Uniform 7-polytope

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Graphs of three regular and related uniform polytopes

7-simplex	Rectified 7-simplex		Truncated 7-simplex
Cantellated 7-simplex	Runcinated 7-simplex		Stericated 7-simplex
Pentellated 7-simplex		Hexicated 7-simplex
7-orthoplex	Truncated 7-orthoplex		Rectified 7-orthoplex
Cantellated 7-orthoplex	Runcinated 7-orthoplex		Stericated 7-orthoplex
Pentellated 7-orthoplex	Hexicated 7-cube		Pentellated 7-cube
Stericated 7-cube	Cantellated 7-cube		Runcinated 7-cube
7-cube	Truncated 7-cube		Rectified 7-cube
7-demicube	Truncated 7-demicube		Cantellated 7-demicube
Runcinated 7-demicube	Stericated 7-demicube		Pentellated 7-demicube
3₂₁	2₃₁		1₃₂

In seven-dimensional geometry, a 7-polytope is a polytope contained by 6-polytope facets. Each 5-polytope ridge being shared by exactly two 6-polytope facets.

A uniform 7-polytope is one 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		Regular and semiregular forms	Uniform count
1	A₇	[3⁶]	7-simplex - {3⁶},	71
2	B₇	[4,3⁵]	7-cube - {4,3⁵}, 7-orthoplex - {3⁵,4},	127
3	D₇	[3^4,1,1]	7-demicube, {3^1,4,1}, 7-orthoplex, {3^4,1,1},	95 (32 unique)
4	E₇	[3^3,2,1]	3₂₁ - 1₃₂ - 2₃₁ -	127

