Graphs of three regular and related uniform polytope

Uniform polytope

A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....

s

7-simplex				Rectified 7-simplex Rectified 7-simplex In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the rectified 7-simplex are located at the edge-centers of the...				Truncated 7-simplex Truncated 7-simplex In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.There are unique 3 degrees of truncation. Vertices of the truncation 7-simplex are located as pairs on the edge of the 7-simplex. Vertices of the bitruncated 7-simplex...
Cantellated 7-simplex Cantellated 7-simplex In seven-dimensional geometry, a cantellated 7-simplex is a convex uniform 7-polytope, being a cantellation of the regular 7-simplex.There are unique 6 degrees of cantellation for the 7-simplex, including truncations.- Cantellated 7-simplex:...				Runcinated 7-simplex Runcinated 7-simplex In seven-dimensional geometry, a runcinated 7-simplex is a convex uniform 7-polytope with 3rd order truncations of the regular 7-simplex.There are 8 unique runcinations of the 7-simplex with permutations of truncations, and cantellations....				Stericated 7-simplex Stericated 7-simplex In seven-dimensional geometry, a stericated 7-simplex is a convex uniform 7-polytope with 4th order truncations of the regular 7-simplex....
Pentellated 7-simplex Pentellated 7-simplex In seven-dimensional geometry, a pentellated 7-simplex is a convex uniform 7-polytope with 5th order truncations of the regular 7-simplex....						Hexicated 7-simplex Hexicated 7-simplex In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations from the regular 7-simplex....
7-orthoplex				Truncated 7-orthoplex Truncated 7-orthoplex In seven-dimensional geometry, a truncated 7-orthoplex is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex.There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated...				Rectified 7-orthoplex
Cantellated 7-orthoplex Cantellated 7-orthoplex In seven-dimensional geometry, a cantellated 7-orthoplex is a convex uniform 7-polytope, being a cantellation of the regular 7-orthoplex.There are ten degrees of cantellation for the 7-orthoplex, including truncations...				Runcinated 7-orthoplex Runcinated 7-orthoplex In seven-dimensional geometry, a runcinated 7-orthoplex is a convex uniform 7-polytope with 3rd order truncations of the regular 7-orthoplex.There are 16 unique runcinations of the 7-orthoplex with permutations of truncations, and cantellations...				Stericated 7-orthoplex Stericated 7-orthoplex In seven-dimensional geometry, a stericated 7-orthoplex is a convex uniform 7-polytope with 4th order truncations of the regular 7-orthoplex....
Pentellated 7-orthoplex Pentellated 7-orthoplex In seven-dimensional geometry, a pentellated 7-orthoplex is a convex uniform 7-polytope with 5th order truncations of the regular 7-orthoplex....				Hexicated 7-cube Hexicated 7-cube In seven-dimensional geometry, a hexicated 7-cube is a convex uniform 7-polytope, including 6th-order truncations from the regular 7-cube....				Pentellated 7-cube Pentellated 7-cube In seven-dimensional geometry, a pentellated 7-cube is a convex uniform 7-polytope with 5th order truncations of the regular 7-cube. There are 32 unique pentellations of the 7-cube with permutations of truncations, cantellations, runcinations, and sterications...
Stericated 7-cube Stericated 7-cube In seven-dimensional geometry, a stericated 7-cube is a convex uniform 7-polytope with 4th order truncations of the regular 7-cube.There are 24 unique sterication for the 7-cube with permutations of truncations, cantellations, and runcinations...				Cantellated 7-cube Cantellated 7-cube In seven-dimensional geometry, a cantellated 7-cube is a convex uniform 7-polytope, being a cantellation of the regular 7-cube.There are 10 degrees of cantellation for the 7-cube, including truncations...				Runcinated 7-cube Runcinated 7-cube In seven-dimensional geometry, a runcinated 7-cube is a convex uniform 7-polytope with 3rd order truncations of the regular 7-cube.There are 16 unique runcinations of the 7-cube with permutations of truncations, and cantellations...
7-cube				Truncated 7-cube Truncated 7-cube In seven-dimensional geometry, a truncated 7-cube is a convex uniform 7-polytope, being a truncation of the regular 7-cube.There are 6 truncations for the 7-cube. Vertices of the truncated 7-cube are located as pairs on the edge of the 7-cube. Vertices of the bitruncated 7-cube are located on the...				Rectified 7-cube Rectified 7-cube In seven-dimensional geometry, a rectified 7-cube is a convex uniform 7-polytope, being a rectification of the regular 7-cube.There are unique 7 degrees of rectifications, the zeroth being the 7-cube, and the 6th and last being the 7-cube. Vertices of the rectified 7-cube are located at the...
7-demicube				Truncated 7-demicube Truncated 7-demicube In seven-dimensional geometry, a truncated 7-demicube is a uniform 7-polytope, being a truncation of the 7-demicube.- Cartesian coordinates :The Cartesian coordinates for the 1344 vertices of a truncated 7-demicube centered at the origin and edge length 6√2 are coordinate permutations:with an odd...				Cantellated 7-demicube Cantellated 7-demicube In seven-dimensional geometry, a cantellated 7-demicube is a convex uniform 7-polytope, being a cantellation of the uniform 7-demicube. There are 2 unique cantellation for the 7-demicube including a truncation.- Cantellated 7-demicube:...
Runcinated 7-demicube				Stericated 7-demicube				Pentellated 7-demicube
321				231				132

In seven-dimensional

Seven-dimensional space

In physics and mathematics, a sequence of n numbers can also be understood as a location in n-dimensional space. When n = 7, the set of all such locations is called 7-dimensional Euclidean space...

 geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a 7-polytope is a polytope

Polytope

In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...

 contained by 6-polytope facets. Each 5-polytope

5-polytope

In five-dimensional geometry, a 5-polytope is a 5-dimensional polytope, bounded by facets. Each polyhedral cell being shared by exactly two polychoron facets. A proposed name for 5-polytopes is polyteron.-Definition:...

 ridge

Ridge (geometry)

In geometry, a ridge is an -dimensional element of an n-dimensional polytope. It is also sometimes called a subfacet for having one lower dimension than a facet.By dimension, this corresponds to:*a vertex of a polygon;...

 being shared by exactly two 6-polytope

6-polytope

In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform polytera....

 facets

Facet (mathematics)

A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...

.

A uniform 7-polytope is one which is vertex-transitive

Vertex-transitive

In geometry, a polytope is isogonal or vertex-transitive if, loosely speaking, all its vertices are the same...

