# Uniform 8-polytope

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.

 8-simplex Rectified 8-simplex Truncated 8-simplex Cantellated 8-simplex Runcinated 8-simplex Stericated 8-simplex Pentellated 8-simplex Hexicated 8-simplex Heptellated 8-simplex 8-orthoplex Rectified 8-orthoplex Truncated 8-orthoplex Cantellated 8-orthoplex Runcinated 8-orthoplex Hexicated 8-orthoplex Cantellated 8-cube Runcinated 8-cube Stericated 8-cube Pentellated 8-cube Hexicated 8-cube Heptellated 8-cube 8-cube Rectified 8-cube Truncated 8-cube 8-demicube Truncated 8-demicube Cantellated 8-demicube Runcinated 8-demicube Stericated 8-demicube Pentellated 8-demicube Hexicated 8-demicube 421 142 241

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]135
2BC8[4,36]255
3D8[35,1,1]191 (64 unique)
4E8[34,2,1]255

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

1. Simplex family: A8 [37] -
• 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 -
2. Hypercube/orthoplex family: B8 [4,36] -
• 255 uniform 8-polytopes as permutations of rings in the group diagram, including two regular ones:
1. {4,36} - 8-cube or octeract-
2. {36,4} - 8-orthoplex or octacross -
3. Demihypercube D8 family: [35,1,1] -
• 191 uniform 8-polytopes as permutations of rings in the group diagram, including:
1. {3,35,1} - 8-demicube or demiocteract, 151 - ; also as h{4,36} .
2. {3,3,3,3,3,31,1} - 8-orthoplex, 511 -
4. E-polytope family E8 family: [34,1,1] -
• 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,
2. {3,34,2} - the uniform 142, ,
3. {3,3,34,1} - the uniform 241,

### 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

There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 7-space:

# Coxeter group Coxeter diagram Forms
1${\displaystyle {\tilde {A}}_{7}}$[3[8]]29
2${\displaystyle {\tilde {C}}_{7}}$[4,35,4]135
3${\displaystyle {\tilde {B}}_{7}}$[4,34,31,1]191 (64 new)
4${\displaystyle {\tilde {D}}_{7}}$[31,1,33,31,1]77 (10 new)
5${\displaystyle {\tilde {E}}_{7}}$[33,3,1]143

Regular and uniform tessellations include:

• ${\displaystyle {\tilde {A}}_{7}}$ 29 uniquely ringed forms, including:
• ${\displaystyle {\tilde {C}}_{7}}$ 135 uniquely ringed forms, including:
• ${\displaystyle {\tilde {B}}_{7}}$ 191 uniquely ringed forms, 127 shared with ${\displaystyle {\tilde {C}}_{7}}$, and 64 new, including:
• ${\displaystyle {\tilde {D}}_{7}}$, [31,1,33,31,1]: 77 unique ring permutations, and 10 are new, the first Coxeter called a quarter 7-cubic honeycomb.
• , , , , , , , , ,
• ${\displaystyle {\tilde {E}}_{7}}$ 143 uniquely ringed forms, including:

### 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.

 ${\displaystyle {\bar {P}}_{7}}$ = [3,3[7]]: ${\displaystyle {\bar {Q}}_{7}}$ = [31,1,32,32,1]: ${\displaystyle {\bar {S}}_{7}}$ = [4,33,32,1]: ${\displaystyle {\bar {T}}_{7}}$ = [33,2,2]:

## References

1. Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topoplogy, Princeton, 2008.
• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
• H.S.M. Coxeter:
• H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
• H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
• Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
• (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
• (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
• (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• Klitzing, Richard. "8D uniform polytopes (polyzetta)".
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