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

The A₇ family

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		t₀	7-simplex (oca)	(0,0,0,0,0,0,0,1)	8	28	56	70	56	28	8
2		t₁	Rectified 7-simplex (roc)	(0,0,0,0,0,0,1,1)	16	84	224	350	336	168	28
3		t₂	Birectified 7-simplex (broc)	(0,0,0,0,0,1,1,1)	16	112	392	770	840	420	56
4		t₃	Trirectified 7-simplex (he)	(0,0,0,0,1,1,1,1)	16	112	448	980	1120	560	70
5		t_0,1	Truncated 7-simplex (toc)	(0,0,0,0,0,0,1,2)	16	84	224	350	336	196	56
6		t_0,2	Cantellated 7-simplex (saro)	(0,0,0,0,0,1,1,2)	44	308	980	1750	1876	1008	168
7		t_1,2	Bitruncated 7-simplex (bittoc)	(0,0,0,0,0,1,2,2)						588	168
8		t_0,3	Runcinated 7-simplex (spo)	(0,0,0,0,1,1,1,2)	100	756	2548	4830	4760	2100	280
9		t_1,3	Bicantellated 7-simplex (sabro)	(0,0,0,0,1,1,2,2)						2520	420
10		t_2,3	Tritruncated 7-simplex (tattoc)	(0,0,0,0,1,2,2,2)						980	280
11		t_0,4	Stericated 7-simplex (sco)	(0,0,0,1,1,1,1,2)						2240	280
12		t_1,4	Biruncinated 7-simplex (sibpo)	(0,0,0,1,1,1,2,2)						4200	560
13		t_2,4	Tricantellated 7-simplex (stiroh)	(0,0,0,1,1,2,2,2)						3360	560
14		t_0,5	Pentellated 7-simplex (seto)	(0,0,1,1,1,1,1,2)						1260	168
15		t_1,5	Bistericated 7-simplex (sabach)	(0,0,1,1,1,1,2,2)						3360	420
16		t_0,6	Hexicated 7-simplex (suph)	(0,1,1,1,1,1,1,2)						336	56
17		t_0,1,2	Cantitruncated 7-simplex (garo)	(0,0,0,0,0,1,2,3)						1176	336
18		t_0,1,3	Runcitruncated 7-simplex (patto)	(0,0,0,0,1,1,2,3)						4620	840
19		t_0,2,3	Runcicantellated 7-simplex (paro)	(0,0,0,0,1,2,2,3)						3360	840
20		t_1,2,3	Bicantitruncated 7-simplex (gabro)	(0,0,0,0,1,2,3,3)						2940	840
21		t_0,1,4	Steritruncated 7-simplex (cato)	(0,0,0,1,1,1,2,3)						7280	1120
22		t_0,2,4	Stericantellated 7-simplex (caro)	(0,0,0,1,1,2,2,3)						10080	1680
23		t_1,2,4	Biruncitruncated 7-simplex (bipto)	(0,0,0,1,1,2,3,3)						8400	1680
24		t_0,3,4	Steriruncinated 7-simplex (cepo)	(0,0,0,1,2,2,2,3)						5040	1120
25		t_1,3,4	Biruncicantellated 7-simplex (bipro)	(0,0,0,1,2,2,3,3)						7560	1680
26		t_2,3,4	Tricantitruncated 7-simplex (gatroh)	(0,0,0,1,2,3,3,3)						3920	1120
27		t_0,1,5	Pentitruncated 7-simplex (teto)	(0,0,1,1,1,1,2,3)						5460	840
28		t_0,2,5	Penticantellated 7-simplex (tero)	(0,0,1,1,1,2,2,3)						11760	1680
29		t_1,2,5	Bisteritruncated 7-simplex (bacto)	(0,0,1,1,1,2,3,3)						9240	1680
30		t_0,3,5	Pentiruncinated 7-simplex (tepo)	(0,0,1,1,2,2,2,3)						10920	1680
31		t_1,3,5	Bistericantellated 7-simplex (bacroh)	(0,0,1,1,2,2,3,3)						15120	2520
32		t_0,4,5	Pentistericated 7-simplex (teco)	(0,0,1,2,2,2,2,3)						4200	840
33		t_0,1,6	Hexitruncated 7-simplex (puto)	(0,1,1,1,1,1,2,3)						1848	336
34		t_0,2,6	Hexicantellated 7-simplex (puro)	(0,1,1,1,1,2,2,3)						5880	840
35		t_0,3,6	Hexiruncinated 7-simplex (puph)	(0,1,1,1,2,2,2,3)						8400	1120
36		t_0,1,2,3	Runcicantitruncated 7-simplex (gapo)	(0,0,0,0,1,2,3,4)						5880	1680
37		t_0,1,2,4	Stericantitruncated 7-simplex (cagro)	(0,0,0,1,1,2,3,4)						16800	3360
