,6</sub>||[[Omnitruncated 7-simplex]] (guph)||(0,1,2,3,4,5,6,7)||||||||||||141120||40320
+|}
+== The B<sub>7</sub> family ==
+The B<sub>7</sub> family has symmetry of order 645120 (7 [[factorial]] x 2<sup>7</sup>).
+There are 127 forms based on all permutations of the [[Coxeter-Dynkin diagram]]s with one or more rings. Johnson and Bowers names.
+See also a [[list of B7 polytopes]] for symmetric [[Coxeter plane]] graphs of these polytopes.
+{| class=wikitable
+!rowspan=2|#
+!rowspan=2|[[Coxeter-Dynkin diagram]]<BR>t-notation
+!rowspan=2|Name (BSA)
+!rowspan=2|Base point
+!colspan=7|Element counts
+|-
+!6||5||4||3||2||1||0
+|-  style="text-align:center; background:#f0e0e0;"
+!1
+|<!-- [x3o3o3o3o3o4o] -->{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node_1}}<BR>t<sub>0</sub>{3,3,3,3,3,4}||[[7-orthoplex]] (zee)|||(0,0,0,0,0,0,1)√2||128||448||672||560||280||84||14
+|-  style="text-align:center; background:#f0e0e0;"
+!2
+|<!-- [o3x3o3o3o3o4o] -->{{CDD|node|4|node|3|node|3|node|3|node|3|node_1|3|node}}<BR>t<sub>1</sub>{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
+|-  style="text-align:center; background:#f0e0e0;"
+!3
+|<!-- [o3o3x3o3o3o4o] -->{{CDD|node|4|node|3|node|3|node|3|node_1|3|node|3|node}}<BR>t<sub>2</sub>{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
+|-  style="text-align:center; background:#e0f0e0;"
+!4
+|<!-- [o3o3o3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node|3|node|3|node}}<BR>t<sub>3</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!5
+|<!-- [o3o3o3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node|3|node|3|node}}<BR>t<sub>2</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!6
+|<!-- [o3o3o3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node|3|node|3|node}}<BR>t<sub>1</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!7
+|<!-- [o3o3o3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node}}<BR>t<sub>0</sub>{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
+|-  style="text-align:center; background:#f0e0e0;"
+!8
+|<!-- [x3x3o3o3o3o4o] -->{{CDD|node|4|node|3|node|3|node|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1</sub>{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
+|-  style="text-align:center; background:#f0e0e0;"
+!9
+|<!-- [x3o3x3o3o3o4o] -->{{CDD|node|4|node|3|node|3|node|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2</sub>{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
+|-  style="text-align:center; background:#f0e0e0;"
+!10
+|<!-- [o3x3x3o3o3o4o] -->{{CDD|node|4|node|3|node|3|node|3|node_1|3|node_1|3|node}}<BR>t<sub>1,2</sub>{3,3,3,3,3,4}||[[Bitruncated 7-orthoplex]] (Botaz)|||(0,0,0,0,1,2,2)√2|||||| || || ||4200||840
+|-  style="text-align:center; background:#f0e0e0;"
+!11
+|<!-- [x3o3o3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node|3|node|3|node_1}}<BR>t<sub>0,3</sub>{3,3,3,3,3,4}||[[Runcinated 7-orthoplex]] (Spaz)|||(0,0,0,1,1,1,2)√2|||||| || || ||23520||2240
+|-  style="text-align:center; background:#f0e0e0;"
+!12
+|<!-- [o3x3o3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node|3|node_1|3|node}}<BR>t<sub>1,3</sub>{3,3,3,3,3,4}||[[Bicantellated 7-orthoplex]] (Sebraz)|||(0,0,0,1,1,2,2)√2|||||| || || ||26880||3360
+|-  style="text-align:center; background:#f0e0e0;"
+!13
+|<!-- [o3o3x3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node|3|node}}<BR>t<sub>2,3</sub>{3,3,3,3,3,4}||[[Tritruncated 7-orthoplex]] (Totaz)|||(0,0,0,1,2,2,2)√2|||||| || || ||10080||2240
+|-  style="text-align:center; background:#f0e0e0;"
+!14
+|<!-- [x3o3o3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node|3|node|3|node_1}}<BR>t<sub>0,4</sub>{3,3,3,3,3,4}||[[Stericated 7-orthoplex]] (Scaz)|||(0,0,1,1,1,1,2)√2|||||| || || ||33600||3360
+|-  style="text-align:center; background:#f0e0e0;"
+!15
+|<!-- [o3x3o3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node|3|node_1|3|node}}<BR>t<sub>1,4</sub>{3,3,3,3,3,4}||[[Biruncinated 7-orthoplex]] (Sibpaz)|||(0,0,1,1,1,2,2)√2|||||| || || ||60480||6720
+|-  style="text-align:center; background:#e0f0e0;"
+!16
+|<!-- [o3o3x3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node|3|node}}<BR>t<sub>2,4</sub>{4,3,3,3,3,3}||[[Tricantellated 7-cube]] (Strasaz)|||(0,0,1,1,2,2,2)√2|||||| || || ||47040||6720
+|-  style="text-align:center; background:#e0e0f0;"
+!17
+|<!-- [o3o3o3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node|3|node|3|node}}<BR>t<sub>2,3</sub>{4,3,3,3,3,3}||[[Tritruncated 7-cube]] (Tatsa)|||(0,0,1,2,2,2,2)√2|||||| || || ||13440||3360
+|-  style="text-align:center; background:#f0e0e0;"
+!18
+|<!-- [x3o3o3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node|3|node|3|node_1}}<BR>t<sub>0,5</sub>{3,3,3,3,3,4}||[[Pentellated 7-orthoplex]] (Staz)|||(0,1,1,1,1,1,2)√2|||||| || || ||20160||2688
+|-  style="text-align:center; background:#e0f0e0;"
+!19
+|<!-- [o3x3o3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node|3|node_1|3|node}}<BR>t<sub>1,5</sub>{4,3,3,3,3,3}||[[Bistericated 7-cube]] (Sabcosaz)|||(0,1,1,1,1,2,2)√2|||||| || || ||53760||6720
+|-  style="text-align:center; background:#e0e0f0;"
+!20
+|<!-- [o3o3x3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node|3|node}}<BR>t<sub>1,4</sub>{4,3,3,3,3,3}||[[Biruncinated 7-cube]] (Sibposa)|||(0,1,1,1,2,2,2)√2|||||| || || ||67200||8960
+|-  style="text-align:center; background:#e0e0f0;"
+!21
+|<!-- [o3o3o3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node|3|node}}<BR>t<sub>1,3</sub>{4,3,3,3,3,3}||[[Bicantellated 7-cube]] (Sibrosa)|||(0,1,1,2,2,2,2)√2|||||| || || ||40320||6720
+|-  style="text-align:center; background:#e0e0f0;"
+!22
+|<!-- [o3o3o3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node|3|node}}<BR>t<sub>1,2</sub>{4,3,3,3,3,3}||[[Bitruncated 7-cube]] (Betsa)|||(0,1,2,2,2,2,2)√2|||||| || || ||9408||2688
+|-  style="text-align:center; background:#e0f0e0;"
+!23
+|<!-- [x3o3o3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node_1}}<BR>t<sub>0,6</sub>{4,3,3,3,3,3}||[[Hexicated 7-cube]] (Suposaz)|||(0,0,0,0,0,0,1)√2 + (1,1,1,1,1,1,1)|||||| || || ||5376||896
+|-  style="text-align:center; background:#e0e0f0;"
+!24
+|<!-- [o3x3o3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node|3|node_1|3|node}}<BR>t<sub>0,5</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!25
+|<!-- [o3o3x3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node|3|node}}<BR>t<sub>0,4</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!26
+|<!-- [o3o3o3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node|3|node}}<BR>t<sub>0,3</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!27
+|<!-- [o3o3o3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node|3|node}}<BR>t<sub>0,2</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!28
+|<!-- [o3o3o3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node|3|node}}<BR>t<sub>0,1</sub>{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
+|-  style="text-align:center; background:#f0e0e0;"
+!29
+|<!-- [x3x3x3o3o3o4o] -->{{CDD|node|4|node|3|node|3|node|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2</sub>{3,3,3,3,3,4}||[[Cantitruncated 7-orthoplex]] (Garz)|||(0,1,2,3,3,3,3)√2|||||| || || ||8400||1680
+|-  style="text-align:center; background:#f0e0e0;"
+!30
+|<!-- [x3x3o3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,3</sub>{3,3,3,3,3,4}||[[Runcitruncated 7-orthoplex]] (Potaz)|||(0,1,2,2,3,3,3)√2|||||| || || ||50400||6720
+|-  style="text-align:center; background:#f0e0e0;"
+!31
+|<!-- [x3o3x3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,3</sub>{3,3,3,3,3,4}||[[Runcicantellated 7-orthoplex]] (Parz)|||(0,1,1,2,3,3,3)√2|||||| || || ||33600||6720
+|-  style="text-align:center; background:#f0e0e0;"
+!32
+|<!-- [o3x3x3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node}}<BR>t<sub>1,2,3</sub>{3,3,3,3,3,4}||[[Bicantitruncated 7-orthoplex]] (Gebraz)|||(0,0,1,2,3,3,3)√2|||||| || || ||30240||6720
+|-  style="text-align:center; background:#f0e0e0;"
+!33
+|<!-- [x3x3o3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,4</sub>{3,3,3,3,3,4}||[[Steritruncated 7-orthoplex]] (Catz)|||(0,0,1,1,1,2,3)√2|||||| || || ||107520||13440
+|-  style="text-align:center; background:#f0e0e0;"
+!34
+|<!-- [x3o3x3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,4</sub>{3,3,3,3,3,4}||[[Stericantellated 7-orthoplex]] (Craze)|||(0,0,1,1,2,2,3)√2|||||| || || ||141120||20160