, and constructed from uniform 6-polytope

6-polytope

In six-dimensional geometry, a uniform polypeton is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform polytera....

 facets.

A proposed name for 7-polytopes is 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

Facet (mathematics)

A facet of a simplicial complex is a maximal simplex.In the general theory of polyhedra and polytopes, two conflicting meanings are currently jostling for acceptability:...

 around each 4-face.

There are exactly three such convex regular 7-polytopes:

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

There are no nonconvex regular 7-polytopes.

Euler characteristic

The Euler characteristic

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...

 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 diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

s:

#	Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...		Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...	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	[34,1,1]		7-demicube, {31,4,1}, 7-orthoplex, {34,1,1},	95 (32 unique)
4	E7	[33,2,1]		321 Gosset 3 21 polytope In 7-dimensional geometry, the 321 polytope is a uniform 6-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper...  - 132 Gosset 1 32 polytope In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.Coxeter named it 132 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences....  - 132 Gosset 1 32 polytope In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.Coxeter named it 132 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 1-node sequences....  -	127

Prismatic finite Coxeter groups

#	Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...		Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
6+1
1	A6×A1	[35]×[ ]
2	BC6×A1	[4,34]×[ ]
3	D6×A1	[33,1,1]×[ ]
4	E6×A1	[32,2,1]×[ ]
5+2
1	A5×I2(p)	[3,3,3]×[p]
2	BC5×I2(p)	[4,3,3]×[p]
3	D5×I2(p)	[32,1,1]×[p]
5+1+1
1	A5×A12	[3,3,3]×[ ]2
2	BC5×A12	[4,3,3]×[ ]2
3	D5×A12	[32,1,1]×[ ]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	[31,1,1]×[3,3]
17	D4×BC3	[31,1,1]×[4,3]
18	D4×H3	[31,1,1]×[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	[31,1,1]×[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	[31,1,1]×[ ]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

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

).

There are 71 (64+8-1) forms based on all permutations of the Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