38		t_0,1,3,4	Steriruncitruncated 7-simplex (capto)	(0,0,0,1,2,2,3,4)						13440	3360
39		t_0,2,3,4	Steriruncicantellated 7-simplex (capro)	(0,0,0,1,2,3,3,4)						13440	3360
40		t_1,2,3,4	Biruncicantitruncated 7-simplex (gibpo)	(0,0,0,1,2,3,4,4)						11760	3360
41		t_0,1,2,5	Penticantitruncated 7-simplex (tegro)	(0,0,1,1,1,2,3,4)						18480	3360
42		t_0,1,3,5	Pentiruncitruncated 7-simplex (tapto)	(0,0,1,1,2,2,3,4)						27720	5040
43		t_0,2,3,5	Pentiruncicantellated 7-simplex (tapro)	(0,0,1,1,2,3,3,4)						25200	5040
44		t_1,2,3,5	Bistericantitruncated 7-simplex (bacogro)	(0,0,1,1,2,3,4,4)						22680	5040
45		t_0,1,4,5	Pentisteritruncated 7-simplex (tecto)	(0,0,1,2,2,2,3,4)						15120	3360
46		t_0,2,4,5	Pentistericantellated 7-simplex (tecro)	(0,0,1,2,2,3,3,4)						25200	5040
47		t_1,2,4,5	Bisteriruncitruncated 7-simplex (bicpath)	(0,0,1,2,2,3,4,4)						20160	5040
48		t_0,3,4,5	Pentisteriruncinated 7-simplex (tacpo)	(0,0,1,2,3,3,3,4)						15120	3360
49		t_0,1,2,6	Hexicantitruncated 7-simplex (pugro)	(0,1,1,1,1,2,3,4)						8400	1680
50		t_0,1,3,6	Hexiruncitruncated 7-simplex (pugato)	(0,1,1,1,2,2,3,4)						20160	3360
51		t_0,2,3,6	Hexiruncicantellated 7-simplex (pugro)	(0,1,1,1,2,3,3,4)						16800	3360
52		t_0,1,4,6	Hexisteritruncated 7-simplex (pucto)	(0,1,1,2,2,2,3,4)						20160	3360
53		t_0,2,4,6	Hexistericantellated 7-simplex (pucroh)	(0,1,1,2,2,3,3,4)						30240	5040
54		t_0,1,5,6	Hexipentitruncated 7-simplex (putath)	(0,1,2,2,2,2,3,4)						8400	1680
55		t_0,1,2,3,4	Steriruncicantitruncated 7-simplex (gecco)	(0,0,0,1,2,3,4,5)						23520	6720
56		t_0,1,2,3,5	Pentiruncicantitruncated 7-simplex (tegapo)	(0,0,1,1,2,3,4,5)						45360	10080
57		t_0,1,2,4,5	Pentistericantitruncated 7-simplex (tecagro)	(0,0,1,2,2,3,4,5)						40320	10080
58		t_0,1,3,4,5	Pentisteriruncitruncated 7-simplex (tacpeto)	(0,0,1,2,3,3,4,5)						40320	10080
59		t_0,2,3,4,5	Pentisteriruncicantellated 7-simplex (tacpro)	(0,0,1,2,3,4,4,5)						40320	10080
60		t_1,2,3,4,5	Bisteriruncicantitruncated 7-simplex (gabach)	(0,0,1,2,3,4,5,5)						35280	10080
61		t_0,1,2,3,6	Hexiruncicantitruncated 7-simplex (pugopo)	(0,1,1,1,2,3,4,5)						30240	6720
62		t_0,1,2,4,6	Hexistericantitruncated 7-simplex (pucagro)	(0,1,1,2,2,3,4,5)						50400	10080
63		t_0,1,3,4,6	Hexisteriruncitruncated 7-simplex (pucpato)	(0,1,1,2,3,3,4,5)						45360	10080
64		t_0,2,3,4,6	Hexisteriruncicantellated 7-simplex (pucproh)	(0,1,1,2,3,4,4,5)						45360	10080
65		t_0,1,2,5,6	Hexipenticantitruncated 7-simplex (putagro)	(0,1,2,2,2,3,4,5)						30240	6720
66		t_0,1,3,5,6	Hexipentiruncitruncated 7-simplex (putpath)	(0,1,2,2,3,3,4,5)						50400	10080
67		t_0,1,2,3,4,5	Pentisteriruncicantitruncated 7-simplex (geto)	(0,0,1,2,3,4,5,6)						70560	20160
68		t_0,1,2,3,4,6	Hexisteriruncicantitruncated 7-simplex (pugaco)	(0,1,1,2,3,4,5,6)						80640	20160
69		t_0,1,2,3,5,6	Hexipentiruncicantitruncated 7-simplex (putgapo)	(0,1,2,2,3,4,5,6)						80640	20160
70		t_0,1,2,4,5,6	Hexipentistericantitruncated 7-simplex (putcagroh)	(0,1,2,3,3,4,5,6)						80640	20160
