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Uniform 7-polytope

Seven-dimensional geometric object

[[File:E7 graph.svg	100px]]
321	[[File:Gosset 2 31 polytope.svg	100px]]
231	[[File:Gosset 1 32 petrie.svg	100px]]
132

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

A uniform 7-polytope is one whose symmetry group is transitive on vertices and whose facets are uniform 6-polytopes.

Regular 7-polytopes

Regular 7-polytopes are represented by the Schläfli symbol {p,q,r,s,t,u} with u {p,q,r,s,t} 6-polytopes facets around each 4-face.

There are exactly three such convex regular 7-polytopes:

{3,3,3,3,3,3} - 7-simplex
{4,3,3,3,3,3} - 7-cube
{3,3,3,3,3,4} - 7-orthoplex

There are no nonconvex regular 7-polytopes.

Characteristics

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

The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology. This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.

Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.

Uniform 7-polytopes by fundamental Coxeter groups

Uniform 7-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:

#	Coxeter group	Regular and semiregular forms
1	A7	[36]
2	B7	[4,35]
3	D7	[33,1,1]
4	E7	[33,2,1]

Prismatic finite Coxeter groups	#	Coxeter group
1	A6A1	[35]×[ ]
2	BC6A1	[4,34]×[ ]
3	D6A1	[33,1,1]×[ ]
4	E6A1	[32,2,1]×[ ]
1	A5I2(p)	[3,3,3]×[p]
2	BC5I2(p)	[4,3,3]×[p]
3	D5I2(p)	[32,1,1]×[p]
1	A5A12	[3,3,3]×[ ]2
2	BC5A12	[4,3,3]×[ ]2
3	D5A12	[32,1,1]×[ ]2
1	A4A3	[3,3,3]×[3,3]
2	A4B3	[3,3,3]×[4,3]
3	A4H3	[3,3,3]×[5,3]
4	BC4A3	[4,3,3]×[3,3]
5	BC4B3	[4,3,3]×[4,3]
6	BC4H3	[4,3,3]×[5,3]
7	H4A3	[5,3,3]×[3,3]
8	H4B3	[5,3,3]×[4,3]
9	H4H3	[5,3,3]×[5,3]
10	F4A3	[3,4,3]×[3,3]
11	F4B3	[3,4,3]×[4,3]
12	F4H3	[3,4,3]×[5,3]
13	D4A3	[31,1,1]×[3,3]
14	D4B3	[31,1,1]×[4,3]
15	D4H3	[31,1,1]×[5,3]
1	A4I2(p)A1	[3,3,3]×[p]×[ ]
2	BC4I2(p)A1	[4,3,3]×[p]×[ ]
3	F4I2(p)A1	[3,4,3]×[p]×[ ]
4	H4I2(p)A1	[5,3,3]×[p]×[ ]
5	D4I2(p)A1	[31,1,1]×[p]×[ ]
1	A4A13	[3,3,3]×[ ]3
2	BC4A13	[4,3,3]×[ ]3
3	F4A13	[3,4,3]×[ ]3
4	H4A13	[5,3,3]×[ ]3
5	D4A13	[31,1,1]×[ ]3
1	A3A3A1	[3,3]×[3,3]×[ ]
2	A3B3A1	[3,3]×[4,3]×[ ]
3	A3H3A1	[3,3]×[5,3]×[ ]
4	BC3B3A1	[4,3]×[4,3]×[ ]
5	BC3H3A1	[4,3]×[5,3]×[ ]
6	H3A3A1	[5,3]×[5,3]×[ ]
1	A3I2(p)I2(q)	[3,3]×[p]×[q]
2	BC3I2(p)I2(q)	[4,3]×[p]×[q]
3	H3I2(p)I2(q)	[5,3]×[p]×[q]
1	A3I2(p)A12	[3,3]×[p]×[ ]2
2	BC3I2(p)A12	[4,3]×[p]×[ ]2
3	H3I2(p)A12	[5,3]×[p]×[ ]2
1	A3A14	[3,3]×[ ]4
2	BC3A14	[4,3]×[ ]4
3	H3A14	[5,3]×[ ]4
1	I2(p)I2(q)I2(r)A1	[p]×[q]×[r]×[ ]
1	I2(p)I2(q)A13	[p]×[q]×[ ]3
1	I2(p)A15	[p]×[ ]5
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.