s 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 Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...	Truncation indices	Johnson name Bowers name (and acronym)	Basepoint	Element counts
					6	5	4	3	2	1	0
1		t0	7-simplex	(0,0,0,0,0,0,0,1)	8	28	56	70	56	28	8
2		t1	Rectified 7-simplex Rectified 7-simplex In seven-dimensional geometry, a rectified 7-simplex is a convex uniform 7-polytope, being a rectification of the regular 7-simplex.There are four unique degrees of rectifications, including the zeroth, the 7-simplex itself. Vertices of the rectified 7-simplex are located at the edge-centers of the...	(0,0,0,0,0,0,1,1)	16	84	224	350	336	168	28
3		t2	Birectified 7-simplex	(0,0,0,0,0,1,1,1)	16	112	392	770	840	420	56
4		t3	Trirectified 7-simplex	(0,0,0,0,1,1,1,1)	16	112	448	980	1120	560	70
5		t0,1	Truncated 7-simplex Truncated 7-simplex In seven-dimensional geometry, a truncated 7-simplex is a convex uniform 7-polytope, being a truncation of the regular 7-simplex.There are unique 3 degrees of truncation. Vertices of the truncation 7-simplex are located as pairs on the edge of the 7-simplex. Vertices of the bitruncated 7-simplex...	(0,0,0,0,0,0,1,2)				350	336	196	56
6		t0,2	Cantellated 7-simplex Cantellated 7-simplex In seven-dimensional geometry, a cantellated 7-simplex is a convex uniform 7-polytope, being a cantellation of the regular 7-simplex.There are unique 6 degrees of cantellation for the 7-simplex, including truncations.- Cantellated 7-simplex:...	(0,0,0,0,0,1,1,2)						1008	168
7		t1,2	Bitruncated 7-simplex	(0,0,0,0,0,1,2,2)						588	168
8		t0,3	Runcinated 7-simplex Runcinated 7-simplex In seven-dimensional geometry, a runcinated 7-simplex is a convex uniform 7-polytope with 3rd order truncations of the regular 7-simplex.There are 8 unique runcinations of the 7-simplex with permutations of truncations, and cantellations....	(0,0,0,0,1,1,1,2)						2100	280
9		t1,3	Bicantellated 7-simplex	(0,0,0,0,1,1,2,2)						2520	420
10		t2,3	Tritruncated 7-simplex	(0,0,0,0,1,2,2,2)						980	280
11		t0,4	Stericated 7-simplex Stericated 7-simplex In seven-dimensional geometry, a stericated 7-simplex is a convex uniform 7-polytope with 4th order truncations of the regular 7-simplex....	(0,0,0,1,1,1,1,2)						2240	280
12		t1,4	Biruncinated 7-simplex	(0,0,0,1,1,1,2,2)						4200	560
13		t2,4	Tricantellated 7-simplex	(0,0,0,1,1,2,2,2)						3360	560
14		t0,5	Pentellated 7-simplex Pentellated 7-simplex In seven-dimensional geometry, a pentellated 7-simplex is a convex uniform 7-polytope with 5th order truncations of the regular 7-simplex....	(0,0,1,1,1,1,1,2)						1260	168
15		t1,5	Bistericated 7-simplex	(0,0,1,1,1,1,2,2)						3360	420
16		t0,6	Hexicated 7-simplex Hexicated 7-simplex In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations from the regular 7-simplex....	(0,1,1,1,1,1,1,2)						336	56
17		t0,1,2	Cantitruncated 7-simplex	(0,0,0,0,0,1,2,3)						1176	336
18		t0,1,3	Runcitruncated 7-simplex	(0,0,0,0,1,1,2,3)						4620	840
19		t0,2,3	Runcicantellated 7-simplex	(0,0,0,0,1,2,2,3)						3360	840
20		t1,2,3	Bicantitruncated 7-simplex	(0,0,0,0,1,2,3,3)						2940	840
21		t0,1,4	Steritruncated 7-simplex	(0,0,0,1,1,1,2,3)						7280	1120
22		t0,2,4	Stericantellated 7-simplex	(0,0,0,1,1,2,2,3)						10080	1680
23		t1,2,4	Biruncitruncated 7-simplex	(0,0,0,1,1,2,3,3)						8400	1680
24		t0,3,4	Steriruncinated 7-simplex	(0,0,0,1,2,2,2,3)						5040	1120
25		t1,3,4	Biruncicantellated 7-simplex	(0,0,0,1,2,2,3,3)						7560	1680
26		t2,3,4	Tricantitruncated 7-simplex	(0,0,0,1,2,3,3,3)						3920	1120
27		t0,1,5	Pentitruncated 7-simplex	(0,0,1,1,1,1,2,3)						5460	840
28		t0,2,5	Penticantellated 7-simplex	(0,0,1,1,1,2,2,3)						11760	1680
29		t1,2,5	Bisteritruncated 7-simplex	(0,0,1,1,1,2,3,3)						9240	1680
30		t0,3,5	Pentiruncinated 7-simplex	(0,0,1,1,2,2,2,3)						10920	1680
31		t1,3,5	Bistericantellated 7-simplex	(0,0,1,1,2,2,3,3)						15120	2520
32		t0,4,5	Pentistericated 7-simplex	(0,0,1,2,2,2,2,3)						4200	840
33		t0,1,6	Hexitruncated 7-simplex	(0,1,1,1,1,1,2,3)						1848	336
34		t0,2,6	Hexicantellated 7-simplex	(0,1,1,1,1,2,2,3)						5880	840
35		t0,3,6	Hexiruncinated 7-simplex	(0,1,1,1,2,2,2,3)						8400	1120
36		t0,1,2,3	Runcicantitruncated 7-simplex	(0,0,0,0,1,2,3,4)						5880	1680
37		t0,1,2,4	Stericantitruncated 7-simplex	(0,0,0,1,1,2,3,4)						16800	3360
38		t0,1,3,4	Steriruncitruncated 7-simplex	(0,0,0,1,2,2,3,4)						13440	3360
39		t0,2,3,4	Steriruncicantellated 7-simplex	(0,0,0,1,2,3,3,4)						13440	3360
40		t1,2,3,4	Biruncicantitruncated 7-simplex	(0,0,0,1,2,3,4,4)						11760	3360
41		t0,1,2,5	Penticantitruncated 7-simplex	(0,0,1,1,1,2,3,4)						18480	3360
42		t0,1,3,5	Pentiruncitruncated 7-simplex	(0,0,1,1,2,2,3,4)						27720	5040
43		t0,2,3,5	Pentiruncicantellated 7-simplex	(0,0,1,1,2,3,3,4)						25200	5040
44		t1,2,3,5	Bistericantitruncated 7-simplex	(0,0,1,1,2,3,4,4)						22680	5040
45		t0,1,4,5	Pentisteritruncated 7-simplex	(0,0,1,2,2,2,3,4)						15120	3360
46		t0,2,4,5	Pentistericantellated 7-simplex	(0,0,1,2,2,3,3,4)						25200	5040
47		t1,2,4,5	Bisteriruncitruncated 7-simplex	(0,0,1,2,2,3,4,4)						20160	5040
48		t0,3,4,5	Pentisteriruncinated 7-simplex	(0,0,1,2,3,3,3,4)						15120	3360
49		t0,1,2,6	Hexicantitruncated 7-simplex	(0,1,1,1,1,2,3,4)						8400	1680
50		t0,1,3,6	Hexiruncitruncated 7-simplex	(0,1,1,1,2,2,3,4)						20160	3360
51		t0,2,3,6	Hexiruncicantellated 7-simplex	(0,1,1,1,2,3,3,4)						16800	3360
52		t0,1,4,6	Hexisteritruncated 7-simplex	(0,1,1,2,2,2,3,4)						20160	3360
53		t0,2,4,6	Hexistericantellated 7-simplex	(0,1,1,2,2,3,3,4)						30240	5040
54		t0,1,5,6	Hexipentitruncated 7-simplex	(0,1,2,2,2,2,3,4)						8400	1680
55		t0,1,2,3,4	Steriruncicantitruncated 7-simplex	(0,0,0,1,2,3,4,5)						23520	6720
56		t0,1,2,3,5	Pentiruncicantitruncated 7-simplex	(0,0,1,1,2,3,4,5)						45360	10080
57		t0,1,2,4,5	Pentistericantitruncated 7-simplex	(0,0,1,2,2,3,4,5)						40320	10080
58		t0,1,3,4,5	Pentisteriruncitruncated 7-simplex	(0,0,1,2,3,3,4,5)						40320	10080
59		t0,2,3,4,5	Pentisteriruncicantellated 7-simplex	(0,0,1,2,3,4,4,5)						40320	10080
60		t1,2,3,4,5	Bisteriruncicantitruncated 7-simplex	(0,0,1,2,3,4,5,5)						35280	10080
61		t0,1,2,3,6	Hexiruncicantitruncated 7-simplex	(0,1,1,1,2,3,4,5)						30240	6720
62		t0,1,2,4,6	Hexistericantitruncated 7-simplex	(0,1,1,2,2,3,4,5)						50400	10080
63		t0,1,3,4,6	Hexisteriruncitruncated 7-simplex	(0,1,1,2,3,3,4,5)						45360	10080
64		t0,2,3,4,6	Hexisteriruncicantellated 7-simplex	(0,1,1,2,3,4,4,5)						45360	10080
65		t0,1,2,5,6	Hexipenticantitruncated 7-simplex	(0,1,2,2,2,3,4,5)						30240	6720
66		t0,1,3,5,6	Hexipentiruncitruncated 7-simplex	(0,1,2,2,3,3,4,5)						50400	10080
67		t0,1,2,3,4,5	Pentisteriruncicantitruncated 7-simplex	(0,0,1,2,3,4,5,6)						70560	20160
68		t0,1,2,3,4,6	Hexisteriruncicantitruncated 7-simplex	(0,1,1,2,3,4,5,6)						80640	20160
69		t0,1,2,3,5,6	Hexipentiruncicantitruncated 7-simplex	(0,1,2,2,3,4,5,6)						80640	20160
70		t0,1,2,4,5,6	Hexipentistericantitruncated 7-simplex	(0,1,2,3,3,4,5,6)						80640	20160
71		t0,1,2,3,4,5,6	Omnitruncated 7-simplex	(0,1,2,3,4,5,6,7)						141120	40320

The B₇ family

The B₇ family has symmetry of order 645120 (7 factorial

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

 x 2⁷).

There are 127 forms based on all permutations of the Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

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.