71		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	t₀{3,3,3,3,3,4}	7-orthoplex (zee)	(0,0,0,0,0,0,1)√2	128	448	672	560	280	84	14
2	t₁{3,3,3,3,3,4}	Rectified 7-orthoplex (rez)	(0,0,0,0,0,1,1)√2	142	1344	3360	3920	2520	840	84
3	t₂{3,3,3,3,3,4}	Birectified 7-orthoplex (barz)	(0,0,0,0,1,1,1)√2	142	1428	6048	10640	8960	3360	280
4	t₃{4,3,3,3,3,3}	Trirectified 7-cube (sez)	(0,0,0,1,1,1,1)√2	142	1428	6328	14560	15680	6720	560
5	t₂{4,3,3,3,3,3}	Birectified 7-cube (bersa)	(0,0,1,1,1,1,1)√2	142	1428	5656	11760	13440	6720	672
6	t₁{4,3,3,3,3,3}	Rectified 7-cube (rasa)	(0,1,1,1,1,1,1)√2	142	980	2968	5040	5152	2688	448
7	t₀{4,3,3,3,3,3}	7-cube (hept)	(0,0,0,0,0,0,0)√2 + (1,1,1,1,1,1,1)	14	84	280	560	672	448	128
8	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	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	t_1,2{3,3,3,3,3,4}	Bitruncated 7-orthoplex (Botaz)	(0,0,0,0,1,2,2)√2						4200	840
11	t_0,3{3,3,3,3,3,4}	Runcinated 7-orthoplex (Spaz)	(0,0,0,1,1,1,2)√2						23520	2240
12	t_1,3{3,3,3,3,3,4}	Bicantellated 7-orthoplex (Sebraz)	(0,0,0,1,1,2,2)√2						26880	3360
13	t_2,3{3,3,3,3,3,4}	Tritruncated 7-orthoplex (Totaz)	(0,0,0,1,2,2,2)√2						10080	2240
14	t_0,4{3,3,3,3,3,4}	Stericated 7-orthoplex (Scaz)	(0,0,1,1,1,1,2)√2						33600	3360
15	t_1,4{3,3,3,3,3,4}	Biruncinated 7-orthoplex (Sibpaz)	(0,0,1,1,1,2,2)√2						60480	6720
16	t_2,4{4,3,3,3,3,3}	Tricantellated 7-cube (Strasaz)	(0,0,1,1,2,2,2)√2						47040	6720
17	t_2,3{4,3,3,3,3,3}	Tritruncated 7-cube (Tatsa)	(0,0,1,2,2,2,2)√2						13440	3360
18	t_0,5{3,3,3,3,3,4}	Pentellated 7-orthoplex (Staz)	(0,1,1,1,1,1,2)√2						20160	2688
19	t_1,5{4,3,3,3,3,3}	Bistericated 7-cube (Sabcosaz)	(0,1,1,1,1,2,2)√2						53760	6720
20	t_1,4{4,3,3,3,3,3}	Biruncinated 7-cube (Sibposa)	(0,1,1,1,2,2,2)√2						67200	8960
21	t_1,3{4,3,3,3,3,3}	Bicantellated 7-cube (Sibrosa)	(0,1,1,2,2,2,2)√2						40320	6720
22	t_1,2{4,3,3,3,3,3}	Bitruncated 7-cube (Betsa)	(0,1,2,2,2,2,2)√2						9408	2688
23	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	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	=	7-demicube Demihepteract (Hesa)	(1,1,1,1,1,1,1)	78	532	1624	2800	2240	672	64
2	=	Truncated 7-demicube Truncated demihepteract (Thesa)	(1,1,3,3,3,3,3)	142	1428	5656	11760	13440	7392	1344
3	=	Cantellated 7-demicube Small rhombated demihepteract (Sirhesa)	(1,1,1,3,3,3,3)						16800	2240
4	=	Runcinated 7-demicube Small prismated demihepteract (Sphosa)	(1,1,1,1,3,3,3)						20160	2240
5	=	Stericated 7-demicube Small cellated demihepteract (Sochesa)	(1,1,1,1,1,3,3)						13440	1344
6	=	Pentellated 7-demicube Small terated demihepteract (Suthesa)	(1,1,1,1,1,1,3)						4704	448
7	=	Cantitruncated 7-demicube Great rhombated demihepteract (Girhesa)	(1,1,3,5,5,5,5)						23520	6720
8	=	Runcitruncated 7-demicube Prismatotruncated demihepteract (Pothesa)	(1,1,3,3,5,5,5)						73920	13440
9	=	Runcicantellated 7-demicube Prismatorhomated demihepteract (Prohesa)	(1,1,1,3,5,5,5)						40320	8960