A7 uniform polytopes	#	Coxeter-Dynkin diagram	Truncation
indices	Johnson name
Bowers name (and acronym)	Basepoint	Element counts	6		5		4	3	2	1	0
1		t0	7-simplex (oca)	(0,0,0,0,0,0,0,1)	8	28	56
2		t1	Rectified 7-simplex (roc)	(0,0,0,0,0,0,1,1)	16	84	224
3		t2	Birectified 7-simplex (broc)	(0,0,0,0,0,1,1,1)	16	112	392
4		t3	Trirectified 7-simplex (he)	(0,0,0,0,1,1,1,1)	16	112	448
5		t0,1	Truncated 7-simplex (toc)	(0,0,0,0,0,0,1,2)	16	84	224
6		t0,2	Cantellated 7-simplex (saro)	(0,0,0,0,0,1,1,2)	44	308	980
7		t1,2	Bitruncated 7-simplex (bittoc)	(0,0,0,0,0,1,2,2)
8		t0,3	Runcinated 7-simplex (spo)	(0,0,0,0,1,1,1,2)	100	756	2548
9		t1,3	Bicantellated 7-simplex (sabro)	(0,0,0,0,1,1,2,2)
10		t2,3	Tritruncated 7-simplex (tattoc)	(0,0,0,0,1,2,2,2)
11		t0,4	Stericated 7-simplex (sco)	(0,0,0,1,1,1,1,2)
12		t1,4	Biruncinated 7-simplex (sibpo)	(0,0,0,1,1,1,2,2)
13		t2,4	Tricantellated 7-simplex (stiroh)	(0,0,0,1,1,2,2,2)
14		t0,5	Pentellated 7-simplex (seto)	(0,0,1,1,1,1,1,2)
15		t1,5	Bistericated 7-simplex (sabach)	(0,0,1,1,1,1,2,2)
16		t0,6	Hexicated 7-simplex (suph)	(0,1,1,1,1,1,1,2)
17		t0,1,2	Cantitruncated 7-simplex (garo)	(0,0,0,0,0,1,2,3)
18		t0,1,3	Runcitruncated 7-simplex (patto)	(0,0,0,0,1,1,2,3)
19		t0,2,3	Runcicantellated 7-simplex (paro)	(0,0,0,0,1,2,2,3)
20		t1,2,3	Bicantitruncated 7-simplex (gabro)	(0,0,0,0,1,2,3,3)
21		t0,1,4	Steritruncated 7-simplex (cato)	(0,0,0,1,1,1,2,3)
22		t0,2,4	Stericantellated 7-simplex (caro)	(0,0,0,1,1,2,2,3)
23		t1,2,4	Biruncitruncated 7-simplex (bipto)	(0,0,0,1,1,2,3,3)
24		t0,3,4	Steriruncinated 7-simplex (cepo)	(0,0,0,1,2,2,2,3)
25		t1,3,4	Biruncicantellated 7-simplex (bipro)	(0,0,0,1,2,2,3,3)
26		t2,3,4	Tricantitruncated 7-simplex (gatroh)	(0,0,0,1,2,3,3,3)
27		t0,1,5	Pentitruncated 7-simplex (teto)	(0,0,1,1,1,1,2,3)
28		t0,2,5	Penticantellated 7-simplex (tero)	(0,0,1,1,1,2,2,3)
29		t1,2,5	Bisteritruncated 7-simplex (bacto)	(0,0,1,1,1,2,3,3)
30		t0,3,5	Pentiruncinated 7-simplex (tepo)	(0,0,1,1,2,2,2,3)
31		t1,3,5	Bistericantellated 7-simplex (bacroh)	(0,0,1,1,2,2,3,3)
32		t0,4,5	Pentistericated 7-simplex (teco)	(0,0,1,2,2,2,2,3)
33		t0,1,6	Hexitruncated 7-simplex (puto)	(0,1,1,1,1,1,2,3)
34		t0,2,6	Hexicantellated 7-simplex (puro)	(0,1,1,1,1,2,2,3)
35		t0,3,6	Hexiruncinated 7-simplex (puph)	(0,1,1,1,2,2,2,3)
36		t0,1,2,3	Runcicantitruncated 7-simplex (gapo)	(0,0,0,0,1,2,3,4)
37		t0,1,2,4	Stericantitruncated 7-simplex (cagro)	(0,0,0,1,1,2,3,4)
38		t0,1,3,4	Steriruncitruncated 7-simplex (capto)	(0,0,0,1,2,2,3,4)
39		t0,2,3,4	Steriruncicantellated 7-simplex (capro)	(0,0,0,1,2,3,3,4)
40		t1,2,3,4	Biruncicantitruncated 7-simplex (gibpo)	(0,0,0,1,2,3,4,4)
41		t0,1,2,5	Penticantitruncated 7-simplex (tegro)	(0,0,1,1,1,2,3,4)
42		t0,1,3,5	Pentiruncitruncated 7-simplex (tapto)	(0,0,1,1,2,2,3,4)
43		t0,2,3,5	Pentiruncicantellated 7-simplex (tapro)	(0,0,1,1,2,3,3,4)
44		t1,2,3,5	Bistericantitruncated 7-simplex (bacogro)	(0,0,1,1,2,3,4,4)
45		t0,1,4,5	Pentisteritruncated 7-simplex (tecto)	(0,0,1,2,2,2,3,4)
46		t0,2,4,5	Pentistericantellated 7-simplex (tecro)	(0,0,1,2,2,3,3,4)
47		t1,2,4,5	Bisteriruncitruncated 7-simplex (bicpath)	(0,0,1,2,2,3,4,4)
48		t0,3,4,5	Pentisteriruncinated 7-simplex (tacpo)	(0,0,1,2,3,3,3,4)
49		t0,1,2,6	Hexicantitruncated 7-simplex (pugro)	(0,1,1,1,1,2,3,4)
50		t0,1,3,6	Hexiruncitruncated 7-simplex (pugato)	(0,1,1,1,2,2,3,4)
51		t0,2,3,6	Hexiruncicantellated 7-simplex (pugro)	(0,1,1,1,2,3,3,4)
52		t0,1,4,6	Hexisteritruncated 7-simplex (pucto)	(0,1,1,2,2,2,3,4)
53		t0,2,4,6	Hexistericantellated 7-simplex (pucroh)	(0,1,1,2,2,3,3,4)
54		t0,1,5,6	Hexipentitruncated 7-simplex (putath)	(0,1,2,2,2,2,3,4)
55		t0,1,2,3,4	Steriruncicantitruncated 7-simplex (gecco)	(0,0,0,1,2,3,4,5)
56		t0,1,2,3,5	Pentiruncicantitruncated 7-simplex (tegapo)	(0,0,1,1,2,3,4,5)
57		t0,1,2,4,5	Pentistericantitruncated 7-simplex (tecagro)	(0,0,1,2,2,3,4,5)
58		t0,1,3,4,5	Pentisteriruncitruncated 7-simplex (tacpeto)	(0,0,1,2,3,3,4,5)
59		t0,2,3,4,5	Pentisteriruncicantellated 7-simplex (tacpro)	(0,0,1,2,3,4,4,5)
60		t1,2,3,4,5	Bisteriruncicantitruncated 7-simplex (gabach)	(0,0,1,2,3,4,5,5)
61		t0,1,2,3,6	Hexiruncicantitruncated 7-simplex (pugopo)	(0,1,1,1,2,3,4,5)
62		t0,1,2,4,6	Hexistericantitruncated 7-simplex (pucagro)	(0,1,1,2,2,3,4,5)
63		t0,1,3,4,6	Hexisteriruncitruncated 7-simplex (pucpato)	(0,1,1,2,3,3,4,5)
64		t0,2,3,4,6	Hexisteriruncicantellated 7-simplex (pucproh)	(0,1,1,2,3,4,4,5)
65		t0,1,2,5,6	Hexipenticantitruncated 7-simplex (putagro)	(0,1,2,2,2,3,4,5)
66		t0,1,3,5,6	Hexipentiruncitruncated 7-simplex (putpath)	(0,1,2,2,3,3,4,5)
67		t0,1,2,3,4,5	Pentisteriruncicantitruncated 7-simplex (geto)	(0,0,1,2,3,4,5,6)
68		t0,1,2,3,4,6	Hexisteriruncicantitruncated 7-simplex (pugaco)	(0,1,1,2,3,4,5,6)
69		t0,1,2,3,5,6	Hexipentiruncicantitruncated 7-simplex (putgapo)	(0,1,2,2,3,4,5,6)
70		t0,1,2,4,5,6	Hexipentistericantitruncated 7-simplex (putcagroh)	(0,1,2,3,3,4,5,6)
71		t0,1,2,3,4,5,6	Omnitruncated 7-simplex (guph)	(0,1,2,3,4,5,6,7)