+|-  style="text-align:center; background:#f0e0e0;"
+!35
+|<!-- [o3x3x3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node}}<BR>t<sub>1,2,4</sub>{3,3,3,3,3,4}||[[Biruncitruncated 7-orthoplex]] (Baptize)|||(0,0,1,1,2,3,3)√2|||||| || || ||120960||20160
+|-  style="text-align:center; background:#f0e0e0;"
+!36
+|<!-- [x3o3o3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node|3|node|3|node_1}}<BR>t<sub>0,3,4</sub>{3,3,3,3,3,4}||[[Steriruncinated 7-orthoplex]] (Copaz)|||(0,1,1,1,2,3,3)√2|||||| || || ||67200||13440
+|-  style="text-align:center; background:#f0e0e0;"
+!37
+|<!-- [o3x3o3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node}}<BR>t<sub>1,3,4</sub>{3,3,3,3,3,4}||[[Biruncicantellated 7-orthoplex]] (Boparz)|||(0,0,1,2,2,3,3)√2|||||| || || ||100800||20160
+|-  style="text-align:center; background:#e0f0e0;"
+!38
+|<!-- [o3o3x3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}<BR>t<sub>2,3,4</sub>{4,3,3,3,3,3}||[[Tricantitruncated 7-cube]] (Gotrasaz)|||(0,0,0,1,2,3,3)√2|||||| || || ||53760||13440
+|-  style="text-align:center; background:#f0e0e0;"
+!39
+|<!-- [x3x3o3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,5</sub>{3,3,3,3,3,4}||[[Pentitruncated 7-orthoplex]] (Tetaz)|||(0,1,1,1,1,2,3)√2|||||| || || ||87360||13440
+|-  style="text-align:center; background:#f0e0e0;"
+!40
+|<!-- [x3o3x3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,5</sub>{3,3,3,3,3,4}||[[Penticantellated 7-orthoplex]] (Teroz)|||(0,1,1,1,2,2,3)√2|||||| || || ||188160||26880
+|-  style="text-align:center; background:#f0e0e0;"
+!41
+|<!-- [o3x3x3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node_1|3|node}}<BR>t<sub>1,2,5</sub>{3,3,3,3,3,4}||[[Bisteritruncated 7-orthoplex]] (Boctaz)|||(0,1,1,1,2,3,3)√2|||||| || || ||147840||26880
+|-  style="text-align:center; background:#f0e0e0;"
+!42
+|<!-- [x3o3o3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node|3|node_1}}<BR>t<sub>0,3,5</sub>{3,3,3,3,3,4}||[[Pentiruncinated 7-orthoplex]] (Topaz)|||(0,1,1,2,2,2,3)√2|||||| || || ||174720||26880
+|-  style="text-align:center; background:#e0f0e0;"
+!43
+|<!-- [o3x3o3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node_1|3|node}}<BR>t<sub>1,3,5</sub>{4,3,3,3,3,3}||[[Bistericantellated 7-cube]] (Bacresaz)|||(0,1,1,2,2,3,3)√2|||||| || || ||241920||40320
+|-  style="text-align:center; background:#e0e0f0;"
+!44
+|<!-- [o3o3x3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node|3|node}}<BR>t<sub>1,3,4</sub>{4,3,3,3,3,3}||[[Biruncicantellated 7-cube]] (Bopresa)|||(0,1,1,2,3,3,3)√2|||||| || || ||120960||26880
+|-  style="text-align:center; background:#f0e0e0;"
+!45
+|<!-- [x3o3o3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node|3|node_1}}<BR>t<sub>0,4,5</sub>{3,3,3,3,3,4}||[[Pentistericated 7-orthoplex]] (Tocaz)|||(0,1,2,2,2,2,3)√2|||||| || || ||67200||13440
+|-  style="text-align:center; background:#e0e0f0;"
+!46
+|<!-- [o3x3o3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node_1|3|node}}<BR>t<sub>1,2,5</sub>{4,3,3,3,3,3}||[[Bisteritruncated 7-cube]] (Bactasa)|||(0,1,2,2,2,3,3)√2|||||| || || ||147840||26880
+|-  style="text-align:center; background:#e0e0f0;"
+!47
+|<!-- [o3o3x3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node|3|node}}<BR>t<sub>1,2,4</sub>{4,3,3,3,3,3}||[[Biruncitruncated 7-cube]] (Biptesa)|||(0,1,2,2,3,3,3)√2|||||| || || ||134400||26880
+|-  style="text-align:center; background:#e0e0f0;"
+!48
+|<!-- [o3o3o3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}<BR>t<sub>1,2,3</sub>{4,3,3,3,3,3}||[[Bicantitruncated 7-cube]] (Gibrosa)|||(0,1,2,3,3,3,3)√2|||||| || || ||47040||13440
+|-  style="text-align:center; background:#f0e0e0;"
+!49
+|<!-- [x3x3o3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,6</sub>{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
+|-  style="text-align:center; background:#f0e0e0;"
+!50
+|<!-- [x3o3x3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!51
+|<!-- [o3x3x3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node_1|3|node}}<BR>t<sub>0,4,5</sub>{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
+|-  style="text-align:center; background:#e0f0e0;"
+!52
+|<!-- [x3o3o3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node|3|node_1}}<BR>t<sub>0,3,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!53
+|<!-- [o3x3o3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node_1|3|node}}<BR>t<sub>0,3,5</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!54
+|<!-- [o3o3x3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node|3|node}}<BR>t<sub>0,3,4</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!55
+|<!-- [x3o3o3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node|3|node_1}}<BR>t<sub>0,2,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!56
+|<!-- [o3x3o3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node_1|3|node}}<BR>t<sub>0,2,5</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!57
+|<!-- [o3o3x3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node|3|node}}<BR>t<sub>0,2,4</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!58
+|<!-- [o3o3o3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node|3|node}}<BR>t<sub>0,2,3</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!59
+|<!-- [x3o3o3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node|3|node_1}}<BR>t<sub>0,1,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!60
+|<!-- [o3x3o3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node_1|3|node}}<BR>t<sub>0,1,5</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!61
+|<!-- [o3o3x3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node_1|3|node|3|node}}<BR>t<sub>0,1,4</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!62
+|<!-- [o3o3o3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node|3|node}}<BR>t<sub>0,1,3</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!63
+|<!-- [o3o3o3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node|3|node}}<BR>t<sub>0,1,2</sub>{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
+|-  style="text-align:center; background:#f0e0e0;"
+!64
+|<!-- [x3x3x3x3o3o4o] -->{{CDD|node|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,3</sub>{3,3,3,3,3,4}||[[Runcicantitruncated 7-orthoplex]] (Gopaz)|||(0,1,2,3,4,4,4)√2|||||| || || ||60480||13440
+|-  style="text-align:center; background:#f0e0e0;"
+!65
+|<!-- [x3x3x3o3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,4</sub>{3,3,3,3,3,4}||[[Stericantitruncated 7-orthoplex]] (Cogarz)|||(0,0,1,1,2,3,4)√2|||||| || || ||241920||40320
+|-  style="text-align:center; background:#f0e0e0;"
+!66
+|<!-- [x3x3o3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,3,4</sub>{3,3,3,3,3,4}||[[Steriruncitruncated 7-orthoplex]] (Captaz)|||(0,0,1,2,2,3,4)√2|||||| || || ||181440||40320
+|-  style="text-align:center; background:#f0e0e0;"
+!67
+|<!-- [x3o3x3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,3,4</sub>{3,3,3,3,3,4}||[[Steriruncicantellated 7-orthoplex]] (Caparz)|||(0,0,1,2,3,3,4)√2|||||| || || ||181440||40320
+|-  style="text-align:center; background:#f0e0e0;"
+!68
+|<!-- [o3x3x3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}<BR>t<sub>1,2,3,4</sub>{3,3,3,3,3,4}||[[Biruncicantitruncated 7-orthoplex]] (Gibpaz)|||(0,0,1,2,3,4,4)√2|||||| || || ||161280||40320
+|-  style="text-align:center; background:#f0e0e0;"
+!69
+|<!-- [x3x3x3o3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,5</sub>{3,3,3,3,3,4}||[[Penticantitruncated 7-orthoplex]] (Tograz)|||(0,1,1,1,2,3,4)√2|||||| || || ||295680||53760
+|-  style="text-align:center; background:#f0e0e0;"
+!70
+|<!-- [x3x3o3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,3,5</sub>{3,3,3,3,3,4}||[[Pentiruncitruncated 7-orthoplex]] (Toptaz)|||(0,1,1,2,2,3,4)√2|||||| || || ||443520||80640
+|-  style="text-align:center; background:#f0e0e0;"
+!71
+|<!-- [x3o3x3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,3,5</sub>{3,3,3,3,3,4}||[[Pentiruncicantellated 7-orthoplex]] (Toparz)|||(0,1,1,2,3,3,4)√2|||||| || || ||403200||80640
+|-  style="text-align:center; background:#f0e0e0;"
+!72
+|<!-- [o3x3x3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node}}<BR>t<sub>1,2,3,5</sub>{3,3,3,3,3,4}||[[Bistericantitruncated 7-orthoplex]] (Becogarz)|||(0,1,1,2,3,4,4)√2|||||| || || ||362880||80640
