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.

Contents

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

Regular 8-polytopes

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

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

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
1A8[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.png135
2BC8[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.png255
3D8[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.png191 (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.png255

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

There are many uniform prismatic families, including:

The A8 family

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

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.

The D8 family

The D8 family has symmetry of order 5,160,960 (8 factorial x 27).

This family has 191 Wythoffian uniform polytopes, from 3x64-1 permutations of the D8 Coxeter-Dynkin diagram with one or more rings. 127 (2x64-1) are repeated from the B8 family and 64 are unique to this family, all listed below.

See list of D8 polytopes for Coxeter plane graphs of these polytopes.

The E8 family

The E8 family has symmetry order 696,729,600.

There are 255 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Eight forms are shown below, 4 single-ringed, 3 truncations (2 rings), and the final omnitruncation are given below. Bowers-style acronym names are given for cross-referencing.

See also list of E8 polytopes for Coxeter plane graphs of this family.

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. Coxeter diagram affine rank8 correspondence.png
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 that generate regular and uniform tessellations in 7-space:

# Coxeter group Coxeter diagram Forms
1[3[8]]CDel node.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.png29
2[4,35,4]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 4.pngCDel node.png135
3[4,34,31,1]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png191 (64 new)
4[31,1,33,31,1]CDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png77 (10 new)
5[33,3,1]CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png143

Regular and uniform tessellations include:

Regular and uniform hyperbolic honeycombs

There are no compact hyperbolic Coxeter groups of rank 8, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 4 paracompact hyperbolic Coxeter groups of rank 8, each generating uniform honeycombs in 7-space as permutations of rings of the Coxeter diagrams.

= [3,3[7]]:
CDel branch.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.png
= [31,1,32,32,1]:
CDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
= [4,33,32,1]:
CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 4a.pngCDel nodea.png
= [33,2,2]:
CDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

References

  1. 1 2 3 Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton, 2008.
  2. Klitzing.
  3. Klitzing, (x3o3x3o3o3o3o3x3 - xorene).
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compoundsPolytope operations