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.

B7 uniform polytopes	#	Name (BSA)	Base point	Element counts	6		5		4
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
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
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
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
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
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
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
t0,1{3,3,3,3,3,4}	Truncated 7-orthoplex (Taz)	(0,0,0,0,0,1,2)√2	142	1344	3360	4760	2520	924	168
t0,2{3,3,3,3,3,4}	Cantellated 7-orthoplex (Sarz)	(0,0,0,0,1,1,2)√2	226	4200	15456	24080	19320	7560	840
t1,2{3,3,3,3,3,4}	Bitruncated 7-orthoplex (Botaz)	(0,0,0,0,1,2,2)√2						4200	840
t0,3{3,3,3,3,3,4}	Runcinated 7-orthoplex (Spaz)	(0,0,0,1,1,1,2)√2						23520	2240
t1,3{3,3,3,3,3,4}	Bicantellated 7-orthoplex (Sebraz)	(0,0,0,1,1,2,2)√2						26880	3360
t2,3{3,3,3,3,3,4}	Tritruncated 7-orthoplex (Totaz)	(0,0,0,1,2,2,2)√2						10080	2240
t0,4{3,3,3,3,3,4}	Stericated 7-orthoplex (Scaz)	(0,0,1,1,1,1,2)√2						33600	3360
t1,4{3,3,3,3,3,4}	Biruncinated 7-orthoplex (Sibpaz)	(0,0,1,1,1,2,2)√2						60480	6720
t2,4{4,3,3,3,3,3}	Tricantellated 7-cube (Strasaz)	(0,0,1,1,2,2,2)√2						47040	6720
t2,3{4,3,3,3,3,3}	Tritruncated 7-cube (Tatsa)	(0,0,1,2,2,2,2)√2						13440	3360
t0,5{3,3,3,3,3,4}	Pentellated 7-orthoplex (Staz)	(0,1,1,1,1,1,2)√2						20160	2688
t1,5{4,3,3,3,3,3}	Bistericated 7-cube (Sabcosaz)	(0,1,1,1,1,2,2)√2						53760	6720
t1,4{4,3,3,3,3,3}	Biruncinated 7-cube (Sibposa)	(0,1,1,1,2,2,2)√2						67200	8960
t1,3{4,3,3,3,3,3}	Bicantellated 7-cube (Sibrosa)	(0,1,1,2,2,2,2)√2						40320	6720
t1,2{4,3,3,3,3,3}	Bitruncated 7-cube (Betsa)	(0,1,2,2,2,2,2)√2						9408	2688
t0,6{4,3,3,3,3,3}	Hexicated 7-cube (Supposaz)	(0,0,0,0,0,0,1)√2 + (1,1,1,1,1,1,1)						5376	896
t0,5{4,3,3,3,3,3}	Pentellated 7-cube (Stesa)	(0,0,0,0,0,1,1)√2 + (1,1,1,1,1,1,1)						20160	2688
t0,4{4,3,3,3,3,3}	Stericated 7-cube (Scosa)	(0,0,0,0,1,1,1)√2 + (1,1,1,1,1,1,1)						35840	4480
t0,3{4,3,3,3,3,3}	Runcinated 7-cube (Spesa)	(0,0,0,1,1,1,1)√2 + (1,1,1,1,1,1,1)						33600	4480
t0,2{4,3,3,3,3,3}	Cantellated 7-cube (Sersa)	(0,0,1,1,1,1,1)√2 + (1,1,1,1,1,1,1)						16128	2688
t0,1{4,3,3,3,3,3}	Truncated 7-cube (Tasa)	(0,1,1,1,1,1,1)√2 + (1,1,1,1,1,1,1)	142	980	2968	5040	5152	3136	896
t0,1,2{3,3,3,3,3,4}	Cantitruncated 7-orthoplex (Garz)	(0,1,2,3,3,3,3)√2						8400	1680
t0,1,3{3,3,3,3,3,4}	Runcitruncated 7-orthoplex (Potaz)	(0,1,2,2,3,3,3)√2						50400	6720
t0,2,3{3,3,3,3,3,4}	Runcicantellated 7-orthoplex (Parz)	(0,1,1,2,3,3,3)√2						33600	6720
t1,2,3{3,3,3,3,3,4}	Bicantitruncated 7-orthoplex (Gebraz)	(0,0,1,2,3,3,3)√2						30240	6720
t0,1,4{3,3,3,3,3,4}	Steritruncated 7-orthoplex (Catz)	(0,0,1,1,1,2,3)√2						107520	13440
t0,2,4{3,3,3,3,3,4}	Stericantellated 7-orthoplex (Craze)	(0,0,1,1,2,2,3)√2						141120	20160
t1,2,4{3,3,3,3,3,4}	Biruncitruncated 7-orthoplex (Baptize)	(0,0,1,1,2,3,3)√2						120960	20160
t0,3,4{3,3,3,3,3,4}	Steriruncinated 7-orthoplex (Copaz)	(0,1,1,1,2,3,3)√2						67200	13440
t1,3,4{3,3,3,3,3,4}	Biruncicantellated 7-orthoplex (Boparz)	(0,0,1,2,2,3,3)√2						100800	20160
t2,3,4{4,3,3,3,3,3}	Tricantitruncated 7-cube (Gotrasaz)	(0,0,0,1,2,3,3)√2						53760	13440
