Uniform 7-polytope

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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 which is vertex-transitive, and constructed from uniform 6-polytope facets.

Regular 7-polytopes[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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.

The D7 family[edit]

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.

The E7 family[edit]

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.

Regular and uniform honeycombs[edit]

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

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

# Coxeter group Coxeter diagram Forms
1