#	Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... t-notation	Name (BSA)	Base point	Element counts
				6	5	4	3	2	1	0
1	t0{3,3,3,3,3,4}	7-orthoplex (zee)	|(0,0,0,0,0,0,1)√2	128	448	672	560	280	84	14
2	t1{3,3,3,3,3,4}	Rectified 7-orthoplex (rez)	|(0,0,0,0,0,1,1)√2	142	1344	3360	3920	2520	840	84
3	t2{3,3,3,3,3,4}	Birectified 7-orthoplex (barz)	|(0,0,0,0,1,1,1)√2	142	1428	6048	10640	8960	3360	280
4	t3{4,3,3,3,3,3}	Trirectified 7-cube (sez)	|(0,0,0,1,1,1,1)√2	142	1428	6328	14560	15680	6720	560
5	t2{4,3,3,3,3,3}	Birectified 7-cube (bersa)	|(0,0,1,1,1,1,1)√2	142	1428	5656	11760	13440	6720	672
6	t1{4,3,3,3,3,3}	Rectified 7-cube Rectified 7-cube In seven-dimensional geometry, a rectified 7-cube is a convex uniform 7-polytope, being a rectification of the regular 7-cube.There are unique 7 degrees of rectifications, the zeroth being the 7-cube, and the 6th and last being the 7-cube. Vertices of the rectified 7-cube are located at the...  (rasa)	|(0,1,1,1,1,1,1)√2	142	980	2968	5040	5152	2688	448
7	t0{4,3,3,3,3,3}	7-cube (hept)	|(0,0,0,0,0,0,0)√2 + (1,1,1,1,1,1,1)	14	84	280	560	672	448	128
8	t0,1{3,3,3,3,3,4}	Truncated 7-orthoplex Truncated 7-orthoplex In seven-dimensional geometry, a truncated 7-orthoplex is a convex uniform 7-polytope, being a truncation of the regular 7-orthoplex.There are 6 truncations of the 7-orthoplex. Vertices of the truncation 7-orthoplex are located as pairs on the edge of the 7-orthoplex. Vertices of the bitruncated...  (Taz)	|(0,0,0,0,0,1,2)√2						924	168
9	t0,2{3,3,3,3,3,4}	Cantellated 7-orthoplex Cantellated 7-orthoplex In seven-dimensional geometry, a cantellated 7-orthoplex is a convex uniform 7-polytope, being a cantellation of the regular 7-orthoplex.There are ten degrees of cantellation for the 7-orthoplex, including truncations...  (Sarz)	|(0,0,0,0,1,1,2)√2						7560	840
10	t1,2{3,3,3,3,3,4}	Bitruncated 7-orthoplex (Botaz)	|(0,0,0,0,1,2,2)√2						4200	840
11	t0,3{3,3,3,3,3,4}	Runcinated 7-orthoplex Runcinated 7-orthoplex In seven-dimensional geometry, a runcinated 7-orthoplex is a convex uniform 7-polytope with 3rd order truncations of the regular 7-orthoplex.There are 16 unique runcinations of the 7-orthoplex with permutations of truncations, and cantellations...  (Spaz)	|(0,0,0,1,1,1,2)√2						23520	2240
12	t1,3{3,3,3,3,3,4}	Bicantellated 7-orthoplex (Sebraz)	|(0,0,0,1,1,2,2)√2						26880	3360
13	t2,3{3,3,3,3,3,4}	Tritruncated 7-orthoplex (Totaz)	|(0,0,0,1,2,2,2)√2						10080	2240
14	t0,4{3,3,3,3,3,4}	Stericated 7-orthoplex Stericated 7-orthoplex In seven-dimensional geometry, a stericated 7-orthoplex is a convex uniform 7-polytope with 4th order truncations of the regular 7-orthoplex....  (Scaz)	|(0,0,1,1,1,1,2)√2						33600	3360
15	t1,4{3,3,3,3,3,4}	Biruncinated 7-orthoplex (Sibpaz)	|(0,0,1,1,1,2,2)√2						60480	6720
16	t2,4{4,3,3,3,3,3}	Tricantellated 7-cube (Strasaz)	|(0,0,1,1,2,2,2)√2						47040	6720
17	t2,3{4,3,3,3,3,3}	Tritruncated 7-cube (Tatsa)	|(0,0,1,2,2,2,2)√2						13440	3360
18	t0,5{3,3,3,3,3,4}	Pentellated 7-orthoplex Pentellated 7-orthoplex In seven-dimensional geometry, a pentellated 7-orthoplex is a convex uniform 7-polytope with 5th order truncations of the regular 7-orthoplex....  (Staz)	|(0,1,1,1,1,1,2)√2						20160	2688
19	t1,5{4,3,3,3,3,3}	Bistericated 7-cube (Sabcosaz)	|(0,1,1,1,1,2,2)√2						53760	6720
20	t1,4{4,3,3,3,3,3}	Biruncinated 7-cube (Sibposa)	|(0,1,1,1,2,2,2)√2						67200	8960
21	t1,3{4,3,3,3,3,3}	Bicantellated 7-cube (Sibrosa)	|(0,1,1,2,2,2,2)√2						40320	6720
22	t1,2{4,3,3,3,3,3}	Bitruncated 7-cube (Betsa)	|(0,1,2,2,2,2,2)√2						9408	2688
23	t0,6{4,3,3,3,3,3}	Hexicated 7-cube Hexicated 7-cube In seven-dimensional geometry, a hexicated 7-cube is a convex uniform 7-polytope, including 6th-order truncations from the regular 7-cube....  (Suposaz)	|(0,0,0,0,0,0,1)√2 + (1,1,1,1,1,1,1)						5376	896
24	t0,5{4,3,3,3,3,3}	Pentellated 7-cube Pentellated 7-cube In seven-dimensional geometry, a pentellated 7-cube is a convex uniform 7-polytope with 5th order truncations of the regular 7-cube. There are 32 unique pentellations of the 7-cube with permutations of truncations, cantellations, runcinations, and sterications...  (Stesa)	|(0,0,0,0,0,1,1)√2 + (1,1,1,1,1,1,1)						20160	2688
