Graphs of three regular and related uniform polytopes
7-simplex t0.svg

7-simplex
7-simplex t1.svg

Rectified 7-simplex
7-simplex t01.svg

Truncated 7-simplex
7-simplex t02.svg

Cantellated 7-simplex
7-simplex t03.svg

Runcinated 7-simplex
7-simplex t04.svg

Stericated 7-simplex
7-simplex t05.svg

Pentellated 7-simplex
7-simplex t06.svg

Hexicated 7-simplex
7-cube t6.svg

7-orthoplex
7-cube t56.svg

Truncated 7-orthoplex
7-cube t5.svg

Rectified 7-orthoplex
7-cube t46.svg

Cantellated 7-orthoplex
7-cube t36.svg

Runcinated 7-orthoplex
7-cube t26.svg

Stericated 7-orthoplex
7-cube t16.svg

Pentellated 7-orthoplex
7-cube t06.svg

Hexicated 7-cube
7-cube t05.svg

Pentellated 7-cube
7-cube t04.svg

Stericated 7-cube
7-cube t02.svg

Cantellated 7-cube
7-cube t03.svg

Runcinated 7-cube
7-cube t0.svg

7-cube
7-cube t01.svg

Truncated 7-cube
7-cube t1.svg

Rectified 7-cube
7-demicube t0 D7.svg

7-demicube
7-demicube t01 D7.svg

Cantic 7-cube
7-demicube t02 D7.svg

Runcic 7-cube
7-demicube t03 D7.svg

Steric 7-cube
7-demicube t04 D7.svg

Pentic 7-cube
7-demicube t05 D7.svg

Hexic 7-cube
E7 graph.svg

321
Gosset 2 31 polytope.svg

231
Gosset 1 32 petrie.svg

132

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

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

Regular 7-polytopes

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

There are exactly three such convex regular 7-polytopes:

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

There are no nonconvex regular 7-polytopes.

Characteristics

The topology of any given 7-polytope is defined by its Betti numbers and torsion coefficients.[1]

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.[1]

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.[1]

Uniform 7-polytopes by fundamental Coxeter groups

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

# Coxeter group Regular and semiregular forms Uniform count
1 A7 [36] CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 71
2 B7 [4,35] CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 127 + 32
3 D7 [33,1,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 95 (0 unique)
4 E7 [33,2,1] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 127

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.

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.