10	=	Steritruncated 7-demicube Cellitruncated demihepteract (Cothesa)	(1,1,3,3,3,5,5)						87360	13440
11	=	Stericantellated 7-demicube Cellirhombated demihepteract (Crohesa)	(1,1,1,3,3,5,5)						87360	13440
12	=	Steriruncinated 7-demicube Celliprismated demihepteract (Caphesa)	(1,1,1,1,3,5,5)						40320	6720
13	=	Pentitruncated 7-demicube Teritruncated demihepteract (Tuthesa)	(1,1,3,3,3,3,5)						43680	6720
14	=	Penticantellated 7-demicube Terirhombated demihepteract (Turhesa)	(1,1,1,3,3,3,5)						67200	8960
15	=	Pentiruncinated 7-demicube Teriprismated demihepteract (Tuphesa)	(1,1,1,1,3,3,5)						53760	6720
16	=	Pentistericated 7-demicube Tericellated demihepteract (Tuchesa)	(1,1,1,1,1,3,5)						21504	2688
17	=	Runcicantitruncated 7-demicube Great prismated demihepteract (Gephosa)	(1,1,3,5,7,7,7)						94080	26880
18	=	Stericantitruncated 7-demicube Celligreatorhombated demihepteract (Cagrohesa)	(1,1,3,5,5,7,7)						181440	40320
19	=	Steriruncitruncated 7-demicube Celliprismatotruncated demihepteract (Capthesa)	(1,1,3,3,5,7,7)						181440	40320
20	=	Steriruncicantellated 7-demicube Celliprismatorhombated demihepteract (Coprahesa)	(1,1,1,3,5,7,7)						120960	26880
21	=	Penticantitruncated 7-demicube Terigreatorhombated demihepteract (Tugrohesa)	(1,1,3,5,5,5,7)						120960	26880
22	=	Pentiruncitruncated 7-demicube Teriprismatotruncated demihepteract (Tupthesa)	(1,1,3,3,5,5,7)						221760	40320
23	=	Pentiruncicantellated 7-demicube Teriprismatorhombated demihepteract (Tuprohesa)	(1,1,1,3,5,5,7)						134400	26880
24	=	Pentisteritruncated 7-demicube Tericellitruncated demihepteract (Tucothesa)	(1,1,3,3,3,5,7)						147840	26880
25	=	Pentistericantellated 7-demicube Tericellirhombated demihepteract (Tucrohesa)	(1,1,1,3,3,5,7)						161280	26880
26	=	Pentisteriruncinated 7-demicube Tericelliprismated demihepteract (Tucophesa)	(1,1,1,1,3,5,7)						80640	13440
27	=	Steriruncicantitruncated 7-demicube Great cellated demihepteract (Gochesa)	(1,1,3,5,7,9,9)						282240	80640
28	=	Pentiruncicantitruncated 7-demicube Terigreatoprimated demihepteract (Tugphesa)	(1,1,3,5,7,7,9)						322560	80640
29	=	Pentistericantitruncated 7-demicube Tericelligreatorhombated demihepteract (Tucagrohesa)	(1,1,3,5,5,7,9)						322560	80640
30	=	Pentisteriruncitruncated 7-demicube Tericelliprismatotruncated demihepteract (Tucpathesa)	(1,1,3,3,5,7,9)						362880	80640
31	=	Pentisteriruncicantellated 7-demicube Tericellprismatorhombated demihepteract (Tucprohesa)	(1,1,1,3,5,7,9)						241920	53760
32	=	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		2₃₁ (laq)	632	4788	16128	20160	10080	2016	126
2		Rectified 2₃₁ (rolaq)	758	10332	47880	100800	90720	30240	2016
3		Rectified 1₃₂ (rolin)	758	12348	72072	191520	241920	120960	10080
4		1₃₂ (lin)	182	4284	23688	50400	40320	10080	576
5		Birectified 3₂₁ (branq)	758	12348	68040	161280	161280	60480	4032
6		Rectified 3₂₁ (ranq)	758	44352	70560	48384	11592	12096	756
7		3₂₁ (naq)	702	6048	12096	10080	4032	756	56