25	t0,4{4,3,3,3,3,3}	Stericated 7-cube Stericated 7-cube In seven-dimensional geometry, a stericated 7-cube is a convex uniform 7-polytope with 4th order truncations of the regular 7-cube.There are 24 unique sterication for the 7-cube with permutations of truncations, cantellations, and runcinations...  (Scosa)	|(0,0,0,0,1,1,1)√2 + (1,1,1,1,1,1,1)						35840	4480
26	t0,3{4,3,3,3,3,3}	Runcinated 7-cube Runcinated 7-cube In seven-dimensional geometry, a runcinated 7-cube is a convex uniform 7-polytope with 3rd order truncations of the regular 7-cube.There are 16 unique runcinations of the 7-cube with permutations of truncations, and cantellations...  (Spesa)	|(0,0,0,1,1,1,1)√2 + (1,1,1,1,1,1,1)						33600	4480
27	t0,2{4,3,3,3,3,3}	Cantellated 7-cube Cantellated 7-cube In seven-dimensional geometry, a cantellated 7-cube is a convex uniform 7-polytope, being a cantellation of the regular 7-cube.There are 10 degrees of cantellation for the 7-cube, including truncations...  (Sersa)	|(0,0,1,1,1,1,1)√2 + (1,1,1,1,1,1,1)						16128	2688
28	t0,1{4,3,3,3,3,3}	Truncated 7-cube Truncated 7-cube In seven-dimensional geometry, a truncated 7-cube is a convex uniform 7-polytope, being a truncation of the regular 7-cube.There are 6 truncations for the 7-cube. Vertices of the truncated 7-cube are located as pairs on the edge of the 7-cube. Vertices of the bitruncated 7-cube are located on the...  (Tasa)	|(0,1,1,1,1,1,1)√2 + (1,1,1,1,1,1,1)						3136	896
29	t0,1,2{3,3,3,3,3,4}	Cantitruncated 7-orthoplex (Garz)	|(0,1,2,3,3,3,3)√2						8400	1680
30	t0,1,3{3,3,3,3,3,4}	Runcitruncated 7-orthoplex (Potaz)	|(0,1,2,2,3,3,3)√2						50400	6720
31	t0,2,3{3,3,3,3,3,4}	Runcicantellated 7-orthoplex (Parz)	|(0,1,1,2,3,3,3)√2						33600	6720
32	t1,2,3{3,3,3,3,3,4}	Bicantitruncated 7-orthoplex (Gebraz)	|(0,0,1,2,3,3,3)√2						30240	6720
33	t0,1,4{3,3,3,3,3,4}	Steritruncated 7-orthoplex (Catz)	|(0,0,1,1,1,2,3)√2						107520	13440
34	t0,2,4{3,3,3,3,3,4}	Stericantellated 7-orthoplex (Craze)	|(0,0,1,1,2,2,3)√2						141120	20160
35	t1,2,4{3,3,3,3,3,4}	Biruncitruncated 7-orthoplex (Baptize)	|(0,0,1,1,2,3,3)√2						120960	20160
36	t0,3,4{3,3,3,3,3,4}	Steriruncinated 7-orthoplex (Copaz)	|(0,1,1,1,2,3,3)√2						67200	13440
37	t1,3,4{3,3,3,3,3,4}	Biruncicantellated 7-orthoplex (Boparz)	|(0,0,1,2,2,3,3)√2						100800	20160
38	t2,3,4{4,3,3,3,3,3}	Tricantitruncated 7-cube (Gotrasaz)	|(0,0,0,1,2,3,3)√2						53760	13440
39	t0,1,5{3,3,3,3,3,4}	Pentitruncated 7-orthoplex (Tetaz)	|(0,1,1,1,1,2,3)√2						87360	13440
40	t0,2,5{3,3,3,3,3,4}	Penticantellated 7-orthoplex (Teroz)	|(0,1,1,1,2,2,3)√2						188160	26880
41	t1,2,5{3,3,3,3,3,4}	Bisteritruncated 7-orthoplex (Boctaz)	|(0,1,1,1,2,3,3)√2						147840	26880
42	t0,3,5{3,3,3,3,3,4}	Pentiruncinated 7-orthoplex (Topaz)	|(0,1,1,2,2,2,3)√2						174720	26880
43	t1,3,5{4,3,3,3,3,3}	Bistericantellated 7-cube (Bacresaz)	|(0,1,1,2,2,3,3)√2						241920	40320
44	t1,3,4{4,3,3,3,3,3}	Biruncicantellated 7-cube (Bopresa)	|(0,1,1,2,3,3,3)√2						120960	26880
45	t0,4,5{3,3,3,3,3,4}	Pentistericated 7-orthoplex (Tocaz)	|(0,1,2,2,2,2,3)√2						67200	13440
46	t1,2,5{4,3,3,3,3,3}	Bisteritruncated 7-cube (Bactasa)	|(0,1,2,2,2,3,3)√2						147840	26880
47	t1,2,4{4,3,3,3,3,3}	Biruncitruncated 7-cube (Biptesa)	|(0,1,2,2,3,3,3)√2						134400	26880
48	t1,2,3{4,3,3,3,3,3}	Bicantitruncated 7-cube (Gibrosa)	|(0,1,2,3,3,3,3)√2						47040	13440
49	t0,1,6{3,3,3,3,3,4}	Hexitruncated 7-orthoplex (Putaz)	|(0,0,0,0,0,1,2)√2 + (1,1,1,1,1,1,1)						29568	5376
50	t0,2,6{3,3,3,3,3,4}	Hexicantellated 7-orthoplex (Puraz)	|(0,0,0,0,1,1,2)√2 + (1,1,1,1,1,1,1)						94080	13440
51	t0,4,5{4,3,3,3,3,3}	Pentistericated 7-cube (Tacosa)	|(0,0,0,0,1,2,2)√2 + (1,1,1,1,1,1,1)						67200	13440
52	t0,3,6{4,3,3,3,3,3}	Hexiruncinated 7-cube (Pupsez)	|(0,0,0,1,1,1,2)√2 + (1,1,1,1,1,1,1)						134400	17920
53	t0,3,5{4,3,3,3,3,3}	Pentiruncinated 7-cube (Tapsa)	|(0,0,0,1,1,2,2)√2 + (1,1,1,1,1,1,1)						174720	26880
54	t0,3,4{4,3,3,3,3,3}	Steriruncinated 7-cube (Capsa)	|(0,0,0,1,2,2,2)√2 + (1,1,1,1,1,1,1)						80640	17920
55	t0,2,6{4,3,3,3,3,3}	Hexicantellated 7-cube (Purosa)	|(0,0,1,1,1,1,2)√2 + (1,1,1,1,1,1,1)						94080	13440