+|-  style="text-align:center; background:#f0e0e0;"
+!73
+|<!-- [x3x3o3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,4,5</sub>{3,3,3,3,3,4}||[[Pentisteritruncated 7-orthoplex]] (Tacotaz)|||(0,1,2,2,2,3,4)√2|||||| || || ||241920||53760
+|-  style="text-align:center; background:#f0e0e0;"
+!74
+|<!-- [x3o3x3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,4,5</sub>{3,3,3,3,3,4}||[[Pentistericantellated 7-orthoplex]] (Tocarz)|||(0,1,2,2,3,3,4)√2|||||| || || ||403200||80640
+|-  style="text-align:center; background:#e0f0e0;"
+!75
+|<!-- [o3x3x3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node}}<BR>t<sub>1,2,4,5</sub>{4,3,3,3,3,3}||[[Bisteriruncitruncated 7-cube]] (Bocaptosaz)|||(0,1,2,2,3,4,4)√2|||||| |||| ||322560||80640
+|-  style="text-align:center; background:#f0e0e0;"
+!76
+|<!-- [x3o3o3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1}}<BR>t<sub>0,3,4,5</sub>{3,3,3,3,3,4}||[[Pentisteriruncinated 7-orthoplex]] (Tecpaz)|||(0,1,2,3,3,3,4)√2|||||| || || ||241920||53760
+|-  style="text-align:center; background:#e0e0f0;"
+!77
+|<!-- [o3x3o3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node}}<BR>t<sub>1,2,3,5</sub>{4,3,3,3,3,3}||[[Bistericantitruncated 7-cube]] (Becgresa)|||(0,1,2,3,3,4,4)√2|||||| || || ||362880||80640
+|-  style="text-align:center; background:#e0e0f0;"
+!78
+|<!-- [o3o3x3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node}}<BR>t<sub>1,2,3,4</sub>{4,3,3,3,3,3}||[[Biruncicantitruncated 7-cube]] (Gibposa)|||(0,1,2,3,4,4,4)√2|||||| || || ||188160||53760
+|-  style="text-align:center; background:#f0e0e0;"
+!79
+|<!-- [x3x3x3o3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,6</sub>{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
+|-  style="text-align:center; background:#f0e0e0;"
+!80
+|<!-- [x3x3o3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,3,6</sub>{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
+|-  style="text-align:center; background:#f0e0e0;"
+!81
+|<!-- [x3o3x3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,3,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!82
+|<!-- [o3x3x3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node}}<BR>t<sub>0,3,4,5</sub>{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
+|-  style="text-align:center; background:#f0e0e0;"
+!83
+|<!-- [x3x3o3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,4,6</sub>{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
+|-  style="text-align:center; background:#e0f0e0;"
+!84
+|<!-- [x3o3x3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,4,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!85
+|<!-- [o3x3x3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node}}<BR>t<sub>0,2,4,5</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!86
+|<!-- [x3o3o3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node|3|node_1}}<BR>t<sub>0,2,3,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!87
+|<!-- [o3x3o3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node}}<BR>t<sub>0,2,3,5</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!88
+|<!-- [o3o3x3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}<BR>t<sub>0,2,3,4</sub>{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
+|-  style="text-align:center; background:#e0f0e0;"
+!89
+|<!-- [x3x3o3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,5,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!90
+|<!-- [x3o3x3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node_1|3|node|3|node_1}}<BR>t<sub>0,1,4,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!91
+|<!-- [o3x3x3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node_1|3|node_1|3|node}}<BR>t<sub>0,1,4,5</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!92
+|<!-- [x3o3o3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node|3|node_1}}<BR>t<sub>0,1,3,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!93
+|<!-- [o3x3o3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node_1|3|node}}<BR>t<sub>0,1,3,5</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!94
+|<!-- [o3o3x3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1|3|node|3|node}}<BR>t<sub>0,1,3,4</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!95
+|<!-- [x3o3o3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node|3|node_1}}<BR>t<sub>0,1,2,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!96
+|<!-- [o3x3o3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node_1|3|node}}<BR>t<sub>0,1,2,5</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!97
+|<!-- [o3o3x3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1|3|node|3|node}}<BR>t<sub>0,1,2,4</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!98
+|<!-- [o3o3o3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}<BR>t<sub>0,1,2,3</sub>{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
+|-  style="text-align:center; background:#f0e0e0;"
+!99
+|<!-- [x3x3x3x3x3o4o] -->{{CDD|node|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,3,4</sub>{3,3,3,3,3,4}||[[Steriruncicantitruncated 7-orthoplex]] (Gocaz)|||(0,0,1,2,3,4,5)√2|||||| || || ||322560||80640
+|-  style="text-align:center; background:#f0e0e0;"
+!100
+|<!-- [x3x3x3x3o3x4o] -->{{CDD|node|4|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,3,5</sub>{3,3,3,3,3,4}||[[Pentiruncicantitruncated 7-orthoplex]] (Tegopaz)|||(0,1,1,2,3,4,5)√2|||||| || || ||725760||161280
+|-  style="text-align:center; background:#f0e0e0;"
+!101
+|<!-- [x3x3x3o3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,4,5</sub>{3,3,3,3,3,4}||[[Pentistericantitruncated 7-orthoplex]] (Tecagraz)|||(0,1,2,2,3,4,5)√2|||||| || || ||645120||161280
+|-  style="text-align:center; background:#f0e0e0;"
+!102
+|<!-- [x3x3o3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,3,4,5</sub>{3,3,3,3,3,4}||[[Pentisteriruncitruncated 7-orthoplex]] (Tecpotaz)|||(0,1,2,3,3,4,5)√2|||||| || || ||645120||161280
+|-  style="text-align:center; background:#f0e0e0;"
+!103
+|<!-- [x3o3x3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,3,4,5</sub>{3,3,3,3,3,4}||[[Pentisteriruncicantellated 7-orthoplex]] (Tacparez)|||(0,1,2,3,4,4,5)√2|||||| || || ||645120||161280
+|-  style="text-align:center; background:#e0f0e0;"
+!104
+|<!-- [o3x3x3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}<BR>t<sub>1,2,3,4,5</sub>{4,3,3,3,3,3}||[[Bisteriruncicantitruncated 7-cube]] (Gabcosaz)|||(0,1,2,3,4,5,5)√2|||||| || || ||564480||161280
+|-  style="text-align:center; background:#f0e0e0;"
+!105
+|<!-- [x3x3x3x3o3o4x] -->{{CDD|node_1|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,3,6</sub>{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
+|-  style="text-align:center; background:#f0e0e0;"
+!106
+|<!-- [x3x3x3o3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,4,6</sub>{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
+|-  style="text-align:center; background:#f0e0e0;"
+!107
+|<!-- [x3x3o3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,3,4,6</sub>{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
+|-  style="text-align:center; background:#e0f0e0;"
+!108
+|<!-- [x3o3x3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}<BR>t<sub>0,2,3,4,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!109
+|<!-- [o3x3x3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}<BR>t<sub>0,2,3,4,5</sub>{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
+|-  style="text-align:center; background:#f0e0e0;"
+!110
+|<!-- [x3x3x3o3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,5,6</sub>{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
+|-  style="text-align:center; background:#e0f0e0;"
+!111
+|<!-- [x3x3o3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,3,5,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!112
+|<!-- [x3o3x3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1|3|node|3|node_1}}<BR>t<sub>0,1,3,4,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!113
+|<!-- [o3x3x3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node}}<BR>t<sub>0,1,3,4,5</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!114
+|<!-- [x3x3o3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,2,5,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!115
