# Uniform 8-polytope

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Polytope contained by 7-polytope facets

Graphs of three regular and related uniform polytopes. 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

In [eight-dimensional](/source/Eight-dimensional_space) [geometry](/source/Geometry), an **eight-dimensional polytope** or **8-polytope** is a [polytope](/source/Polytope) contained by 7-polytope facets, each [6-polytope](/source/6-polytope) [ridge](/source/Ridge_(geometry)) being shared by exactly two [7-polytope](/source/7-polytope) [facets](/source/Facet_(mathematics)).

A **uniform 8-polytope** is one which is [vertex-transitive](/source/Vertex-transitive), and constructed from [uniform 7-polytope](/source/Uniform_7-polytope) facets.

## Regular 8-polytopes

Regular 8-polytopes can be represented by the [Schläfli symbol](/source/Schl%C3%A4fli_symbol) {p,q,r,s,t,u,v}, with **v** {p,q,r,s,t,u} 7-polytope [facets](/source/Facet_(mathematics)) around each [peak](/source/Peak_(geometry)).

There are exactly three such [convex regular 8-polytopes](/source/List_of_regular_polytopes#Convex):

1. {3,3,3,3,3,3,3} - [8-simplex](/source/8-simplex)

1. {4,3,3,3,3,3,3} - [8-cube](/source/8-cube)

1. {3,3,3,3,3,3,4} - [8-orthoplex](/source/8-orthoplex)

There are no nonconvex regular 8-polytopes.

## Characteristics

The topology of any given 8-polytope is defined by its [Betti numbers](/source/Betti_number) and [torsion coefficients](/source/Torsion_coefficient_(topology)).[1]

The value of the [Euler characteristic](/source/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](/source/Coxeter-Dynkin_diagram):

# Coxeter group Forms 1 A8 [37] 135 2 BC8 [4,36] 255 3 D8 [35,1,1] 191 (64 unique) 4 E8 [34,2,1] 255

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

1. [Simplex](/source/Simplex) family: A8 [37] - - 135 uniform 8-polytopes as permutations of rings in the group diagram, including one regular: 1. {37} - [8-simplex](/source/8-simplex) or ennea-9-tope or enneazetton -

1. [Hypercube](/source/Hypercube)/[orthoplex](/source/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](/source/8-cube) or *octeract* - 1. {36,4} - [8-orthoplex](/source/8-orthoplex) or *octacross* -

1. [Demihypercube](/source/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](/source/8-demicube) or *demiocteract*, **151** - ; also as h{4,36} - 1. {3,3,3,3,3,31,1} - [8-orthoplex](/source/8-orthoplex), **511** -

1. [E-polytope family](/source/Semiregular_E-polytope) 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](/source/Thorold_Gosset)'s semiregular **[421](/source/Gosset_4_21_polytope)**, 1. {3,34,2} - the uniform **[142](/source/Gosset_1_42_polytope)**, , 1. {3,3,34,1} - the uniform **[241](/source/Gosset_2_41_polytope)**,

### Uniform prismatic forms

There are many [uniform](/source/Uniform_polytope) [prismatic](/source/Prismatic_polytope) families, including:

Uniform 8-polytope prism families # Coxeter group Coxeter-Dynkin diagram 7+1 1 A7A1 [3,3,3,3,3,3]×[ ] 2 B7A1 [4,3,3,3,3,3]×[ ] 3 D7A1 [34,1,1]×[ ] 4 E7A1 [33,2,1]×[ ] 6+2 1 A6I2(p) [3,3,3,3,3]×[p] 2 B6I2(p) [4,3,3,3,3]×[p] 3 D6I2(p) [33,1,1]×[p] 4 E6I2(p) [3,3,3,3,3]×[p] 6+1+1 1 A6A1A1 [3,3,3,3,3]×[ ]x[ ] 2 B6A1A1 [4,3,3,3,3]×[ ]x[ ] 3 D6A1A1 [33,1,1]×[ ]x[ ] 4 E6A1A1 [3,3,3,3,3]×[ ]x[ ] 5+3 1 A5A3 [34]×[3,3] 2 B5A3 [4,33]×[3,3] 3 D5A3 [32,1,1]×[3,3] 4 A5B3 [34]×[4,3] 5 B5B3 [4,33]×[4,3] 6 D5B3 [32,1,1]×[4,3] 7 A5H3 [34]×[5,3] 8 B5H3 [4,33]×[5,3] 9 D5H3 [32,1,1]×[5,3] 5+2+1 1 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+1 1 A5A1A1A1 [3,3,3]×[ ]×[ ]×[ ] 2 B5A1A1A1 [4,3,3]×[ ]×[ ]×[ ] 3 D5A1A1A1 [32,1,1]×[ ]×[ ]×[ ] 4+4 1 A4A4 [3,3,3]×[3,3,3] 2 B4A4 [4,3,3]×[3,3,3] 3 D4A4 [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] 7 D4B4 [31,1,1]×[4,3,3] 8 F4B4 [3,4,3]×[4,3,3] 9 H4B4 [5,3,3]×[4,3,3] 10 D4D4 [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+1 1 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+2 1 A3A3I2(p) [3,3]×[3,3]×[p] 2 B3A3I2(p) [4,3]×[3,3]×[p] 3 H3A3I2(p) [5,3]×[3,3]×[p] 4 B3B3I2(p) [4,3]×[4,3]×[p] 5 H3B3I2(p) [5,3]×[4,3]×[p] 6 H3H3I2(p) [5,3]×[5,3]×[p] 3+3+1+1 1 A32A12 [3,3]×[3,3]×[ ]×[ ] 2 B3A3A12 [4,3]×[3,3]×[ ]×[ ] 3 H3A3A12 [5,3]×[3,3]×[ ]×[ ] 4 B3B3A12 [4,3]×[4,3]×[ ]×[ ] 5 H3B3A12 [5,3]×[4,3]×[ ]×[ ] 6 H3H3A12 [5,3]×[5,3]×[ ]×[ ] 3+2+2+1 1 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+1 1 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+1 1 A3A15 [3,3]×[ ]x[ ]×[ ]x[ ]×[ ] 2 B3A15 [4,3]×[ ]x[ ]×[ ]x[ ]×[ ] 3 H3A15 [5,3]×[ ]x[ ]×[ ]x[ ]×[ ] 2+2+2+2 1 I2(p)I2(q)I2(r)I2(s) [p]×[q]×[r]×[s] 2+2+2+1+1 1 I2(p)I2(q)I2(r)A12 [p]×[q]×[r]×[ ]×[ ] 2+2+1+1+1+1 2 I2(p)I2(q)A14 [p]×[q]×[ ]×[ ]×[ ]×[ ] 2+1+1+1+1+1+1 1 I2(p)A16 [p]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ] 1+1+1+1+1+1+1+1 1 A18 [ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]