8		Truncated 2₃₁ (talq)	758	10332	47880	100800	90720	32256	4032
9		Cantellated 2₃₁ (sirlaq)						131040	20160
10		Bitruncated 2₃₁ (botlaq)							30240
11		small demified 2₃₁ (shilq)	2774	22428	78120	151200	131040	42336	4032
12		demirectified 2₃₁ (hirlaq)							12096
13		truncated 1₃₂ (tolin)							20160
14		small demiprismated 2₃₁ (shiplaq)							20160
15		birectified 1₃₂ (berlin)	758	22428	142632	403200	544320	302400	40320
16		tritruncated 3₂₁ (totanq)							40320
17		demibirectified 3₂₁ (hobranq)							20160
18		small cellated 2₃₁ (scalq)							7560
19		small biprismated 2₃₁ (sobpalq)							30240
20		small birhombated 3₂₁ (sabranq)							60480
21		demirectified 3₂₁ (harnaq)							12096
22		bitruncated 3₂₁ (botnaq)							12096
23		small terated 3₂₁ (stanq)							1512
24		small demicellated 3₂₁ (shocanq)							12096
25		small prismated 3₂₁ (spanq)							40320
26		small demified 3₂₁ (shanq)							4032
27		small rhombated 3₂₁ (sranq)							12096
28		Truncated 3₂₁ (tanq)	758	11592	48384	70560	44352	12852	1512
29		great rhombated 2₃₁ (girlaq)							60480
30		demitruncated 2₃₁ (hotlaq)							24192
31		small demirhombated 2₃₁ (sherlaq)							60480
32		demibitruncated 2₃₁ (hobtalq)							60480
33		demiprismated 2₃₁ (hiptalq)							80640
34		demiprismatorhombated 2₃₁ (hiprolaq)							120960
35		bitruncated 1₃₂ (batlin)							120960
36		small prismated 2₃₁ (spalq)							80640
37		small rhombated 1₃₂ (sirlin)							120960
38		tritruncated 2₃₁ (tatilq)							80640
39		cellitruncated 2₃₁ (catalaq)							60480
40		cellirhombated 2₃₁ (crilq)							362880
41		biprismatotruncated 2₃₁ (biptalq)							181440
42		small prismated 1₃₂ (seplin)							60480
43		small biprismated 3₂₁ (sabipnaq)							120960
44		small demibirhombated 3₂₁ (shobranq)							120960
45		cellidemiprismated 2₃₁ (chaplaq)							60480
46		demibiprismatotruncated 3₂₁ (hobpotanq)							120960
47		great birhombated 3₂₁ (gobranq)							120960
48		demibitruncated 3₂₁ (hobtanq)							60480
49		teritruncated 2₃₁ (totalq)							24192
50		terirhombated 2₃₁ (trilq)							120960
51		demicelliprismated 3₂₁ (hicpanq)							120960
52		small teridemified 2₃₁ (sethalq)							24192
53		small cellated 3₂₁ (scanq)							60480
54		demiprismated 3₂₁ (hipnaq)							80640
55		terirhombated 3₂₁ (tranq)							60480
56		demicellirhombated 3₂₁ (hocranq)							120960
57		prismatorhombated 3₂₁ (pranq)							120960
58		small demirhombated 3₂₁ (sharnaq)							60480
59		teritruncated 3₂₁ (tetanq)							15120
60		demicellitruncated 3₂₁ (hictanq)							60480
61		prismatotruncated 3₂₁ (potanq)							120960
62		demitruncated 3₂₁ (hotnaq)							24192
63		great rhombated 3₂₁ (granq)							24192
64		great demified 2₃₁ (gahlaq)							120960
65		great demiprismated 2₃₁ (gahplaq)							241920
66		prismatotruncated 2₃₁ (potlaq)							241920
67		prismatorhombated 2₃₁ (prolaq)							241920
68		great rhombated 1₃₂ (girlin)							241920
69		celligreatorhombated 2₃₁ (cagrilq)							362880
70		cellidemitruncated 2₃₁ (chotalq)							241920