+|<!-- [x3o3x3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1}}<BR>t<sub>0,1,2,4,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!116
+|<!-- [o3x3x3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node}}<BR>t<sub>0,1,2,4,5</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!117
+|<!-- [x3o3o3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1}}<BR>t<sub>0,1,2,3,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!118
+|<!-- [o3x3o3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node}}<BR>t<sub>0,1,2,3,5</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!119
+|<!-- [o3o3x3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node}}<BR>t<sub>0,1,2,3,4</sub>{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
+|-  style="text-align:center; background:#f0e0e0;"
+!120
+|<!-- [x3x3x3x3x3x4o] -->{{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,3,4,5</sub>{3,3,3,3,3,4}||[[Pentisteriruncicantitruncated 7-orthoplex]] (Gotaz)|||(0,1,2,3,4,5,6)√2|||||| || || ||1128960||322560
+|-  style="text-align:center; background:#f0e0e0;"
+!121
+|<!-- [x3x3x3x3x3o4x] -->{{CDD|node_1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,3,4,6</sub>{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
+|-  style="text-align:center; background:#f0e0e0;"
+!122
+|<!-- [x3x3x3x3o3x4x] -->{{CDD|node_1|4|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,3,5,6</sub>{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
+|-  style="text-align:center; background:#e0f0e0;"
+!123
+|<!-- [x3x3x3o3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,4,5,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!124
+|<!-- [x3x3o3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}<BR>t<sub>0,1,2,3,5,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!125
+|<!-- [x3o3x3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}<BR>t<sub>0,1,2,3,4,6</sub>{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
+|-  style="text-align:center; background:#e0e0f0;"
+!126
+|<!-- [o3x3x3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}<BR>t<sub>0,1,2,3,4,5</sub>{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
+|-  style="text-align:center; background:#e0f0e0;"
+!127
+|<!-- [x3x3x3x3x3x4x] -->{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}<BR>t<sub>0,1,2,3,4,5,6</sub>{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 D<sub>7</sub> family ==
+The D<sub>7</sub> family has symmetry of order 322560 (7 [[factorial]] x 2<sup>6</sup>).
+This family has 3×32−1=95 Wythoffian uniform polytopes, generated by marking one or more nodes of the D<sub>7</sub> [[Coxeter-Dynkin diagram]]. Of these, 63 (2×32−1) are repeated from the B<sub>7</sub> 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.
+{| class="wikitable"
+!rowspan=2|#
+!rowspan=2|[[Coxeter diagram]]
+!rowspan=2|Names
+!rowspan=2|Base point<BR>(Alternately signed)
+!colspan=7|Element counts
+|-
+!6||5||4||3||2||1||0
+|- align=center
+!1
+||{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node}}||[[7-demicube]]<BR>Demihepteract (Hesa)||(1,1,1,1,1,1,1)||78||532||1624||2800||2240||672||64
+|- align=center
+!2
+||{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node|3|node}}||[[Truncated 7-demicube]]<BR>Truncated demihepteract (Thesa)||(1,1,3,3,3,3,3)||142||1428||5656||11760||13440||7392||1344
+|- align=center
+!3
+||{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node|3|node}}||[[Cantellated 7-demicube]]<BR>Small rhombated demihepteract (Sirhesa)||(1,1,1,3,3,3,3)|| || || || || ||16800||2240
+|- align=center
+!4
+||{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node|3|node}}||[[Runcinated 7-demicube]]<BR>Small prismated demihepteract (Sphosa)||(1,1,1,1,3,3,3)|| || || || || ||20160||2240
+|- align=center
+!5
+||{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node_1|3|node}}||[[Stericated 7-demicube]]<BR>Small cellated demihepteract (Sochesa)||(1,1,1,1,1,3,3)|| || || || || ||13440||1344
+|- align=center
+!6
+||{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node_1}}||[[Pentellated 7-demicube]]<BR>Small terated demihepteract (Suthesa)||(1,1,1,1,1,1,3)|| || || || || ||4704||448
+|- align=center
+!7
+||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node|3|node}}||[[Cantitruncated 7-demicube]]<BR>Great rhombated demihepteract (Girhesa)||(1,1,3,5,5,5,5)|| || || || || ||23520||6720
+|- align=center
+!8
+||{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node|3|node}}||[[Runcitruncated 7-demicube]]<BR>Prismatotruncated demihepteract (Pothesa)||(1,1,3,3,5,5,5)|| || || || || ||73920||13440
+|- align=center
+!9
+||{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node|3|node}}||[[Runcicantellated 7-demicube]]<BR>Prismatorhomated demihepteract (Prohesa)||(1,1,1,3,5,5,5)|| || || || || ||40320||8960
+|- align=center
+!10
+||{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node_1|3|node}}||[[Steritruncated 7-demicube]]<BR>Cellitruncated demihepteract (Cothesa)||(1,1,3,3,3,5,5)|| || || || || ||87360||13440
+|- align=center
+!11
+||{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node_1|3|node}}||[[Stericantellated 7-demicube]]<BR>Cellirhombated demihepteract (Crohesa)||(1,1,1,3,3,5,5)|| || || || || ||87360||13440
+|- align=center
+!12
+||{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node_1|3|node}}||[[Steriruncinated 7-demicube]]<BR>Celliprismated demihepteract (Caphesa)||(1,1,1,1,3,5,5)|| || || || || ||40320||6720
+|- align=center
+!13
+||{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node|3|node_1}}||[[Pentitruncated 7-demicube]]<BR>Teritruncated demihepteract (Tuthesa)||(1,1,3,3,3,3,5)|| || || || || ||43680||6720
+|- align=center
+!14
+||{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node|3|node_1}}||[[Penticantellated 7-demicube]]<BR>Terirhombated demihepteract (Turhesa)||(1,1,1,3,3,3,5)|| || || || || ||67200||8960
+|- align=center
+!15
+||{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node|3|node_1}}||[[Pentiruncinated 7-demicube]]<BR>Teriprismated demihepteract (Tuphesa)||(1,1,1,1,3,3,5)|| || || || || ||53760||6720
+|- align=center
+!16
+||{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node_1|3|node_1}}||[[Pentistericated 7-demicube]]<BR>Tericellated demihepteract (Tuchesa)||(1,1,1,1,1,3,5)|| || || || || ||21504||2688
+|- align=center
+!17
+||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}||[[Runcicantitruncated 7-demicube]]<BR>Great prismated demihepteract (Gephosa)||(1,1,3,5,7,7,7)|| || || || || ||94080||26880
+|- align=center
+!18
+||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node}}||[[Stericantitruncated 7-demicube]]<BR>Celligreatorhombated demihepteract (Cagrohesa)||(1,1,3,5,5,7,7)|| || || || || ||181440||40320
+|- align=center
+!19
+||{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node}}||[[Steriruncitruncated 7-demicube]]<BR>Celliprismatotruncated demihepteract (Capthesa)||(1,1,3,3,5,7,7)|| || || || || ||181440||40320
+|- align=center
+!20
+||{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node}}||[[Steriruncicantellated 7-demicube]]<BR>Celliprismatorhombated demihepteract (Coprahesa)||(1,1,1,3,5,7,7)|| || || || || ||120960||26880
+|- align=center
+!21
+||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node|3|node_1}}||[[Penticantitruncated 7-demicube]]<BR>Terigreatorhombated demihepteract (Tugrohesa)||(1,1,3,5,5,5,7)|| || || || || ||120960||26880
+|- align=center
+!22
+||{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}||[[Pentiruncitruncated 7-demicube]]<BR>Teriprismatotruncated demihepteract (Tupthesa)||(1,1,3,3,5,5,7)|| || || || || ||221760||40320
+|- align=center
+!23
+||{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node|3|node_1}}||[[Pentiruncicantellated 7-demicube]]<BR>Teriprismatorhombated demihepteract (Tuprohesa)||(1,1,1,3,5,5,7)|| || || || || ||134400||26880
+|- align=center
+!24
+||{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node_1|3|node_1}}||[[Pentisteritruncated 7-demicube]]<BR>Tericellitruncated demihepteract (Tucothesa)||(1,1,3,3,3,5,7)|| || || || || ||147840||26880
+|- align=center
+!25
+||{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node_1|3|node_1}}||[[Pentistericantellated 7-demicube]]<BR>Tericellirhombated demihepteract (Tucrohesa)||(1,1,1,3,3,5,7)|| || || || || ||161280||26880
+|- align=center
+!26
