Uniform 8-polytope

+ Graphs of three regular and related uniform polytopes.8-simplexRectified 8-simplexTruncated 8-simplexCantellated 8-simplexRuncinated 8-simplexStericated 8-simplexPentellated 8-simplexHexicated 8-simplexHeptellated 8-simplex8-orthoplexRectified 8-orthoplexTruncated 8-orthoplexCantellated 8-orthoplexRuncinated 8-orthoplexHexicated 8-orthoplexCantellated 8-cubeRuncinated 8-cubeStericated 8-cubePentellated 8-cubeHexicated 8-cubeHeptellated 8-cube8-cubeRectified 8-cubeTruncated 8-cube8-demicubeTruncated 8-demicubeCantellated 8-demicubeRuncinated 8-demicubeStericated 8-demicubePentellated 8-demicubeHexicated 8-demicube421142241 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 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: {3,3,3,3,3,3,3} - 8-simplex {4,3,3,3,3,3,3} - 8-cube {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.Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton, 2008.

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.

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.

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 groupForms1A8 1352BC8[4,36]2553D8[35,1,1]191 (64 unique)4E8[34,2,1]255 Selected regular and uniform 8-polytopes from each family include: Simplex family: A8 - 135 uniform 8-polytopes as permutations of rings in the group diagram, including one regular: {37} - 8-simplex or ennea-9-tope or enneazetton - Hypercube/orthoplex family: B8 [4,36] - 255 uniform 8-polytopes as permutations of rings in the group diagram, including two regular ones: {4,36} - 8-cube or octeract - {36,4} - 8-orthoplex or octacross - Demihypercube D8 family: [35,1,1] - 191 uniform 8-polytopes as permutations of rings in the group diagram, including: {3,35,1} - 8-demicube or demiocteract, 151 - ; also as h{4,36} - {3,3,3,3,3,31,1} - 8-orthoplex, 511 - E-polytope family E8 family: [34,1,1] - 255 uniform 8-polytopes as permutations of rings in the group diagram, including: {3,3,3,3,32,1} - Thorold Gosset's semiregular 421, {3,34,2} - the uniform 142, , {3,3,34,1} - the uniform 241,

Uniform prismatic forms There are many uniform prismatic families, including:

Uniform 8-polytope prism families#Coxeter groupCoxeter-Dynkin diagram7+11A7A1 [3,3,3,3,3,3]×[ ]2B7A1[4,3,3,3,3,3]×[ ]3D7A1[34,1,1]×[ ]4E7A1[33,2,1]×[ ]6+21A6I2(p) [3,3,3,3,3]×[p] 2B6I2(p) [4,3,3,3,3]×[p] 3D6I2(p) [33,1,1]×[p] 4E6I2(p) [3,3,3,3,3]×[p] 6+1+11A6A1A1 [3,3,3,3,3]×[ ]x[ ] 2B6A1A1 [4,3,3,3,3]×[ ]x[ ] 3D6A1A1 [33,1,1]×[ ]x[ ] 4E6A1A1 [3,3,3,3,3]×[ ]x[ ] 5+31A5A3 ×[3,3]2B5A3[4,33]×[3,3]3D5A3[32,1,1]×[3,3]4A5B3 ×[4,3]5B5B3[4,33]×[4,3]6D5B3[32,1,1]×[4,3]7A5H3 ×[5,3]8B5H3[4,33]×[5,3]9D5H3[32,1,1]×[5,3]5+2+11 A5I2(p)A1 [3,3,3]×[p]×[ ] 2 B5I2(p)A1 [4,3,3]×[p]×[ ] 3 D5I2(p)A1 [32,1,1]×[p]×[ ] 5+1+1+11 A5A1A1A1 [3,3,3]×[ ]×[ ]×[ ] 2 B5A1A1A1 [4,3,3]×[ ]×[ ]×[ ] 3 D5A1A1A1 [32,1,1]×[ ]×[ ]×[ ] 4+41A4A4[3,3,3]×[3,3,3]2 B4A4[4,3,3]×[3,3,3]3D4A4[31,1,1]×[3,3,3]4 F4A4[3,4,3]×[3,3,3]5 H4A4[5,3,3]×[3,3,3]6 B4B4[4,3,3]×[4,3,3]7D4B4[31,1,1]×[4,3,3]8 F4B4[3,4,3]×[4,3,3]9 H4B4[5,3,3]×[4,3,3]10D4D4[31,1,1]×[31,1,1]11 F4D4[3,4,3]×[31,1,1]12 H4D4[5,3,3]×[31,1,1]13 F4×F4[3,4,3]×[3,4,3]14 H4×F4[5,3,3]×[3,4,3]15 H4H4[5,3,3]×[5,3,3]4+3+11 A4A3A1 [3,3,3]×[3,3]×[ ] 2 A4B3A1 [3,3,3]×[4,3]×[ ] 3 A4H3A1 [3,3,3]×[5,3]×[ ] 4 B4A3A1 [4,3,3]×[3,3]×[ ] 5 B4B3A1 [4,3,3]×[4,3]×[ ] 6 B4H3A1 [4,3,3]×[5,3]×[ ] 7 H4A3A1 [5,3,3]×[3,3]×[ ] 8 H4B3A1 [5,3,3]×[4,3]×[ ] 9 H4H3A1 [5,3,3]×[5,3]×[ ] 10 F4A3A1 [3,4,3]×[3,3]×[ ] 11 F4B3A1 [3,4,3]×[4,3]×[ ] 12 F4H3A1 [3,4,3]×[5,3]×[ ] 13 D4A3A1 [31,1,1]×[3,3]×[ ] 14 D4B3A1 [31,1,1]×[4,3]×[ ] 15 D4H3A1 [31,1,1]×[5,3]×[ ] 4+2+2...4+2+1+1...4+1+1+1+1...3+3+21 A3A3I2(p)[3,3]×[3,3]×[p]2 B3A3I2(p)[4,3]×[3,3]×[p]3H3A3I2(p)[5,3]×[3,3]×[p]4 B3B3I2(p)[4,3]×[4,3]×[p]5H3B3I2(p)[5,3]×[4,3]×[p]6H3H3I2(p)[5,3]×[5,3]×[p]3+3+1+11 A32A12[3,3]×[3,3]×[ ]×[ ]2 B3A3A12[4,3]×[3,3]×[ ]×[ ]3H3A3A12[5,3]×[3,3]×[ ]×[ ]4 B3B3A12[4,3]×[4,3]×[ ]×[ ]5H3B3A12[5,3]×[4,3]×[ ]×[ ]6H3H3A12[5,3]×[5,3]×[ ]×[ ]3+2+2+11 A3I2(p)I2(q)A1 [3,3]×[p]×[q]×[ ] 2 B3I2(p)I2(q)A1 [4,3]×[p]×[q]×[ ] 3 H3I2(p)I2(q)A1 [5,3]×[p]×[q]×[ ] 3+2+1+1+11 A3I2(p)A13 [3,3]×[p]×[ ]x[ ]×[ ] 2 B3I2(p)A13 [4,3]×[p]×[ ]x[ ]×[ ] 3 H3I2(p)A13 [5,3]×[p]×[ ]x[ ]×[ ] 3+1+1+1+1+11 A3A15 [3,3]×[ ]x[ ]×[ ]x[ ]×[ ] 2 B3A15 [4,3]×[ ]x[ ]×[ ]x[ ]×[ ] 3 H3A15 [5,3]×[ ]x[ ]×[ ]x[ ]×[ ] 2+2+2+21 I2(p)I2(q)I2(r)I2(s) [p]×[q]×[r]×[s] 2+2+2+1+11 I2(p)I2(q)I2(r)A12 [p]×[q]×[r]×[ ]×[ ] 2+2+1+1+1+12 I2(p)I2(q)A14 [p]×[q]×[ ]×[ ]×[ ]×[ ] 2+1+1+1+1+1+11 I2(p)A16 [p]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ] 1+1+1+1+1+1+1+11 A18 [ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]

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.