t0,1,5{3,3,3,3,3,4}	Pentitruncated 7-orthoplex (Tetaz)	(0,1,1,1,1,2,3)√2						87360	13440
t0,2,5{3,3,3,3,3,4}	Penticantellated 7-orthoplex (Teroz)	(0,1,1,1,2,2,3)√2						188160	26880
t1,2,5{3,3,3,3,3,4}	Bisteritruncated 7-orthoplex (Boctaz)	(0,1,1,1,2,3,3)√2						147840	26880
t0,3,5{3,3,3,3,3,4}	Pentiruncinated 7-orthoplex (Topaz)	(0,1,1,2,2,2,3)√2						174720	26880
t1,3,5{4,3,3,3,3,3}	Bistericantellated 7-cube (Bacresaz)	(0,1,1,2,2,3,3)√2						241920	40320
t1,3,4{4,3,3,3,3,3}	Biruncicantellated 7-cube (Bopresa)	(0,1,1,2,3,3,3)√2						120960	26880
t0,4,5{3,3,3,3,3,4}	Pentistericated 7-orthoplex (Tocaz)	(0,1,2,2,2,2,3)√2						67200	13440
t1,2,5{4,3,3,3,3,3}	Bisteritruncated 7-cube (Bactasa)	(0,1,2,2,2,3,3)√2						147840	26880
t1,2,4{4,3,3,3,3,3}	Biruncitruncated 7-cube (Biptesa)	(0,1,2,2,3,3,3)√2						134400	26880
t1,2,3{4,3,3,3,3,3}	Bicantitruncated 7-cube (Gibrosa)	(0,1,2,3,3,3,3)√2						47040	13440
t0,1,6{3,3,3,3,3,4}	Hexitruncated 7-orthoplex (Putaz)	(0,0,0,0,0,1,2)√2 + (1,1,1,1,1,1,1)						29568	5376
t0,2,6{3,3,3,3,3,4}	Hexicantellated 7-orthoplex (Puraz)	(0,0,0,0,1,1,2)√2 + (1,1,1,1,1,1,1)						94080	13440
t0,4,5{4,3,3,3,3,3}	Pentistericated 7-cube (Tacosa)	(0,0,0,0,1,2,2)√2 + (1,1,1,1,1,1,1)						67200	13440
t0,3,6{4,3,3,3,3,3}	Hexiruncinated 7-cube (Pupsez)	(0,0,0,1,1,1,2)√2 + (1,1,1,1,1,1,1)						134400	17920
t0,3,5{4,3,3,3,3,3}	Pentiruncinated 7-cube (Tapsa)	(0,0,0,1,1,2,2)√2 + (1,1,1,1,1,1,1)						174720	26880
t0,3,4{4,3,3,3,3,3}	Steriruncinated 7-cube (Capsa)	(0,0,0,1,2,2,2)√2 + (1,1,1,1,1,1,1)						80640	17920
t0,2,6{4,3,3,3,3,3}	Hexicantellated 7-cube (Purosa)	(0,0,1,1,1,1,2)√2 + (1,1,1,1,1,1,1)						94080	13440
t0,2,5{4,3,3,3,3,3}	Penticantellated 7-cube (Tersa)	(0,0,1,1,1,2,2)√2 + (1,1,1,1,1,1,1)						188160	26880
t0,2,4{4,3,3,3,3,3}	Stericantellated 7-cube (Carsa)	(0,0,1,1,2,2,2)√2 + (1,1,1,1,1,1,1)						161280	26880
t0,2,3{4,3,3,3,3,3}	Runcicantellated 7-cube (Parsa)	(0,0,1,2,2,2,2)√2 + (1,1,1,1,1,1,1)						53760	13440
t0,1,6{4,3,3,3,3,3}	Hexitruncated 7-cube (Putsa)	(0,1,1,1,1,1,2)√2 + (1,1,1,1,1,1,1)						29568	5376
t0,1,5{4,3,3,3,3,3}	Pentitruncated 7-cube (Tetsa)	(0,1,1,1,1,2,2)√2 + (1,1,1,1,1,1,1)						87360	13440
t0,1,4{4,3,3,3,3,3}	Steritruncated 7-cube (Catsa)	(0,1,1,1,2,2,2)√2 + (1,1,1,1,1,1,1)						116480	17920
t0,1,3{4,3,3,3,3,3}	Runcitruncated 7-cube (Petsa)	(0,1,1,2,2,2,2)√2 + (1,1,1,1,1,1,1)						73920	13440
t0,1,2{4,3,3,3,3,3}	Cantitruncated 7-cube (Gersa)	(0,1,2,2,2,2,2)√2 + (1,1,1,1,1,1,1)						18816	5376
t0,1,2,3{3,3,3,3,3,4}	Runcicantitruncated 7-orthoplex (Gopaz)	(0,1,2,3,4,4,4)√2						60480	13440
t0,1,2,4{3,3,3,3,3,4}	Stericantitruncated 7-orthoplex (Cogarz)	(0,0,1,1,2,3,4)√2						241920	40320
t0,1,3,4{3,3,3,3,3,4}	Steriruncitruncated 7-orthoplex (Captaz)	(0,0,1,2,2,3,4)√2						181440	40320
t0,2,3,4{3,3,3,3,3,4}	Steriruncicantellated 7-orthoplex (Caparz)	(0,0,1,2,3,3,4)√2						181440	40320
t1,2,3,4{3,3,3,3,3,4}	Biruncicantitruncated 7-orthoplex (Gibpaz)	(0,0,1,2,3,4,4)√2						161280	40320
t0,1,2,5{3,3,3,3,3,4}	Penticantitruncated 7-orthoplex (Tograz)	(0,1,1,1,2,3,4)√2						295680	53760
t0,1,3,5{3,3,3,3,3,4}	Pentiruncitruncated 7-orthoplex (Toptaz)	(0,1,1,2,2,3,4)√2						443520	80640
t0,2,3,5{3,3,3,3,3,4}	Pentiruncicantellated 7-orthoplex (Toparz)	(0,1,1,2,3,3,4)√2						403200	80640