+||{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node_1|3|node_1}}||[[Pentisteriruncinated 7-demicube]]<BR>Tericelliprismated demihepteract (Tucophesa)||(1,1,1,1,3,5,7)|| || || || || ||80640||13440
+|- align=center
+!27
+||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}||[[Steriruncicantitruncated 7-demicube]]<BR>Great cellated demihepteract (Gochesa)||(1,1,3,5,7,9,9)|| || || || || ||282240||80640
+|- align=center
+!28
+||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}||[[Pentiruncicantitruncated 7-demicube]]<BR>Terigreatoprimated demihepteract (Tugphesa)||(1,1,3,5,7,7,9)|| || || || || ||322560||80640
+|- align=center
+!29
+||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}||[[Pentistericantitruncated 7-demicube]]<BR>Tericelligreatorhombated demihepteract (Tucagrohesa)||(1,1,3,5,5,7,9)|| || || || || ||322560||80640
+|- align=center
+!30
+||{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}||[[Pentisteriruncitruncated 7-demicube]]<BR>Tericelliprismatotruncated demihepteract (Tucpathesa)||(1,1,3,3,5,7,9)|| || || || || ||362880||80640
+|- align=center
+!31
+||{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}||[[Pentisteriruncicantellated 7-demicube]]<BR>Tericellprismatorhombated demihepteract (Tucprohesa)||(1,1,1,3,5,7,9)|| || || || || ||241920||53760
+|- align=center
+!32
+||{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}||[[Pentisteriruncicantitruncated 7-demicube]]<BR>Great terated demihepteract (Guthesa)||(1,1,3,5,7,9,11)|| || || || || ||564480||161280
+|}
+== The E<sub>7</sub> family ==
+The E<sub>7</sub> [[Coxeter group]] has order 2,903,040.
+There are 127 forms based on all permutations of the [[Coxeter-Dynkin diagram]]s with one or more rings.
+See also a [[list of E7 polytopes]] for symmetric Coxeter plane graphs of these polytopes.
+{| class="wikitable"
+|-
+!rowspan=2|#
+!rowspan=2|[[Coxeter-Dynkin diagram]]<br />[[Schläfli symbol]]
+!rowspan=2|Names
+!colspan=7|Element counts
+|-
+! 6|| 5|| 4|| 3|| 2|| 1|| 0
+|- style="text-align:center;"
+|1||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}||[[Gosset 2 31 polytope|2<sub>31</sub>]] (laq)||632||4788||16128||20160||10080||2016||126
+|- style="text-align:center;"
+|2||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea_1|3a|nodea}}||[[Rectified 2 31 polytope|Rectified 2<sub>31</sub>]] (rolaq)||758||10332||47880||100800||90720||30240||2016
+|- style="text-align:center;"
+|3||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}||[[Rectified 1 32 polytope|Rectified 1<sub>32</sub>]] (rolin)||758||12348||72072||191520||241920||120960||10080
+|- style="text-align:center;"
+|4||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}||[[1 32 polytope|1<sub>32</sub>]] (lin)||182||4284||23688||50400||40320||10080||576
+|- style="text-align:center;"
+|5||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}||[[Birectified 3 21 polytope|Birectified 3<sub>21</sub>]] (branq)||758||12348||68040||161280||161280||60480||4032
+|- style="text-align:center;"
+|6||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea}}||[[Rectified 3 21 polytope|Rectified 3<sub>21</sub>]] (ranq)||758||44352||70560||48384||11592||12096||756
+|- style="text-align:center;"
+|7||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea}}||[[Gosset 3 21 polytope|3<sub>21</sub>]] (naq)||702||6048||12096||10080||4032||756||56
+|- align=center
+|8||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch|3a|nodea_1|3a|nodea_1}}||[[Truncated 2 31 polytope|Truncated 2<sub>31</sub>]] (talq)||758||10332||47880||100800||90720||32256||4032
+|- align=center
+|9||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea_1}}||[[Cantellated 2 31 polytope|Cantellated 2<sub>31</sub>]] (sirlaq)|| || || || || ||131040||20160
+|- align=center
+|10||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea}}||Bitruncated 2<sub>31</sub> (botlaq)||||||||||||||30240
+|- align=center
+|11||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea_1}}||small demified 2<sub>31</sub> (shilq)||2774||22428||78120||151200||131040||42336||4032
+|- align=center
+|12||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea}}||demirectified 2<sub>31</sub> (hirlaq)||||||||||||||12096
+|- align=center
+|13||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}||truncated 1<sub>32</sub> (tolin)||||||||||||||20160
+|- align=center
+|14||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea_1}}||small demiprismated 2<sub>31</sub> (shiplaq)||||||||||||||20160
+|- align=center
+|15||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}||birectified 1<sub>32</sub> (berlin)||758||22428||142632||403200||544320||302400||40320
+|- align=center
+|16||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea}}||tritruncated 3<sub>21</sub> (totanq)||||||||||||||40320
+|- align=center
+|17||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea}}||demibirectified 3<sub>21</sub> (hobranq)||||||||||||||20160
+|- align=center
+|18||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}||small cellated 2<sub>31</sub> (scalq)||||||||||||||7560
+|- align=center
+|19||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch|3a|nodea_1|3a|nodea}}||small biprismated 2<sub>31</sub> (sobpalq)||||||||||||||30240
+|- align=center
+|20||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}||small birhombated 3<sub>21</sub> (sabranq)||||||||||||||60480
+|- align=center
+|21||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}||demirectified 3<sub>21</sub> (harnaq)||||||||||||||12096
+|- align=center
+|22||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}||bitruncated 3<sub>21</sub> (botnaq)||||||||||||||12096
+|- align=center
+|23||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}||small terated 3<sub>21</sub> (stanq)||||||||||||||1512
+|- align=center
+|24||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea_1|3a|nodea}}||small demicellated 3<sub>21</sub> (shocanq)||||||||||||||12096
+|- align=center
+|25||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}||small prismated 3<sub>21</sub> (spanq)||||||||||||||40320
+|- align=center
+|26||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}||small demified 3<sub>21</sub> (shanq)||||||||||||||4032
+|- align=center
+|27||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}||small rhombated 3<sub>21</sub> (sranq)||||||||||||||12096
+|- align=center
+|28||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea}}||[[Truncated 3 21 polytope|Truncated 3<sub>21</sub>]] (tanq)||758||11592||48384||70560||44352||12852||1512
+|- align=center
+|29||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea_1}}||great rhombated 2<sub>31</sub> (girlaq)||||||||||||||60480
+|- align=center
+|30||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||demitruncated 2<sub>31</sub> (hotlaq)||||||||||||||24192
+|- align=center
+|31||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea_1}}||small demirhombated 2<sub>31</sub> (sherlaq)||||||||||||||60480
+|- align=center
+|32||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea}}||demibitruncated 2<sub>31</sub> (hobtalq)||||||||||||||60480
+|- align=center
+|33||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea_1}}||demiprismated 2<sub>31</sub> (hiptalq)||||||||||||||80640
+|- align=center
+|34||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea_1}}||demiprismatorhombated 2<sub>31</sub> (hiprolaq)||||||||||||||120960
+|- align=center
+|35||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea}}||bitruncated 1<sub>32</sub> (batlin)||||||||||||||120960
+|- align=center
+|36||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea_1}}||small prismated 2<sub>31</sub> (spalq)||||||||||||||80640
+|- align=center
+|37||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea}}||small rhombated 1<sub>32</sub> (sirlin)||||||||||||||120960
+|- align=center
+|38||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea}}||tritruncated 2<sub>31</sub> (tatilq)||||||||||||||80640
+|- align=center
+|39||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch|3a|nodea_1|3a|nodea_1}}||cellitruncated 2<sub>31</sub> (catalaq)||||||||||||||60480
+|- align=center
+|40||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea|3a|nodea_1}}||cellirhombated 2<sub>31</sub> (crilq)||||||||||||||362880
+|- align=center
