Uniform 8-polytope

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Graphs of three regular and related uniform polytopes.
8-simplex t0.svg
8-simplex
8-simplex t1.svg
Rectified 8-simplex
8-simplex t01.svg
Truncated 8-simplex
8-simplex t02.svg
Cantellated 8-simplex
8-simplex t03.svg
Runcinated 8-simplex
8-simplex t04.svg
Stericated 8-simplex
8-simplex t05.svg
Pentellated 8-simplex
8-simplex t06.svg
Hexicated 8-simplex
8-simplex t07.svg
Heptellated 8-simplex
8-cube t7.svg
8-orthoplex
8-cube t6.svg
Rectified 8-orthoplex
8-cube t67.svg
Truncated 8-orthoplex
8-cube t57.svg
Cantellated 8-orthoplex
8-cube t47.svg
Runcinated 8-orthoplex
8-cube t17.svg
Hexicated 8-orthoplex
8-cube t02.svg
Cantellated 8-cube
8-cube t03.svg
Runcinated 8-cube
8-cube t04.svg
Stericated 8-cube
8-cube t05.svg
Pentellated 8-cube
8-cube t06.svg
Hexicated 8-cube
8-cube t07.svg
Heptellated 8-cube
8-cube t0.svg
8-cube
8-cube t1.svg
Rectified 8-cube
8-cube t01.svg
Truncated 8-cube
8-demicube t0 D7.svg
8-demicube
8-demicube t01 D7.svg
Truncated 8-demicube
8-demicube t02 D7.svg
Cantellated 8-demicube
8-demicube t03 D7.svg
Runcinated 8-demicube
8-demicube t04 D7.svg
Stericated 8-demicube
8-demicube t05 D7.svg
Pentellated 8-demicube
8-demicube t06 D7.svg
Hexicated 8-demicube
Gosset 4 21 polytope petrie.svg
421
Gosset 1 42 polytope petrie.svg
142
2 41 polytope petrie.svg
241

In eight-dimensional geometry, an eight-dimensional polytope or 8-polytope is a polytope contained by 7-polytope facets. Each 6-polytope ridge being shared by exactly two 7-polytope facets.

A uniform 8-polytope is one which is vertex-transitive, and constructed from uniform 7-polytope facets.

Regular 8-polytopes[edit]

Regular 8-polytopes can be represented by the Schläfli symbol {p,q,r,s,t,u,v}, with v {p,q,r,s,t,u} 7-polytope facets around each peak.

There are exactly three such convex regular 8-polytopes:

  1. {3,3,3,3,3,3,3} - 8-simplex
  2. {4,3,3,3,3,3,3} - 8-cube
  3. {3,3,3,3,3,3,4} - 8-orthoplex

There are no nonconvex regular 8-polytopes.

Characteristics[edit]

The topology of any given 8-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, and is zero for all 8-polytopes, 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 8-polytopes by fundamental Coxeter groups[edit]

Uniform 8-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams:

# Coxeter group Forms
1 A8 [37] 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.pngCDel 3.pngCDel node.png 135
2 BC8 [4,36] 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.pngCDel 3.pngCDel node.png 255
3 D8 [35,1,1] CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 191 (64 unique)
4 E8 [34,2,1] CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 255

Selected regular and uniform 8-polytopes from each family include:

  1. Simplex family: A8 [37] - 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.pngCDel 3.pngCDel node.png
    • 135 uniform 8-polytopes as permutations of rings in the group diagram, including one regular:
      1. {37} - 8-simplex or ennea-9-tope or enneazetton - CDel node 1.pngCDel 3.pngCDel 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
  2. Hypercube/orthoplex family: B8 [4,36] - 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.pngCDel 3.pngCDel node.png
    • 255 uniform 8-polytopes as permutations of rings in the group diagram, including two regular ones:
      1. {4,36} - 8-cube or octeract- CDel node 1.pngCDel 4.pngCDel 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
      2. {36,4} - 8-orthoplex or octacross - CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
  3. Demihypercube D8 family: [35,1,1] - CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
    • 191 uniform 8-polytopes as permutations of rings in the group diagram, including:
      1. {3,35,1} - 8-demicube or demiocteract, 151 - CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png; also as h{4,36} CDel node h.pngCDel 4.pngCDel 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.
      2. {3,3,3,3,3,31,1} - 8-orthoplex, 511 - CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
  4. E-polytope family E8 family: [34,1,1] - CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
    • 255 uniform 8-polytopes as permutations of rings in the group diagram, including:
      1. {3,3,3,3,32,1} - Thorold Gosset's semiregular 421, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
      2. {3,34,2} - the uniform 142, CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png,
      3. {3,3,34,1} - the uniform 241, CDel nodea 1.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png

Uniform prismatic forms[edit]

There are many uniform prismatic families, including:

The A8 family[edit]

The A8 family has symmetry of order 362880 (9 factorial).

There are 135 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. (128+8-1 cases) These are all enumerated below. Bowers-style acronym names are given in parentheses for cross-referencing.

See also a list of 8-simplex polytopes for symmetric Coxeter plane graphs of these polytopes.

The B8 family[edit]

The B8 family has symmetry of order 10321920 (8 factorial x 28). There are 255 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.

See also a list of B8 polytopes for symmetric Coxeter plane graphs of these polytopes.