71		prismatotruncated 1₃₂ (patlin)							362880
72		biprismatorhombated 3₂₁ (bipirnaq)							362880
73		tritruncated 1₃₂ (tatlin)							241920
74		cellidemiprismatorhombated 2₃₁ (chopralq)							362880
75		great demibiprismated 3₂₁ (ghobipnaq)							362880
76		celliprismated 2₃₁ (caplaq)							241920
77		biprismatotruncated 3₂₁ (boptanq)							362880
78		great trirhombated 2₃₁ (gatralaq)							241920
79		terigreatorhombated 2₃₁ (togrilq)							241920
80		teridemitruncated 2₃₁ (thotalq)							120960
81		teridemirhombated 2₃₁ (thorlaq)							241920
82		celliprismated 3₂₁ (capnaq)							241920
83		teridemiprismatotruncated 2₃₁ (thoptalq)							241920
84		teriprismatorhombated 3₂₁ (tapronaq)							362880
85		demicelliprismatorhombated 3₂₁ (hacpranq)							362880
86		teriprismated 2₃₁ (toplaq)							241920
87		cellirhombated 3₂₁ (cranq)							362880
88		demiprismatorhombated 3₂₁ (hapranq)							241920
89		tericellitruncated 2₃₁ (tectalq)							120960
90		teriprismatotruncated 3₂₁ (toptanq)							362880
91		demicelliprismatotruncated 3₂₁ (hecpotanq)							362880
92		teridemitruncated 3₂₁ (thotanq)							120960
93		cellitruncated 3₂₁ (catnaq)							241920
94		demiprismatotruncated 3₂₁ (hiptanq)							241920
95		terigreatorhombated 3₂₁ (tagranq)							120960
96		demicelligreatorhombated 3₂₁ (hicgarnq)							241920
97		great prismated 3₂₁ (gopanq)							241920
98		great demirhombated 3₂₁ (gahranq)							120960
99		great prismated 2₃₁ (gopalq)							483840
100		great cellidemified 2₃₁ (gechalq)							725760
101		great birhombated 1₃₂ (gebrolin)							725760
102		prismatorhombated 1₃₂ (prolin)							725760
103		celliprismatorhombated 2₃₁ (caprolaq)							725760
104		great biprismated 2₃₁ (gobpalq)							725760
105		tericelliprismated 3₂₁ (ticpanq)							483840
106		teridemigreatoprismated 2₃₁ (thegpalq)							725760
107		teriprismatotruncated 2₃₁ (teptalq)							725760
108		teriprismatorhombated 2₃₁ (topralq)							725760
109		cellipriemsatorhombated 3₂₁ (copranq)							725760
110		tericelligreatorhombated 2₃₁ (tecgrolaq)							725760
111		tericellitruncated 3₂₁ (tectanq)							483840
112		teridemiprismatotruncated 3₂₁ (thoptanq)							725760
113		celliprismatotruncated 3₂₁ (coptanq)							725760
114		teridemicelligreatorhombated 3₂₁ (thocgranq)							483840
115		terigreatoprismated 3₂₁ (tagpanq)							725760
116		great demicellated 3₂₁ (gahcnaq)							725760
117		tericelliprismated laq (tecpalq)							483840
118		celligreatorhombated 3₂₁ (cogranq)							725760
119		great demified 3₂₁ (gahnq)							483840
120		great cellated 2₃₁ (gocalq)							1451520
121		terigreatoprismated 2₃₁ (tegpalq)							1451520
122		tericelliprismatotruncated 3₂₁ (tecpotniq)							1451520
123		tericellidemigreatoprismated 2₃₁ (techogaplaq)							1451520
124		tericelligreatorhombated 3₂₁ (tacgarnq)							1451520
125		tericelliprismatorhombated 2₃₁ (tecprolaq)							1451520
126		great cellated 3₂₁ (gocanq)							1451520
127		great terated 3₂₁ (gotanq)							2903040