+|41||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea}}||biprismatotruncated 2<sub>31</sub> (biptalq)||||||||||||||181440
+|- align=center
+|42||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea_1}}||small prismated 1<sub>32</sub> (seplin)||||||||||||||60480
+|- align=center
+|43||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea}}||small biprismated 3<sub>21</sub> (sabipnaq)||||||||||||||120960
+|- align=center
+|44||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}||small demibirhombated 3<sub>21</sub> (shobranq)||||||||||||||120960
+|- align=center
+|45||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea|3a|nodea_1}}||cellidemiprismated 2<sub>31</sub> (chaplaq)||||||||||||||60480
+|- align=center
+|46||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}||demibiprismatotruncated 3<sub>21</sub> (hobpotanq)||||||||||||||120960
+|- align=center
+|47||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea}}||great birhombated 3<sub>21</sub> (gobranq)||||||||||||||120960
+|- align=center
+|48||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea}}||demibitruncated 3<sub>21</sub> (hobtanq)||||||||||||||60480
+|- align=center
+|49||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch|3a|nodea_1|3a|nodea_1}}||teritruncated 2<sub>31</sub> (totalq)||||||||||||||24192
+|- align=center
+|50||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_10|3a|nodea|3a|nodea_1}}||terirhombated 2<sub>31</sub> (trilq)||||||||||||||120960
+|- align=center
+|51||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea}}||demicelliprismated 3<sub>21</sub> (hicpanq)||||||||||||||120960
+|- align=center
+|52||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea_1}}||small teridemified 2<sub>31</sub> (sethalq)||||||||||||||24192
+|- align=center
+|53||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea}}||small cellated 3<sub>21</sub> (scanq)||||||||||||||60480
+|- align=center
+|54||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}||demiprismated 3<sub>21</sub> (hipnaq)||||||||||||||80640
+|- align=center
+|55||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch|3a|nodea|3a|nodea_1}}||terirhombated 3<sub>21</sub> (tranq)||||||||||||||60480
+|- align=center
+|56||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}||demicellirhombated 3<sub>21</sub> (hocranq)||||||||||||||120960
+|- align=center
+|57||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea}}||prismatorhombated 3<sub>21</sub> (pranq)||||||||||||||120960
+|- align=center
+|58||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea}}||small demirhombated 3<sub>21</sub> (sharnaq)||||||||||||||60480
+|- align=center
+|59||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch|3a|nodea|3a|nodea_1}}||teritruncated 3<sub>21</sub> (tetanq)||||||||||||||15120
+|- align=center
+|60||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch|3a|nodea_1|3a|nodea}}||demicellitruncated 3<sub>21</sub> (hictanq)||||||||||||||60480
+|- align=center
+|61||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea|3a|nodea}}||prismatotruncated 3<sub>21</sub> (potanq)||||||||||||||120960
+|- align=center
+|62||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea}}||demitruncated 3<sub>21</sub> (hotnaq)||||||||||||||24192
+|- align=center
+|63||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea|3a|nodea}}||great rhombated 3<sub>21</sub> (granq)||||||||||||||24192
+|- align=center
+|64||{{CDD|nodea|3a|nodea|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea_1}}||great demified 2<sub>31</sub> (gahlaq)||||||||||||||120960
+|- align=center
+|65||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea_1}}||great demiprismated 2<sub>31</sub> (gahplaq)||||||||||||||241920
+|- align=center
+|66||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||prismatotruncated 2<sub>31</sub> (potlaq)||||||||||||||241920
+|- align=center
+|67||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea_1}}||prismatorhombated 2<sub>31</sub> (prolaq)||||||||||||||241920
+|- align=center
+|68||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea}}||great rhombated 1<sub>32</sub> (girlin)||||||||||||||241920
+|- align=center
+|69||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea_1}}||celligreatorhombated 2<sub>31</sub> (cagrilq)||||||||||||||362880
+|- align=center
+|70||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||cellidemitruncated 2<sub>31</sub> (chotalq)||||||||||||||241920
+|- align=center
+|71||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea|3a|nodea_1}}||prismatotruncated 1<sub>32</sub> (patlin)||||||||||||||362880
+|- align=center
+|72||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea}}||biprismatorhombated 3<sub>21</sub> (bipirnaq)||||||||||||||362880
+|- align=center
+|73||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea_1}}||tritruncated 1<sub>32</sub> (tatlin)||||||||||||||241920
+|- align=center
+|74||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea_1}}||cellidemiprismatorhombated 2<sub>31</sub> (chopralq)||||||||||||||362880
+|- align=center
+|75||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea}}||great demibiprismated 3<sub>21</sub> (ghobipnaq)||||||||||||||362880
+|- align=center
+|76||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea_1}}||celliprismated 2<sub>31</sub> (caplaq)||||||||||||||241920
+|- align=center
+|77||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea}}||biprismatotruncated 3<sub>21</sub> (boptanq)||||||||||||||362880
+|- align=center
+|78||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea}}||great trirhombated 2<sub>31</sub> (gatralaq)||||||||||||||241920
+|- align=center
+|79||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea_1}}||terigreatorhombated 2<sub>31</sub> (togrilq)||||||||||||||241920
+|- align=center
+|80||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||teridemitruncated 2<sub>31</sub> (thotalq)||||||||||||||120960
+|- align=center
+|81||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_11|3a|nodea|3a|nodea_1}}||teridemirhombated 2<sub>31</sub> (thorlaq)||||||||||||||241920
+|- align=center
+|82||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea}}||celliprismated 3<sub>21</sub> (capnaq)||||||||||||||241920
+|- align=center
+|83||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea_1}}||teridemiprismatotruncated 2<sub>31</sub> (thoptalq)||||||||||||||241920
+|- align=center
+|84||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea_1}}||teriprismatorhombated 3<sub>21</sub> (tapronaq)||||||||||||||362880
+|- align=center
+|85||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea}}||demicelliprismatorhombated 3<sub>21</sub> (hacpranq)||||||||||||||362880
+|- align=center
+|86||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea_1}}||teriprismated 2<sub>31</sub> (toplaq)||||||||||||||241920
+|- align=center
+|87||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea}}||cellirhombated 3<sub>21</sub> (cranq)||||||||||||||362880
+|- align=center
+|88||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea}}||demiprismatorhombated 3<sub>21</sub> (hapranq)||||||||||||||241920
+|- align=center
+|89||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch|3a|nodea_1|3a|nodea_1}}||tericellitruncated 2<sub>31</sub> (tectalq)||||||||||||||120960
+|- align=center
+|90||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea|3a|nodea_1}}||teriprismatotruncated 3<sub>21</sub> (toptanq)||||||||||||||362880
+|- align=center
+|91||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea}}||demicelliprismatotruncated 3<sub>21</sub> (hecpotanq)||||||||||||||362880
+|- align=center
+|92||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea|3a|nodea_1}}||teridemitruncated 3<sub>21</sub> (thotanq)||||||||||||||120960
+|- align=center
+|93||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea}}||cellitruncated 3<sub>21</sub> (catnaq)||||||||||||||241920
+|- align=center
+|94||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea|3a|nodea}}||demiprismatotruncated 3<sub>21</sub> (hiptanq)||||||||||||||241920
+|- align=center
+|95||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea|3a|nodea_1}}||terigreatorhombated 3<sub>21</sub> (tagranq)||||||||||||||120960
+|- align=center