56	t0,2,5{4,3,3,3,3,3}	Penticantellated 7-cube (Tersa)	|(0,0,1,1,1,2,2)√2 + (1,1,1,1,1,1,1)						188160	26880
57	t0,2,4{4,3,3,3,3,3}	Stericantellated 7-cube (Carsa)	|(0,0,1,1,2,2,2)√2 + (1,1,1,1,1,1,1)						161280	26880
58	t0,2,3{4,3,3,3,3,3}	Runcicantellated 7-cube (Parsa)	|(0,0,1,2,2,2,2)√2 + (1,1,1,1,1,1,1)						53760	13440
59	t0,1,6{4,3,3,3,3,3}	Hexitruncated 7-cube (Putsa)	|(0,1,1,1,1,1,2)√2 + (1,1,1,1,1,1,1)						29568	5376
60	t0,1,5{4,3,3,3,3,3}	Pentitruncated 7-cube (Tetsa)	|(0,1,1,1,1,2,2)√2 + (1,1,1,1,1,1,1)						87360	13440
61	t0,1,4{4,3,3,3,3,3}	Steritruncated 7-cube (Catsa)	|(0,1,1,1,2,2,2)√2 + (1,1,1,1,1,1,1)						116480	17920
62	t0,1,3{4,3,3,3,3,3}	Runcitruncated 7-cube (Petsa)	|(0,1,1,2,2,2,2)√2 + (1,1,1,1,1,1,1)						73920	13440
63	t0,1,2{4,3,3,3,3,3}	Cantitruncated 7-cube (Gersa)	|(0,1,2,2,2,2,2)√2 + (1,1,1,1,1,1,1)						18816	5376
64	t0,1,2,3{3,3,3,3,3,4}	Runcicantitruncated 7-orthoplex (Gopaz)	|(0,1,2,3,4,4,4)√2						60480	13440
65	t0,1,2,4{3,3,3,3,3,4}	Stericantitruncated 7-orthoplex (Cogarz)	|(0,0,1,1,2,3,4)√2						241920	40320
66	t0,1,3,4{3,3,3,3,3,4}	Steriruncitruncated 7-orthoplex (Captaz)	|(0,0,1,2,2,3,4)√2						181440	40320
67	t0,2,3,4{3,3,3,3,3,4}	Steriruncicantellated 7-orthoplex (Caparz)	|(0,0,1,2,3,3,4)√2						181440	40320
68	t1,2,3,4{3,3,3,3,3,4}	Biruncicantitruncated 7-orthoplex (Gibpaz)	|(0,0,1,2,3,4,4)√2						161280	40320
69	t0,1,2,5{3,3,3,3,3,4}	Penticantitruncated 7-orthoplex (Tograz)	|(0,1,1,1,2,3,4)√2						295680	53760
70	t0,1,3,5{3,3,3,3,3,4}	Pentiruncitruncated 7-orthoplex (Toptaz)	|(0,1,1,2,2,3,4)√2						443520	80640
71	t0,2,3,5{3,3,3,3,3,4}	Pentiruncicantellated 7-orthoplex (Toparz)	|(0,1,1,2,3,3,4)√2						403200	80640
72	t1,2,3,5{3,3,3,3,3,4}	Bistericantitruncated 7-orthoplex (Becogarz)	|(0,1,1,2,3,4,4)√2						362880	80640
73	t0,1,4,5{3,3,3,3,3,4}	Pentisteritruncated 7-orthoplex (Tacotaz)	|(0,1,2,2,2,3,4)√2						241920	53760
74	t0,2,4,5{3,3,3,3,3,4}	Pentistericantellated 7-orthoplex (Tocarz)	|(0,1,2,2,3,3,4)√2						403200	80640
75	t1,2,4,5{4,3,3,3,3,3}	Bisteriruncitruncated 7-cube (Bocaptosaz)	|(0,1,2,2,3,4,4)√2						322560	80640
76	t0,3,4,5{3,3,3,3,3,4}	Pentisteriruncinated 7-orthoplex (Tecpaz)	|(0,1,2,3,3,3,4)√2						241920	53760
77	t1,2,3,5{4,3,3,3,3,3}	Bistericantitruncated 7-cube (Becgresa)	|(0,1,2,3,3,4,4)√2						362880	80640
78	t1,2,3,4{4,3,3,3,3,3}	Biruncicantitruncated 7-cube (Gibposa)	|(0,1,2,3,4,4,4)√2						188160	53760
79	t0,1,2,6{3,3,3,3,3,4}	Hexicantitruncated 7-orthoplex (Pugarez)	|(0,0,0,0,1,2,3)√2 + (1,1,1,1,1,1,1)						134400	26880
80	t0,1,3,6{3,3,3,3,3,4}	Hexiruncitruncated 7-orthoplex (Papataz)	|(0,0,0,1,1,2,3)√2 + (1,1,1,1,1,1,1)						322560	53760
81	t0,2,3,6{3,3,3,3,3,4}	Hexiruncicantellated 7-orthoplex (Puparez)	|(0,0,0,1,2,2,3)√2 + (1,1,1,1,1,1,1)						268800	53760
82	t0,3,4,5{4,3,3,3,3,3}	Pentisteriruncinated 7-cube (Tecpasa)	|(0,0,0,1,2,3,3)√2 + (1,1,1,1,1,1,1)						241920	53760
83	t0,1,4,6{3,3,3,3,3,4}	Hexisteritruncated 7-orthoplex (Pucotaz)	|(0,0,1,1,1,2,3)√2 + (1,1,1,1,1,1,1)						322560	53760
84	t0,2,4,6{4,3,3,3,3,3}	Hexistericantellated 7-cube (Pucrosaz)	|(0,0,1,1,2,2,3)√2 + (1,1,1,1,1,1,1)						483840	80640
85	t0,2,4,5{4,3,3,3,3,3}	Pentistericantellated 7-cube (Tecresa)	|(0,0,1,1,2,3,3)√2 + (1,1,1,1,1,1,1)						403200	80640
86	t0,2,3,6{4,3,3,3,3,3}	Hexiruncicantellated 7-cube (Pupresa)	|(0,0,1,2,2,2,3)√2 + (1,1,1,1,1,1,1)						268800	53760
87	t0,2,3,5{4,3,3,3,3,3}	Pentiruncicantellated 7-cube (Topresa)	|(0,0,1,2,2,3,3)√2 + (1,1,1,1,1,1,1)						403200	80640
88	t0,2,3,4{4,3,3,3,3,3}	Steriruncicantellated 7-cube (Copresa)	|(0,0,1,2,3,3,3)√2 + (1,1,1,1,1,1,1)						215040	53760
89	t0,1,5,6{4,3,3,3,3,3}	Hexipentitruncated 7-cube (Putatosez)	|(0,1,1,1,1,2,3)√2 + (1,1,1,1,1,1,1)						134400	26880
90	t0,1,4,6{4,3,3,3,3,3}	Hexisteritruncated 7-cube (Pacutsa)	|(0,1,1,1,2,2,3)√2 + (1,1,1,1,1,1,1)						322560	53760
91	t0,1,4,5{4,3,3,3,3,3}	Pentisteritruncated 7-cube (Tecatsa)	|(0,1,1,1,2,3,3)√2 + (1,1,1,1,1,1,1)						241920	53760
92	t0,1,3,6{4,3,3,3,3,3}	Hexiruncitruncated 7-cube (Pupetsa)	|(0,1,1,2,2,2,3)√2 + (1,1,1,1,1,1,1)						322560	53760
93	t0,1,3,5{4,3,3,3,3,3}	Pentiruncitruncated 7-cube (Toptosa)	|(0,1,1,2,2,3,3)√2 + (1,1,1,1,1,1,1)						443520	80640