t1,2,3,5{3,3,3,3,3,4}	Bistericantitruncated 7-orthoplex (Becogarz)	(0,1,1,2,3,4,4)√2						362880	80640
t0,1,4,5{3,3,3,3,3,4}	Pentisteritruncated 7-orthoplex (Tacotaz)	(0,1,2,2,2,3,4)√2						241920	53760
t0,2,4,5{3,3,3,3,3,4}	Pentistericantellated 7-orthoplex (Tocarz)	(0,1,2,2,3,3,4)√2						403200	80640
t1,2,4,5{4,3,3,3,3,3}	Bisteriruncitruncated 7-cube (Bocaptosaz)	(0,1,2,2,3,4,4)√2						322560	80640
t0,3,4,5{3,3,3,3,3,4}	Pentisteriruncinated 7-orthoplex (Tecpaz)	(0,1,2,3,3,3,4)√2						241920	53760
t1,2,3,5{4,3,3,3,3,3}	Bistericantitruncated 7-cube (Becgresa)	(0,1,2,3,3,4,4)√2						362880	80640
t1,2,3,4{4,3,3,3,3,3}	Biruncicantitruncated 7-cube (Gibposa)	(0,1,2,3,4,4,4)√2						188160	53760
t0,1,2,6{3,3,3,3,3,4}	Hexicantitruncated 7-orthoplex (Pugarez)	(0,0,0,0,1,2,3)√2 + (1,1,1,1,1,1,1)						134400	26880
t0,1,3,6{3,3,3,3,3,4}	Hexiruncitruncated 7-orthoplex (Papataz)	(0,0,0,1,1,2,3)√2 + (1,1,1,1,1,1,1)						322560	53760
t0,2,3,6{3,3,3,3,3,4}	Hexiruncicantellated 7-orthoplex (Puparez)	(0,0,0,1,2,2,3)√2 + (1,1,1,1,1,1,1)						268800	53760
t0,3,4,5{4,3,3,3,3,3}	Pentisteriruncinated 7-cube (Tecpasa)	(0,0,0,1,2,3,3)√2 + (1,1,1,1,1,1,1)						241920	53760
t0,1,4,6{3,3,3,3,3,4}	Hexisteritruncated 7-orthoplex (Pucotaz)	(0,0,1,1,1,2,3)√2 + (1,1,1,1,1,1,1)						322560	53760
t0,2,4,6{4,3,3,3,3,3}	Hexistericantellated 7-cube (Pucrosaz)	(0,0,1,1,2,2,3)√2 + (1,1,1,1,1,1,1)						483840	80640
t0,2,4,5{4,3,3,3,3,3}	Pentistericantellated 7-cube (Tecresa)	(0,0,1,1,2,3,3)√2 + (1,1,1,1,1,1,1)						403200	80640
t0,2,3,6{4,3,3,3,3,3}	Hexiruncicantellated 7-cube (Pupresa)	(0,0,1,2,2,2,3)√2 + (1,1,1,1,1,1,1)						268800	53760
t0,2,3,5{4,3,3,3,3,3}	Pentiruncicantellated 7-cube (Topresa)	(0,0,1,2,2,3,3)√2 + (1,1,1,1,1,1,1)						403200	80640
t0,2,3,4{4,3,3,3,3,3}	Steriruncicantellated 7-cube (Copresa)	(0,0,1,2,3,3,3)√2 + (1,1,1,1,1,1,1)						215040	53760
t0,1,5,6{4,3,3,3,3,3}	Hexipentitruncated 7-cube (Putatosez)	(0,1,1,1,1,2,3)√2 + (1,1,1,1,1,1,1)						134400	26880
t0,1,4,6{4,3,3,3,3,3}	Hexisteritruncated 7-cube (Pacutsa)	(0,1,1,1,2,2,3)√2 + (1,1,1,1,1,1,1)						322560	53760
t0,1,4,5{4,3,3,3,3,3}	Pentisteritruncated 7-cube (Tecatsa)	(0,1,1,1,2,3,3)√2 + (1,1,1,1,1,1,1)						241920	53760
t0,1,3,6{4,3,3,3,3,3}	Hexiruncitruncated 7-cube (Pupetsa)	(0,1,1,2,2,2,3)√2 + (1,1,1,1,1,1,1)						322560	53760
t0,1,3,5{4,3,3,3,3,3}	Pentiruncitruncated 7-cube (Toptosa)	(0,1,1,2,2,3,3)√2 + (1,1,1,1,1,1,1)						443520	80640
t0,1,3,4{4,3,3,3,3,3}	Steriruncitruncated 7-cube (Captesa)	(0,1,1,2,3,3,3)√2 + (1,1,1,1,1,1,1)						215040	53760
t0,1,2,6{4,3,3,3,3,3}	Hexicantitruncated 7-cube (Pugrosa)	(0,1,2,2,2,2,3)√2 + (1,1,1,1,1,1,1)						134400	26880
t0,1,2,5{4,3,3,3,3,3}	Penticantitruncated 7-cube (Togresa)	(0,1,2,2,2,3,3)√2 + (1,1,1,1,1,1,1)						295680	53760
t0,1,2,4{4,3,3,3,3,3}	Stericantitruncated 7-cube (Cogarsa)	(0,1,2,2,3,3,3)√2 + (1,1,1,1,1,1,1)						268800	53760
t0,1,2,3{4,3,3,3,3,3}	Runcicantitruncated 7-cube (Gapsa)	(0,1,2,3,3,3,3)√2 + (1,1,1,1,1,1,1)						94080	26880
t0,1,2,3,4{3,3,3,3,3,4}	Steriruncicantitruncated 7-orthoplex (Gocaz)	(0,0,1,2,3,4,5)√2						322560	80640
t0,1,2,3,5{3,3,3,3,3,4}	Pentiruncicantitruncated 7-orthoplex (Tegopaz)	(0,1,1,2,3,4,5)√2						725760	161280