+|96||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea}}||demicelligreatorhombated 3<sub>21</sub> (hicgarnq)||||||||||||||241920
+|- align=center
+|97||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea}}||great prismated 3<sub>21</sub> (gopanq)||||||||||||||241920
+|- align=center
+|98||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea}}||great demirhombated 3<sub>21</sub> (gahranq)||||||||||||||120960
+|- align=center
+|99||{{CDD|nodea|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea_1}}||great prismated 2<sub>31</sub> (gopalq)||||||||||||||483840
+|- align=center
+|100||{{CDD|nodea|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea_1}}||great cellidemified 2<sub>31</sub> (gechalq)||||||||||||||725760
+|- align=center
+|101||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea_1}}||great birhombated 1<sub>32</sub> (gebrolin)||||||||||||||725760
+|- align=center
+|102||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||prismatorhombated 1<sub>32</sub> (prolin)||||||||||||||725760
+|- align=center
+|103||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea_1}}||celliprismatorhombated 2<sub>31</sub> (caprolaq)||||||||||||||725760
+|- align=center
+|104||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea}}||great biprismated 2<sub>31</sub> (gobpalq)||||||||||||||725760
+|- align=center
+|105||{{CDD|nodea_1|3a|nodea|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea_1}}||tericelliprismated 3<sub>21</sub> (ticpanq)||||||||||||||483840
+|- align=center
+|106||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea_1}}||teridemigreatoprismated 2<sub>31</sub> (thegpalq)||||||||||||||725760
+|- align=center
+|107||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||teriprismatotruncated 2<sub>31</sub> (teptalq)||||||||||||||725760
+|- align=center
+|108||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea_1}}||teriprismatorhombated 2<sub>31</sub> (topralq)||||||||||||||725760
+|- align=center
+|109||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea}}||cellipriemsatorhombated 3<sub>21</sub> (copranq)||||||||||||||725760
+|- align=center
+|110||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_10|3a|nodea_1|3a|nodea_1}}||tericelligreatorhombated 2<sub>31</sub> (tecgrolaq)||||||||||||||725760
+|- align=center
+|111||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||tericellitruncated 3<sub>21</sub> (tectanq)||||||||||||||483840
+|- align=center
+|112||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea|3a|nodea_1}}||teridemiprismatotruncated 3<sub>21</sub> (thoptanq)||||||||||||||725760
+|- align=center
+|113||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea}}||celliprismatotruncated 3<sub>21</sub> (coptanq)||||||||||||||725760
+|- align=center
+|114||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch|3a|nodea_1|3a|nodea_1}}||teridemicelligreatorhombated 3<sub>21</sub> (thocgranq)||||||||||||||483840
+|- align=center
+|115||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea|3a|nodea_1}}||terigreatoprismated 3<sub>21</sub> (tagpanq)||||||||||||||725760
+|- align=center
+|116||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea}}||great demicellated 3<sub>21</sub> (gahcnaq)||||||||||||||725760
+|- align=center
+|117||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea|3a|nodea_1}}||tericelliprismated laq  (tecpalq)||||||||||||||483840
+|- align=center
+|118||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea}}||celligreatorhombated 3<sub>21</sub> (cogranq)||||||||||||||725760
+|- align=center
+|119||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea}}||great demified 3<sub>21</sub> (gahnq)||||||||||||||483840
+|- align=center
+|120||{{CDD|nodea|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea_1}}||great cellated 2<sub>31</sub> (gocalq)||||||||||||||1451520
+|- align=center
+|121||{{CDD|nodea_1|3a|nodea|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea_1}}||terigreatoprismated 2<sub>31</sub> (tegpalq)||||||||||||||1451520
+|- align=center
+|122||{{CDD|nodea_1|3a|nodea_1|3a|nodea|3a|branch_11|3a|nodea_1|3a|nodea_1}}||tericelliprismatotruncated 3<sub>21</sub> (tecpotniq)||||||||||||||1451520
+|- align=center
+|123||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_10|3a|nodea_1|3a|nodea_1}}||tericellidemigreatoprismated 2<sub>31</sub> (techogaplaq)||||||||||||||1451520
+|- align=center
+|124||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_01lr|3a|nodea_1|3a|nodea_1}}||tericelligreatorhombated 3<sub>21</sub> (tacgarnq)||||||||||||||1451520
+|- align=center
+|125||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea|3a|nodea_1}}||tericelliprismatorhombated 2<sub>31</sub> (tecprolaq)||||||||||||||1451520
+|- align=center
+|126||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea}}||great cellated 3<sub>21</sub> (gocanq)||||||||||||||1451520
+|- align=center
+|127||{{CDD|nodea_1|3a|nodea_1|3a|nodea_1|3a|branch_11|3a|nodea_1|3a|nodea_1}}||great terated 3<sub>21</sub> (gotanq)||||||||||||||2903040
+|}
+== Regular and uniform honeycombs ==
+[[File:Coxeter diagram affine rank7 correspondence.png|518px|thumb|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:
+{| class="wikitable"
+|-
+!#
+!colspan=2|[[Coxeter group]]
+![[Coxeter diagram]]
+!Forms
+|- align=center
+|1||<math>{\tilde{A}}_6</math>||[3<sup>[7]</sup>]||{{CDD|branch|3ab|nodes|3ab|nodes|split2|node}}||17
+|- align=center
+|2||<math>{\tilde{C}}_6</math>||[4,3<sup>4</sup>,4]||{{CDD|node|4|node|3|node|3|node|3|node|3|node|4|node}}||71
+|- align=center
+|3||<math>{\tilde{B}}_6</math>||h[4,3<sup>4</sup>,4]<br />[4,3<sup>3</sup>,3<sup>1,1</sup>]||{{CDD|nodes|split2|node|3|node|3|node|3|node|4|node}}||95 (32 new)
+|- align=center
+|4||<math>{\tilde{D}}_6</math>||q[4,3<sup>4</sup>,4]<br />[3<sup>1,1</sup>,3<sup>2</sup>,3<sup>1,1</sup>]||{{CDD|nodes|split2|node|3|node|3|node|split1|nodes}}|| 41 (6 new)
+|- align=center
+|5||<math>{\tilde{E}}_6</math>||[3<sup>2,2,2</sup>]||{{CDD|nodes|3ab|nodes|split2|node|3|node|3|node}}||39
+|}
+Regular and uniform tessellations include:
+* <math>{\tilde{A}}_6</math>, 17 forms
+** Uniform [[6-simplex honeycomb]]: {3<sup>[7]</sup>} {{CDD|node_1|split1|nodes|3ab|nodes|3ab|nodes|3ab|branch}}
+* <math>{\tilde{C}}_6</math>, [4,3<sup>4</sup>,4], 71 forms
+** Regular [[6-cube honeycomb]], represented by symbols {4,3<sup>4</sup>,4}, {{CDD|node_1|4|node|3|node|3|node|3|node|3|node|4|node}}
+* <math>{\tilde{B}}_6</math>, [3<sup>1,1</sup>,3<sup>3</sup>,4], 95 forms, 64 shared with <math>{\tilde{C}}_6</math>, 32 new
+** Uniform [[6-demicube honeycomb]], represented by symbols h{4,3<sup>4</sup>,4} = {3<sup>1,1</sup>,3<sup>3</sup>,4}, {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|4|node}} = {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|4|node}}
+* <math>{\tilde{D}}_6</math>, [3<sup>1,1</sup>,3<sup>2</sup>,3<sup>1,1</sup>], 41 unique ringed permuations, most shared with <math>{\tilde{B}}_6</math> and <math>{\tilde{C}}_6</math>, and 6 are new. Coxeter calls the first one a [[quarter 6-cubic honeycomb]].
+** {{CDD|nodes_10ru|split2|node|3|node|3|node|split1|nodes_10lu}}, {{CDD|nodes_10ru|split2|node_1|3|node|3|node|split1|nodes_10lu}}, {{CDD|nodes_10ru|split2|node|3|node_1|3|node|split1|nodes_10lu}}, {{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|split1|nodes_10lu}}, {{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|split1|nodes_10lu}}, {{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|split1|nodes_10lu}}
+* <math>{\tilde{E}}_6</math>: [3<sup>2,2,2</sup>], 39 forms
+** Uniform [[Gosset 2 22 honeycomb|2<sub>22</sub> honeycomb]]: represented by symbols {3<sup>2,2,2</sup>}, {{CDD|nodes|3ab|nodes|split2|node|3|node|3|node_1}}
+{| class=wikitable
+|+ Prismatic groups
+|-
+!#
+!colspan=2|[[Coxeter group]]
+![[Coxeter-Dynkin diagram]]
+|-
+|1||<math>{\tilde{A}}_5</math>x<math>{\tilde{I}}_1</math>||[3<sup>[6]</sup>,2,∞]||{{CDD|node|split1|nodes|3ab|nodes|split2|node|2|node|infin|node}}
+|-
+|2||<math>{\tilde{B}}_5</math>x<math>{\tilde{I}}_1</math>||[4,3,3<sup>1,1</sup>,2,∞]||{{CDD|node|4|node|3|node|3|node|3|node|4|node|2|node|infin|node}}
+|-
+|3||<math>{\tilde{C}}_5</math>x<math>{\tilde{I}}_1</math>||[4,3<sup>3</sup>,4,2,∞]||{{CDD|nodes|split2|node|3|node|3|node|4|node|2|node|infin|node}}
+|-