94	t0,1,3,4{4,3,3,3,3,3}	Steriruncitruncated 7-cube (Captesa)	|(0,1,1,2,3,3,3)√2 + (1,1,1,1,1,1,1)						215040	53760
95	t0,1,2,6{4,3,3,3,3,3}	Hexicantitruncated 7-cube (Pugrosa)	|(0,1,2,2,2,2,3)√2 + (1,1,1,1,1,1,1)						134400	26880
96	t0,1,2,5{4,3,3,3,3,3}	Penticantitruncated 7-cube (Togresa)	|(0,1,2,2,2,3,3)√2 + (1,1,1,1,1,1,1)						295680	53760
97	t0,1,2,4{4,3,3,3,3,3}	Stericantitruncated 7-cube (Cogarsa)	|(0,1,2,2,3,3,3)√2 + (1,1,1,1,1,1,1)						268800	53760
98	t0,1,2,3{4,3,3,3,3,3}	Runcicantitruncated 7-cube (Gapsa)	|(0,1,2,3,3,3,3)√2 + (1,1,1,1,1,1,1)						94080	26880
99	t0,1,2,3,4{3,3,3,3,3,4}	Steriruncicantitruncated 7-orthoplex (Gocaz)	|(0,0,1,2,3,4,5)√2						322560	80640
100	t0,1,2,3,5{3,3,3,3,3,4}	Pentiruncicantitruncated 7-orthoplex (Tegopaz)	|(0,1,1,2,3,4,5)√2						725760	161280
101	t0,1,2,4,5{3,3,3,3,3,4}	Pentistericantitruncated 7-orthoplex (Tecagraz)	|(0,1,2,2,3,4,5)√2						645120	161280
102	t0,1,3,4,5{3,3,3,3,3,4}	Pentisteriruncitruncated 7-orthoplex (Tecpotaz)	|(0,1,2,3,3,4,5)√2						645120	161280
103	t0,2,3,4,5{3,3,3,3,3,4}	Pentisteriruncicantellated 7-orthoplex (Tacparez)	|(0,1,2,3,4,4,5)√2						645120	161280
104	t1,2,3,4,5{4,3,3,3,3,3}	Bisteriruncicantitruncated 7-cube (Gabcosaz)	|(0,1,2,3,4,5,5)√2						564480	161280
105	t0,1,2,3,6{3,3,3,3,3,4}	Hexiruncicantitruncated 7-orthoplex (Pugopaz)	|(0,0,0,1,2,3,4)√2 + (1,1,1,1,1,1,1)						483840	107520
106	t0,1,2,4,6{3,3,3,3,3,4}	Hexistericantitruncated 7-orthoplex (Pucagraz)	|(0,0,1,1,2,3,4)√2 + (1,1,1,1,1,1,1)						806400	161280
107	t0,1,3,4,6{3,3,3,3,3,4}	Hexisteriruncitruncated 7-orthoplex (Pucpotaz)	|(0,0,1,2,2,3,4)√2 + (1,1,1,1,1,1,1)						725760	161280
108	t0,2,3,4,6{4,3,3,3,3,3}	Hexisteriruncicantellated 7-cube (Pucprosaz)	|(0,0,1,2,3,3,4)√2 + (1,1,1,1,1,1,1)						725760	161280
109	t0,2,3,4,5{4,3,3,3,3,3}	Pentisteriruncicantellated 7-cube (Tocpresa)	|(0,0,1,2,3,4,4)√2 + (1,1,1,1,1,1,1)						645120	161280
110	t0,1,2,5,6{3,3,3,3,3,4}	Hexipenticantitruncated 7-orthoplex (Putegraz)	|(0,1,1,1,2,3,4)√2 + (1,1,1,1,1,1,1)						483840	107520
111	t0,1,3,5,6{4,3,3,3,3,3}	Hexipentiruncitruncated 7-cube (Putpetsaz)	|(0,1,1,2,2,3,4)√2 + (1,1,1,1,1,1,1)						806400	161280
112	t0,1,3,4,6{4,3,3,3,3,3}	Hexisteriruncitruncated 7-cube (Pucpetsa)	|(0,1,1,2,3,3,4)√2 + (1,1,1,1,1,1,1)						725760	161280
113	t0,1,3,4,5{4,3,3,3,3,3}	Pentisteriruncitruncated 7-cube (Tecpetsa)	|(0,1,1,2,3,4,4)√2 + (1,1,1,1,1,1,1)						645120	161280
114	t0,1,2,5,6{4,3,3,3,3,3}	Hexipenticantitruncated 7-cube (Putgresa)	|(0,1,2,2,2,3,4)√2 + (1,1,1,1,1,1,1)						483840	107520
115	t0,1,2,4,6{4,3,3,3,3,3}	Hexistericantitruncated 7-cube (Pucagrosa)	|(0,1,2,2,3,3,4)√2 + (1,1,1,1,1,1,1)						806400	161280
116	t0,1,2,4,5{4,3,3,3,3,3}	Pentistericantitruncated 7-cube (Tecgresa)	|(0,1,2,2,3,4,4)√2 + (1,1,1,1,1,1,1)						645120	161280
117	t0,1,2,3,6{4,3,3,3,3,3}	Hexiruncicantitruncated 7-cube (Pugopsa)	|(0,1,2,3,3,3,4)√2 + (1,1,1,1,1,1,1)						483840	107520
118	t0,1,2,3,5{4,3,3,3,3,3}	Pentiruncicantitruncated 7-cube (Togapsa)	|(0,1,2,3,3,4,4)√2 + (1,1,1,1,1,1,1)						725760	161280
119	t0,1,2,3,4{4,3,3,3,3,3}	Steriruncicantitruncated 7-cube (Gacosa)	|(0,1,2,3,4,4,4)√2 + (1,1,1,1,1,1,1)						376320	107520
120	t0,1,2,3,4,5{3,3,3,3,3,4}	Pentisteriruncicantitruncated 7-orthoplex (Gotaz)	|(0,1,2,3,4,5,6)√2						1128960	322560
121	t0,1,2,3,4,6{3,3,3,3,3,4}	Hexisteriruncicantitruncated 7-orthoplex (Pugacaz)	|(0,0,1,2,3,4,5)√2 + (1,1,1,1,1,1,1)						1290240	322560
122	t0,1,2,3,5,6{3,3,3,3,3,4}	Hexipentiruncicantitruncated 7-orthoplex (Putgapaz)	|(0,1,1,2,3,4,5)√2 + (1,1,1,1,1,1,1)						1290240	322560
123	t0,1,2,4,5,6{4,3,3,3,3,3}	Hexipentistericantitruncated 7-cube (Putcagrasaz)	|(0,1,2,2,3,4,5)√2 + (1,1,1,1,1,1,1)						1290240	322560
124	t0,1,2,3,5,6{4,3,3,3,3,3}	Hexipentiruncicantitruncated 7-cube (Putgapsa)	|(0,1,2,3,3,4,5)√2 + (1,1,1,1,1,1,1)						1290240	322560
125	t0,1,2,3,4,6{4,3,3,3,3,3}	Hexisteriruncicantitruncated 7-cube (Pugacasa)	|(0,1,2,3,4,4,5)√2 + (1,1,1,1,1,1,1)						1290240	322560
126	t0,1,2,3,4,5{4,3,3,3,3,3}	Pentisteriruncicantitruncated 7-cube (Gotesa)	|(0,1,2,3,4,5,5)√2 + (1,1,1,1,1,1,1)						1128960	322560
127	t0,1,2,3,4,5,6{4,3,3,3,3,3}	Omnitruncated 7-cube (Guposaz)	|(0,1,2,3,4,5,6)√2 + (1,1,1,1,1,1,1)						2257920	645120