t0,1,2,4,5{3,3,3,3,3,4}	Pentistericantitruncated 7-orthoplex (Tecagraz)	(0,1,2,2,3,4,5)√2						645120	161280
t0,1,3,4,5{3,3,3,3,3,4}	Pentisteriruncitruncated 7-orthoplex (Tecpotaz)	(0,1,2,3,3,4,5)√2						645120	161280
t0,2,3,4,5{3,3,3,3,3,4}	Pentisteriruncicantellated 7-orthoplex (Tacparez)	(0,1,2,3,4,4,5)√2						645120	161280
t1,2,3,4,5{4,3,3,3,3,3}	Bisteriruncicantitruncated 7-cube (Gabcosaz)	(0,1,2,3,4,5,5)√2						564480	161280
t0,1,2,3,6{3,3,3,3,3,4}	Hexiruncicantitruncated 7-orthoplex (Pugopaz)	(0,0,0,1,2,3,4)√2 + (1,1,1,1,1,1,1)						483840	107520
t0,1,2,4,6{3,3,3,3,3,4}	Hexistericantitruncated 7-orthoplex (Pucagraz)	(0,0,1,1,2,3,4)√2 + (1,1,1,1,1,1,1)						806400	161280
t0,1,3,4,6{3,3,3,3,3,4}	Hexisteriruncitruncated 7-orthoplex (Pucpotaz)	(0,0,1,2,2,3,4)√2 + (1,1,1,1,1,1,1)						725760	161280
t0,2,3,4,6{4,3,3,3,3,3}	Hexisteriruncicantellated 7-cube (Pucprosaz)	(0,0,1,2,3,3,4)√2 + (1,1,1,1,1,1,1)						725760	161280
t0,2,3,4,5{4,3,3,3,3,3}	Pentisteriruncicantellated 7-cube (Tocpresa)	(0,0,1,2,3,4,4)√2 + (1,1,1,1,1,1,1)						645120	161280
t0,1,2,5,6{3,3,3,3,3,4}	Hexipenticantitruncated 7-orthoplex (Putegraz)	(0,1,1,1,2,3,4)√2 + (1,1,1,1,1,1,1)						483840	107520
t0,1,3,5,6{4,3,3,3,3,3}	Hexipentiruncitruncated 7-cube (Putpetsaz)	(0,1,1,2,2,3,4)√2 + (1,1,1,1,1,1,1)						806400	161280
t0,1,3,4,6{4,3,3,3,3,3}	Hexisteriruncitruncated 7-cube (Pucpetsa)	(0,1,1,2,3,3,4)√2 + (1,1,1,1,1,1,1)						725760	161280
t0,1,3,4,5{4,3,3,3,3,3}	Pentisteriruncitruncated 7-cube (Tecpetsa)	(0,1,1,2,3,4,4)√2 + (1,1,1,1,1,1,1)						645120	161280
t0,1,2,5,6{4,3,3,3,3,3}	Hexipenticantitruncated 7-cube (Putgresa)	(0,1,2,2,2,3,4)√2 + (1,1,1,1,1,1,1)						483840	107520
t0,1,2,4,6{4,3,3,3,3,3}	Hexistericantitruncated 7-cube (Pucagrosa)	(0,1,2,2,3,3,4)√2 + (1,1,1,1,1,1,1)						806400	161280
t0,1,2,4,5{4,3,3,3,3,3}	Pentistericantitruncated 7-cube (Tecgresa)	(0,1,2,2,3,4,4)√2 + (1,1,1,1,1,1,1)						645120	161280
t0,1,2,3,6{4,3,3,3,3,3}	Hexiruncicantitruncated 7-cube (Pugopsa)	(0,1,2,3,3,3,4)√2 + (1,1,1,1,1,1,1)						483840	107520
t0,1,2,3,5{4,3,3,3,3,3}	Pentiruncicantitruncated 7-cube (Togapsa)	(0,1,2,3,3,4,4)√2 + (1,1,1,1,1,1,1)						725760	161280
t0,1,2,3,4{4,3,3,3,3,3}	Steriruncicantitruncated 7-cube (Gacosa)	(0,1,2,3,4,4,4)√2 + (1,1,1,1,1,1,1)						376320	107520
t0,1,2,3,4,5{3,3,3,3,3,4}	Pentisteriruncicantitruncated 7-orthoplex (Gotaz)	(0,1,2,3,4,5,6)√2						1128960	322560
t0,1,2,3,4,6{3,3,3,3,3,4}	Hexisteriruncicantitruncated 7-orthoplex (Pugacaz)	(0,0,1,2,3,4,5)√2 + (1,1,1,1,1,1,1)						1290240	322560
t0,1,2,3,5,6{3,3,3,3,3,4}	Hexipentiruncicantitruncated 7-orthoplex (Putgapaz)	(0,1,1,2,3,4,5)√2 + (1,1,1,1,1,1,1)						1290240	322560
t0,1,2,4,5,6{4,3,3,3,3,3}	Hexipentistericantitruncated 7-cube (Putcagrasaz)	(0,1,2,2,3,4,5)√2 + (1,1,1,1,1,1,1)						1290240	322560
t0,1,2,3,5,6{4,3,3,3,3,3}	Hexipentiruncicantitruncated 7-cube (Putgapsa)	(0,1,2,3,3,4,5)√2 + (1,1,1,1,1,1,1)						1290240	322560
t0,1,2,3,4,6{4,3,3,3,3,3}	Hexisteriruncicantitruncated 7-cube (Pugacasa)	(0,1,2,3,4,4,5)√2 + (1,1,1,1,1,1,1)						1290240	322560
t0,1,2,3,4,5{4,3,3,3,3,3}	Pentisteriruncicantitruncated 7-cube (Gotesa)	(0,1,2,3,4,5,5)√2 + (1,1,1,1,1,1,1)						1128960	322560
t0,1,2,3,4,5,6{4,3,3,3,3,3}	Omnitruncated 7-cube (Guposaz)	(0,1,2,3,4,5,6)√2 + (1,1,1,1,1,1,1)						2257920	645120