+|4||<math>{\tilde{D}}_5</math>x<math>{\tilde{I}}_1</math>||[3<sup>1,1</sup>,3,3<sup>1,1</sup>,2,∞]||{{CDD|nodes|split2|node|3|node|split1|nodes|2|node|infin|node}}
+|-
+|5||<math>{\tilde{A}}_4</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[3<sup>[5]</sup>,2,∞,2,∞,2,∞]||{{CDD|branch|3ab|nodes|split2|node|2|node|infin|node|2|node|infin|node}}
+|-
+|6||<math>{\tilde{B}}_4</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[4,3,3<sup>1,1</sup>,2,∞,2,∞]||{{CDD|nodes|split2|node|3|node|4|node|2|node|infin|node|2|node|infin|node}}
+|-
+|7||<math>{\tilde{C}}_4</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[4,3,3,4,2,∞,2,∞]||{{CDD|node|4|node|3|node|3|node|4|node|2|node|infin|node|2|node|infin|node}}
+|-
+|8||<math>{\tilde{D}}_4</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[3<sup>1,1,1,1</sup>,2,∞,2,∞]||{{CDD|nodes|split2|node|split1|nodes|2|node|infin|node|2|node|infin|node}}
+|-
+|9||<math>{\tilde{F}}_4</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[3,4,3,3,2,∞,2,∞]||{{CDD|node|3|node|4|node|3|node|3|node|2|node|infin|node|2|node|infin|node}}
+|-
+|10||<math>{\tilde{C}}_3</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[4,3,4,2,∞,2,∞,2,∞]||{{CDD|node|4|node|3|node|4|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}}
+|-
+|11||<math>{\tilde{B}}_3</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[4,3<sup>1,1</sup>,2,∞,2,∞,2,∞]||{{CDD|nodes|split2|node|4|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}}
+|-
+|12||<math>{\tilde{A}}_3</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[3<sup>[4]</sup>,2,∞,2,∞,2,∞]||{{CDD|branch|3ab|branch|2|node|infin|node|2|node|infin|node|2|node|infin|node}}
+|-
+|13||<math>{\tilde{C}}_2</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[4,4,2,∞,2,∞,2,∞,2,∞]||{{CDD|node|4|node|4|node|2|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}}
+|-
+|14||<math>{\tilde{H}}_2</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[6,3,2,∞,2,∞,2,∞,2,∞]||{{CDD|node|6|node|3|node|2|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}}
+|-
+|15||<math>{\tilde{A}}_2</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[3<sup>[3]</sup>,2,∞,2,∞,2,∞,2,∞]||{{CDD|node|split1|branch|2|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}}
+|-
+|16||<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>x<math>{\tilde{I}}_1</math>||[∞,2,∞,2,∞,2,∞,2,∞]||{{CDD|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node|2|node|infin|node}}
+|}
+=== 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 [[Coxeter-Dynkin_diagram#Rank_4_to_10|3 noncompact hyperbolic Coxeter groups]] of rank 7, each generating uniform honeycombs in 6-space as permutations of rings of the Coxeter diagrams.
+{| class=wikitable
+|align=right|<math>{\bar{P}}_6</math> = [3,3<sup>[6]</sup>]:<BR>{{CDD|node|split1|nodes|3ab|nodes|split2|node|3|node}}
+|align=right|<math>{\bar{Q}}_6</math> = [3<sup>1,1</sup>,3,3<sup>2,1</sup>]:<BR>{{CDD|nodea|3a|branch|3a|branch|3a|nodea|3a|nodea}}
+|align=right|<math>{\bar{S}}_6</math> = [4,3,3,3<sup>2,1</sup>]:<BR>{{CDD|nodea|3a|nodea|3a|branch|3a|nodea|3a|nodea|4a|nodea}}
+|}
+== Notes on the Wythoff construction for the uniform 7-polytopes ==
+The reflective 7-dimensional [[uniform polytope]]s 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 polytope]]s 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.
+{|class="wikitable"
+|-
+!Operation
+!Extended<br/>[[Schläfli symbol]]
+!width=110|[[Coxeter-Dynkin diagram|Coxeter-<br/>Dynkin<br/>diagram]]
+!Description
+|-
+! Parent
+|width=70| t<sub>0</sub>{p,q,r,s,t,u}
+|{{CDD|node_1|p|node|q|node|r|node|s|node|t|node|u|node}}
+| Any regular 7-polytope
+|-
+! [[Rectification (geometry)|Rectified]]
+| t<sub>1</sub>{p,q,r,s,t,u}
+|{{CDD|node|p|node_1|q|node|r|node|s|node|t|node|u|node}}
+|The edges are fully truncated into single points. The 7-polytope now has the combined faces of the parent and dual.
+|-
+! Birectified
+| t<sub>2</sub>{p,q,r,s,t,u}
+|{{CDD|node|p|node|q|node_1|r|node|s|node|t|node|u|node}}
+|Birectification reduces [[Cell (geometry)|cells]] to their [[Dual polytope|duals]].
+|-
+![[Truncation (geometry)|Truncated]]
+| t<sub>0,1</sub>{p,q,r,s,t,u}
+|{{CDD|node_1|p|node_1|q|node|r|node|s|node|t|node|u|node}}
+|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.<br/>[[File:Cube truncation sequence.svg|400px]]
+|-
+![[Bitruncated]]
+| t<sub>1,2</sub>{p,q,r,s,t,u}
+|{{CDD|node|p|node_1|q|node_1|r|node|s|node|t|node|u|node}}
+|Bitrunction transforms cells to their dual truncation.
+|-
+!Tritruncated
+| t<sub>2,3</sub>{p,q,r,s,t,u}
+|{{CDD|node|p|node|q|node_1|r|node_1|s|node|t|node|u|node}}
+|Tritruncation transforms 4-faces to their dual truncation.
+|-
+! [[Cantellation (geometry)|Cantellated]]
+| t<sub>0,2</sub>{p,q,r,s,t,u}
+|{{CDD|node_1|p|node|q|node_1|r|node|s|node|t|node|u|node}}
+|In addition to vertex truncation, each original edge is ''beveled'' with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms.<br/>[[File:Cube cantellation sequence.svg|400px]]
+|-
+! Bicantellated
+| t<sub>1,3</sub>{p,q,r,s,t,u}
+|{{CDD|node|p|node_1|q|node|r|node_1|s|node|t|node|u|node}}
+|In addition to vertex truncation, each original edge is ''beveled'' with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms.
+|-
+! [[Runcination (geometry)|Runcinated]]
+| t<sub>0,3</sub>{p,q,r,s,t,u}
+|{{CDD|node_1|p|node|q|node|r|node_1|s|node|t|node|u|node}}
+|Runcination reduces cells and creates new cells at the vertices and edges.
+|-
+! Biruncinated
+| t<sub>1,4</sub>{p,q,r,s,t,u}
+|{{CDD|node|p|node_1|q|node|r|node|s|node_1|t|node|u|node}}
+|Runcination reduces cells and creates new cells at the vertices and edges.
+|-
+! [[Sterication|Stericated]]
+| t<sub>0,4</sub>{p,q,r,s,t,u}
+|{{CDD|node_1|p|node|q|node|r|node|s|node_1|t|node|u|node}}
+|Sterication reduces 4-faces and creates new 4-faces at the vertices, edges, and faces in the gaps.
+|-
+! Pentellated
+| t<sub>0,5</sub>{p,q,r,s,t,u}
+|{{CDD|node_1|p|node|q|node|r|node|s|node|t|node_1|u|node}}
+|Pentellation reduces 5-faces and creates new 5-faces at the vertices, edges, faces, and cells in the gaps.
+|-
+! Hexicated
+| t<sub>0,6</sub>{p,q,r,s,t,u}
+|{{CDD|node_1|p|node|q|node|r|node|s|node|t|node|u|node_1}}
+|Hexication reduces 6-faces and creates new 6-faces at the vertices, edges, faces, cells, and 4-faces in the gaps. ([[Expansion (geometry)|expansion]] operation for 7-polytopes)
+|-
+![[Omnitruncation (geometry)|Omnitruncated]]
+| t<sub>0,1,2,3,4,5,6</sub>{p,q,r,s,t,u}
+|{{CDD|node_1|p|node_1|q|node_1|r|node_1|s|node_1|t|node_1|u|node_1}}
+|All six operators, truncation, cantellation, runcination, sterication, pentellation, and hexication are applied.
+|}
+== References ==
+* [[Thorold Gosset|T. Gosset]]: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', [[Messenger of Mathematics]], Macmillan, 1900
+* [[Alicia Boole Stott|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
+* [[Harold Scott MacDonald Coxeter|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''', editied 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'', [Math. Zeit. 46 (1940) 380-407, MR 2,10]
+** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', [Math. Zeit. 188 (1985) 559-591]
+** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
+* [[Norman Johnson (mathematician)|N.W. Johnson]]: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
+* {{KlitzingPolytopes|polyexa.htm|7D|uniform polytopes (polyexa)}}
+== External links ==
+* [http://www.steelpillow.com/polyhedra/ditela.html Polytope names]
+* [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
+* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
+* {{PolyCell | urlname = glossary.html| title = Glossary for hyperspace}}
+{{Polytopes}}
+[[Category:7-polytopes]]

Convergence of measures: Difference between revisions

en>Hairer

Revision as of 19:47, 3 February 2014

Regular 7-polytopes

Euler characteristic

Uniform 7-polytopes by fundamental Coxeter groups

The A7 family

The B7 family

The D7 family

The E7 family

Regular and uniform honeycombs

Regular and uniform hyperbolic honeycombs

Notes on the Wythoff construction for the uniform 7-polytopes

References

External links

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The A₇ family

The B₇ family

The D₇ family

The E₇ family