Regular and uniform honeycombs

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

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

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

Regular and uniform tessellations include:

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

Prismatic groups
#	Coxeter group
1	${{\tilde {A}}}_{5}$ x ${{\tilde {I}}}_{1}$	[3^[6],2,∞]
2	${{\tilde {B}}}_{5}$ x ${{\tilde {I}}}_{1}$	[4,3,3^1,1,2,∞]
3	${{\tilde {C}}}_{5}$ x ${{\tilde {I}}}_{1}$	[4,3³,4,2,∞]
4	${{\tilde {D}}}_{5}$ x ${{\tilde {I}}}_{1}$	[3^1,1,3,3^1,1,2,∞]
5	${{\tilde {A}}}_{4}$ x ${{\tilde {I}}}_{1}$ x ${{\tilde {I}}}_{1}$	[3^[5],2,∞,2,∞,2,∞]
6	${{\tilde {B}}}_{4}$ x ${{\tilde {I}}}_{1}$ x ${{\tilde {I}}}_{1}$	[4,3,3^1,1,2,∞,2,∞]
7	${{\tilde {C}}}_{4}$ x ${{\tilde {I}}}_{1}$ x ${{\tilde {I}}}_{1}$	[4,3,3,4,2,∞,2,∞]
8	${{\tilde {D}}}_{4}$ x ${{\tilde {I}}}_{1}$ x ${{\tilde {I}}}_{1}$	[3^1,1,1,1,2,∞,2,∞]
9	${{\tilde {F}}}_{4}$ x ${{\tilde {I}}}_{1}$ x ${{\tilde {I}}}_{1}$	[3,4,3,3,2,∞,2,∞]
10	${{\tilde {C}}}_{3}$ x ${{\tilde {I}}}_{1}$ x ${{\tilde {I}}}_{1}$ x ${{\tilde {I}}}_{1}$	[4,3,4,2,∞,2,∞,2,∞]
11	${{\tilde {B}}}_{3}$ x ${{\tilde {I}}}_{1}$ x ${{\tilde {I}}}_{1}$ x ${{\tilde {I}}}_{1}$	[4,3^1,1,2,∞,2,∞,2,∞]
12	${{\tilde {A}}}_{3}$ x ${{\tilde {I}}}_{1}$ x ${{\tilde {I}}}_{1}$ x ${{\tilde {I}}}_{1}$	[3^[4],2,∞,2,∞,2,∞]
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,∞]
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,∞]
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,∞]
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,∞]

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

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	Description
Parent	t₀{p,q,r,s,t,u}	Any regular 7-polytope
Rectified	t₁{p,q,r,s,t,u}	The edges are fully truncated into single points. The 7-polytope now has the combined faces of the parent and dual.
Birectified	t₂{p,q,r,s,t,u}	Birectification reduces cells to their duals.
Truncated	t_0,1{p,q,r,s,t,u}	Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated 7-polytope. The 7-polytope has its original faces doubled in sides, and contains the faces of the dual.
Bitruncated	t_1,2{p,q,r,s,t,u}	Bitrunction transforms cells to their dual truncation.
Tritruncated	t_2,3{p,q,r,s,t,u}	Tritruncation transforms 4-faces to their dual truncation.
Cantellated	t_0,2{p,q,r,s,t,u}	In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms.
Bicantellated	t_1,3{p,q,r,s,t,u}	In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is half way between both the parent and dual forms.
Runcinated	t_0,3{p,q,r,s,t,u}	Runcination reduces cells and creates new cells at the vertices and edges.
Biruncinated	t_1,4{p,q,r,s,t,u}	Runcination reduces cells and creates new cells at the vertices and edges.
Stericated	t_0,4{p,q,r,s,t,u}	Sterication reduces 4-faces and creates new 4-faces at the vertices, edges, and faces in the gaps.
Pentellated	t_0,5{p,q,r,s,t,u}	Pentellation reduces 5-faces and creates new 5-faces at the vertices, edges, faces, and cells in the gaps.
Hexicated	t_0,6{p,q,r,s,t,u}	Hexication reduces 6-faces and creates new 6-faces at the vertices, edges, faces, cells, and 4-faces in the gaps. (expansion operation for 7-polytopes)
Omnitruncated	t_{0,1,2,3,4,5,6}{p,q,r,s,t,u}	All six operators, truncation, cantellation, runcination, sterication, pentellation, and hexication are applied.

References

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
Richard Klitzing, 7D, uniform polytopes (polyexa)

External links

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	BC_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	5-cell	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes

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