The D₇ family

The D7 family has symmetry of order 322560 (7 factorial x 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	6		5		4
=	7-cubedemihepteract (hesa)	(1,1,1,1,1,1,1)	78	532	1624	2800	2240	672	64
=	cantic 7-cubetruncated demihepteract (thesa)	(1,1,3,3,3,3,3)	142	1428	5656	11760	13440	7392	1344
=	runcic 7-cubesmall rhombated demihepteract (sirhesa)	(1,1,1,3,3,3,3)						16800	2240
=	steric 7-cubesmall prismated demihepteract (sphosa)	(1,1,1,1,3,3,3)						20160	2240
=	pentic 7-cubesmall cellated demihepteract (sochesa)	(1,1,1,1,1,3,3)						13440	1344
=	hexic 7-cubesmall terated demihepteract (suthesa)	(1,1,1,1,1,1,3)						4704	448
=	runcicantic 7-cubegreat rhombated demihepteract (Girhesa)	(1,1,3,5,5,5,5)						23520	6720
=	stericantic 7-cubeprismatotruncated demihepteract (pothesa)	(1,1,3,3,5,5,5)						73920	13440
=	steriruncic 7-cubeprismatorhomated demihepteract (prohesa)	(1,1,1,3,5,5,5)						40320	8960
=	penticantic 7-cubecellitruncated demihepteract (cothesa)	(1,1,3,3,3,5,5)						87360	13440
=	pentiruncic 7-cubecellirhombated demihepteract (crohesa)	(1,1,1,3,3,5,5)						87360	13440
=	pentisteric 7-cubecelliprismated demihepteract (caphesa)	(1,1,1,1,3,5,5)						40320	6720
=	hexicantic 7-cubetericantic demihepteract (tuthesa)	(1,1,3,3,3,3,5)						43680	6720
=	hexiruncic 7-cubeterirhombated demihepteract (turhesa)	(1,1,1,3,3,3,5)						67200	8960
=	hexisteric 7-cubeteriprismated demihepteract (tuphesa)	(1,1,1,1,3,3,5)						53760	6720
=	hexipentic 7-cubetericellated demihepteract (tuchesa)	(1,1,1,1,1,3,5)						21504	2688
=	steriruncicantic 7-cubegreat prismated demihepteract (Gephosa)	(1,1,3,5,7,7,7)						94080	26880
=	pentiruncicantic 7-cubecelligreatorhombated demihepteract (cagrohesa)	(1,1,3,5,5,7,7)						181440	40320
=	pentistericantic 7-cubecelliprismatotruncated demihepteract (capthesa)	(1,1,3,3,5,7,7)						181440	40320
=	pentisteriruncic 7-cubecelliprismatorhombated demihepteract (coprahesa)	(1,1,1,3,5,7,7)						120960	26880
=	hexiruncicantic 7-cubeterigreatorhombated demihepteract (tugrohesa)	(1,1,3,5,5,5,7)						120960	26880
=	hexistericantic 7-cubeteriprismatotruncated demihepteract (tupthesa)	(1,1,3,3,5,5,7)						221760	40320
=	hexisteriruncic 7-cubeteriprismatorhombated demihepteract (tuprohesa)	(1,1,1,3,5,5,7)						134400	26880
=	hexipenticantic 7-cubeteriCellitruncated demihepteract (tucothesa)	(1,1,3,3,3,5,7)						147840	26880
=	hexipentiruncic 7-cubetericellirhombated demihepteract (tucrohesa)	(1,1,1,3,3,5,7)						161280	26880
=	hexipentisteric 7-cubetericelliprismated demihepteract (tucophesa)	(1,1,1,1,3,5,7)						80640	13440
=	pentisteriruncicantic 7-cubegreat cellated demihepteract (gochesa)	(1,1,3,5,7,9,9)						282240	80640
=	hexisteriruncicantic 7-cubeterigreatoprimated demihepteract (tugphesa)	(1,1,3,5,7,7,9)						322560	80640
=	hexipentiruncicantic 7-cubetericelligreatorhombated demihepteract (tucagrohesa)	(1,1,3,5,5,7,9)						322560	80640
=	hexipentistericantic 7-cubetericelliprismatotruncated demihepteract (tucpathesa)	(1,1,3,3,5,7,9)						362880	80640
=	hexipentisteriruncic 7-cubetericellprismatorhombated demihepteract (tucprohesa)	(1,1,1,3,5,7,9)						241920	53760
=	hexipentisteriruncicantic 7-cubegreat terated demihepteract (guthesa)	(1,1,3,5,7,9,11)						564480	161280

The E₇ family

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

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

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

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

Regular and uniform honeycombs

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

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

#	Coxeter group	Coxeter diagram
1	{\tilde{A}}_6	[3[7]]
2	{\tilde{C}}_6	[4,34,4]
3	{\tilde{B}}_6	h[4,34,4]
[4,33,31,1]
4	{\tilde{D}}_6	q[4,34,4]
[31,1,32,31,1]
5	{\tilde{E}}_6	[32,2,2]