The D₇ family

The D₇ family has symmetry of order 322560 (7 factorial

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...

 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

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

. 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-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...	Names	Base point (Alternately signed)	Element counts
				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 7-demicube In seven-dimensional geometry, a truncated 7-demicube is a uniform 7-polytope, being a truncation of the 7-demicube.- Cartesian coordinates :The Cartesian coordinates for the 1344 vertices of a truncated 7-demicube centered at the origin and edge length 6√2 are coordinate permutations:with an odd... Truncated demihepteract (Thesa)	(1,1,3,3,3,3,3)						7392	1344
3		Cantellated 7-demicube Cantellated 7-demicube In seven-dimensional geometry, a cantellated 7-demicube is a convex uniform 7-polytope, being a cantellation of the uniform 7-demicube. There are 2 unique cantellation for the 7-demicube including a truncation.- 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

Coxeter group

In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

 has order 2,903,040.

There are 127 forms based on all permutations of the Coxeter-Dynkin diagram

Coxeter-Dynkin diagram

In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...

s with one or more rings.

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

#	Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... Schläfli symbol	Names	Element counts
			6	5	4	3	2	1	0
1		231 Gosset 2 31 polytope In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.Coxeter named it 231 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of one of the 2-node sequences....	632	4788	16128	20160	10080	2016	126
2		Rectified 231	758	10332	47880	100800	90720	30240	2016
3		Rectified 132	758	12348	72072	191520	241920	120960	10080
4		132	182	4284	23688	50400	40320	10080	576
5		Birectified 321	758	12348	68040	161280	161280	60480	4032
6		Rectified 321	758	44352	70560	48384	11592	12096	756
7		321 Gosset 3 21 polytope In 7-dimensional geometry, the 321 polytope is a uniform 6-polytope, constructed within the symmetry of the E7 group. It was discovered by Thorold Gosset, published in his 1900 paper...	702	6048	12096	10080	4032	756	56
8		Truncated 231						32256	4032
9		Truncated 132						131040	20160
10		Bitruncated 231							30240
11		small demified 231 (shilq)							4032
12		demirectified 231 (hirlaq)							12096
13		truncated 132 (tolin)							20160
14		small demiprismated 231 (shiplaq)							20160
15		birectified 132 (berlin)							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 truncated 321 (tanq)						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

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

#	Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...		Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
1		[3[7]]
2		[4,34,4]
3		h[4,34,4] [4,33,31,1]
4		q[4,34,4] [31,1,32,31,1]
5		[32,2,2]

Prismatic groups

#	Coxeter group Coxeter group In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...		Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors...
1	x	[3[6]]x[∞]
2	x	[4,3,31,1]x[∞]
3	x	[4,33,4]x[∞]
4	x	[31,1,3,31,1]x[∞]
5	xx	[3[5]]x[∞]x[∞]x[∞]
6	xx	[4,3,31,1]x[∞]x[∞]
7	xx	[4,3,3,4]x[∞]x[∞]
8	xx	[31,1,1,1]x[∞]x[∞]
9	xx	[3,4,3,3]x[∞]x[∞]
10	xxx	[4,3,4]x[∞]x[∞]x[∞]
11	xxx	[4,31,1]x[∞]x[∞]x[∞]
12	xxx	[3[4]]x[∞]x[∞]x[∞]
13	xxxx	[4,4]x[∞]x[∞]x[∞]x[∞]
14	xxxx	[6,3]x[∞]x[∞]x[∞]x[∞]
15	xxxx	[3[3]]x[∞]x[∞]x[∞]x[∞]
16	xxxxx	[∞]x[∞]x[∞]x[∞]x[∞]

Regular and uniform tessellations include:

• , [4,3⁴,4]
  • Regular hexeractic honeycomb, represented by symbols {4,3⁴,4},
• , [3^1,1,3³,4]
  • Uniform demihexeractic honeycomb, represented by symbols h{4,3⁴,4} = {3^1,1,3³,4}, =
• : [3^2,2,2]
  • Uniform 2₂₂ honeycomb: represented by symbols {3^2,2,2},

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

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

. 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.

= [3,3[6]]:

= [31,1,3,32,1]:

= [4,3,3,32,1]:

External links

• Polytope names
• Polytopes of Various Dimensions
• Multi-dimensional Glossary