Regular and uniform tessellations include:

{\tilde{A}}_6, 17 forms
- Uniform 6-simplex honeycomb: {3[7]}
- Uniform Cyclotruncated 6-simplex honeycomb: t0,1{3[7]}
- Uniform Omnitruncated 6-simplex honeycomb: t0,1,2,3,4,5,6,7{3[7]}
{\tilde{C}}_6, [4,34,4], 71 forms
- Regular 6-cube honeycomb, represented by symbols {4,34,4},
{\tilde{B}}_6, [31,1,33,4], 95 forms, 64 shared with {\tilde{C}}_6, 32 new
- Uniform 6-demicube honeycomb, represented by symbols h{4,34,4} = {31,1,33,4}, =
{\tilde{D}}_6, [31,1,32,31,1], 41 unique ringed permutations, most shared with {\tilde{B}}_6 and {\tilde{C}}_6, and 6 are new. Coxeter calls the first one a quarter 6-cubic honeycomb.
- =
- =
- =
- =
- =
- =
{\tilde{E}}_6: [32,2,2], 39 forms
- Uniform 222 honeycomb: represented by symbols {3,3,32,2},
- Uniform t4(222) honeycomb: 4r{3,3,32,2},
- Uniform 0222 honeycomb: {32,2,2},
- Uniform t2(0222) honeycomb: 2r{32,2,2},

#	Coxeter group	Coxeter-Dynkin diagram
1	{\tilde{A}}_5x{\tilde{I}}_1	[3[6],2,∞]
2	{\tilde{B}}_5x{\tilde{I}}_1	[4,3,31,1,2,∞]
3	{\tilde{C}}_5x{\tilde{I}}_1	[4,33,4,2,∞]
4	{\tilde{D}}_5x{\tilde{I}}_1	[31,1,3,31,1,2,∞]
5	{\tilde{A}}_4x{\tilde{I}}_1x{\tilde{I}}_1	[3[5],2,∞,2,∞,2,∞]
6	{\tilde{B}}_4x{\tilde{I}}_1x{\tilde{I}}_1	[4,3,31,1,2,∞,2,∞]
7	{\tilde{C}}_4x{\tilde{I}}_1x{\tilde{I}}_1	[4,3,3,4,2,∞,2,∞]
8	{\tilde{D}}_4x{\tilde{I}}_1x{\tilde{I}}_1	[31,1,1,1,2,∞,2,∞]
9	{\tilde{F}}_4x{\tilde{I}}_1x{\tilde{I}}_1	[3,4,3,3,2,∞,2,∞]
10	{\tilde{C}}_3x{\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1	[4,3,4,2,∞,2,∞,2,∞]
11	{\tilde{B}}_3x{\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1	[4,31,1,2,∞,2,∞,2,∞]
12	{\tilde{A}}_3x{\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1	[3[4],2,∞,2,∞,2,∞]
13	{\tilde{C}}_2x{\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1	[4,4,2,∞,2,∞,2,∞,2,∞]
14	{\tilde{H}}_2x{\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1	[6,3,2,∞,2,∞,2,∞,2,∞]
15	{\tilde{A}}_2x{\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1	[3[3],2,∞,2,∞,2,∞,2,∞]
16	{\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1x{\tilde{I}}_1x{\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 paracompact 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 = [31,1,3,32,1]:	{\bar{S}}_6 = [4,3,3,32,1]:

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	Coxeter-
Dynkin
diagram	Description	Parent	Rectified	Birectified	Truncated	Bitruncated	Tritruncated	Cantellated	Bicantellated	Runcinated	Biruncinated	Stericated	Pentellated	Hexicated	Omnitruncated
t0{p,q,r,s,t,u}		Any regular 7-polytope
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.
t2{p,q,r,s,t,u}		Birectification reduces cells to their duals.
t0,1{p,q,r,s,t,u}		Each original vertex is cut off, with a new face filling the gap. Truncation has a degree of freedom, which has one solution that creates a uniform truncated 7-polytope. The 7-polytope has its original faces doubled in sides, and contains the faces of the dual.
[[File:Cube truncation sequence.svg	400px]]
t1,2{p,q,r,s,t,u}		Bitrunction transforms cells to their dual truncation.
t2,3{p,q,r,s,t,u}		Tritruncation transforms 4-faces to their dual truncation.
t0,2{p,q,r,s,t,u}		In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is halfway between both the parent and dual forms.
[[File:Cube cantellation sequence.svg	400px]]
t1,3{p,q,r,s,t,u}		In addition to vertex truncation, each original edge is beveled with new rectangular faces appearing in their place. A uniform cantellation is halfway between both the parent and dual forms.
t0,3{p,q,r,s,t,u}		Runcination reduces cells and creates new cells at the vertices and edges.
t1,4{p,q,r,s,t,u}		Runcination reduces cells and creates new cells at the vertices and edges.
t0,4{p,q,r,s,t,u}		Sterication reduces 4-faces and creates new 4-faces at the vertices, edges, and faces in the gaps.
t0,5{p,q,r,s,t,u}		Pentellation reduces 5-faces and creates new 5-faces at the vertices, edges, faces, and cells in the gaps.
t0,6{p,q,r,s,t,u}		Hexication reduces 6-faces and creates new 6-faces at the vertices, edges, faces, cells, and 4-faces in the gaps. (expansion operation for 7-polytopes)
t0,1,2,3,4,5,6{p,q,r,s,t,u}		All six operators, truncation, cantellation, runcination, sterication, pentellation, and hexication are applied.

References

T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
H.S.M. Coxeter:
- H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, wiley.com,
- (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

References

Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy'', Princeton, 2008.

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Uniform 7-polytope

Regular 7-polytopes

Characteristics

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

References

The A₇ family

The B₇ family

The D₇ family

The E₇ family