### The A8 family

The A8 family has symmetry of order 362880 (9 [factorial](/source/Factorial)).

There are 135 forms based on all permutations of the [Coxeter-Dynkin diagrams](/source/Coxeter-Dynkin_diagram) 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](/source/List_of_8-simplex_polytopes) for symmetric [Coxeter plane](/source/Coxeter_plane) graphs of these polytopes.

A8 uniform polytopes # Coxeter-Dynkin diagram Truncation indices Johnson name (acronym)[2] Basepoint Element counts 7 6 5 4 3 2 1 0 1 t0 8-simplex (ene) (0,0,0,0,0,0,0,0,1) 9 36 84 126 126 84 36 9 2 t1 Rectified 8-simplex (rene) (0,0,0,0,0,0,0,1,1) 18 108 336 630 576 588 252 36 3 t2 Birectified 8-simplex (brene) (0,0,0,0,0,0,1,1,1) 18 144 588 1386 2016 1764 756 84 4 t3 Trirectified 8-simplex (trene) (0,0,0,0,0,1,1,1,1) 1260 126 5 t0,1 Truncated 8-simplex (tene) (0,0,0,0,0,0,0,1,2) 288 72 6 t0,2 Cantellated 8-simplex (srene) (0,0,0,0,0,0,1,1,2) 1764 252 7 t1,2 Bitruncated 8-simplex (batene) (0,0,0,0,0,0,1,2,2) 1008 252 8 t0,3 Runcinated 8-simplex (spene) (0,0,0,0,0,1,1,1,2) 4536 504 9 t1,3 Bicantellated 8-simplex (sabrene) (0,0,0,0,0,1,1,2,2) 5292 756 10 t2,3 Tritruncated 8-simplex (tatene) (0,0,0,0,0,1,2,2,2) 2016 504 11 t0,4 Stericated 8-simplex (secane) (0,0,0,0,1,1,1,1,2) 6300 630 12 t1,4 Biruncinated 8-simplex (sabpene) (0,0,0,0,1,1,1,2,2) 11340 1260 13 t2,4 Tricantellated 8-simplex (satrene) (0,0,0,0,1,1,2,2,2) 8820 1260 14 t3,4 Quadritruncated 8-simplex (be) (0,0,0,0,1,2,2,2,2) 2520 630 15 t0,5 Pentellated 8-simplex (sotane) (0,0,0,1,1,1,1,1,2) 5040 504 16 t1,5 Bistericated 8-simplex (sobcane) (0,0,0,1,1,1,1,2,2) 12600 1260 17 t2,5 Triruncinated 8-simplex (satpeb) (0,0,0,1,1,1,2,2,2) 15120 1680 18 t0,6 Hexicated 8-simplex (supane) (0,0,1,1,1,1,1,1,2) 2268 252 19 t1,6 Bipentellated 8-simplex (sobteb) (0,0,1,1,1,1,1,2,2) 7560 756 20 t0,7 Heptellated 8-simplex (soxeb) (0,1,1,1,1,1,1,1,2) 504 72 21 t0,1,2 Cantitruncated 8-simplex (grene) (0,0,0,0,0,0,1,2,3) 2016 504 22 t0,1,3 Runcitruncated 8-simplex (potane) (0,0,0,0,0,1,1,2,3) 9828 1512 23 t0,2,3 Runcicantellated 8-simplex (prene) (0,0,0,0,0,1,2,2,3) 6804 1512 24 t1,2,3 Bicantitruncated 8-simplex (gabrene) (0,0,0,0,0,1,2,3,3) 6048 1512 25 t0,1,4 Steritruncated 8-simplex (catene) (0,0,0,0,1,1,1,2,3) 20160 2520 26 t0,2,4 Stericantellated 8-simplex (crane) (0,0,0,0,1,1,2,2,3) 26460 3780 27 t1,2,4 Biruncitruncated 8-simplex (biptene) (0,0,0,0,1,1,2,3,3) 22680 3780 28 t0,3,4 Steriruncinated 8-simplex (capene) (0,0,0,0,1,2,2,2,3) 12600 2520 29 t1,3,4 Biruncicantellated 8-simplex (biprene) (0,0,0,0,1,2,2,3,3) 18900 3780 30 t2,3,4 Tricantitruncated 8-simplex (gatrene) (0,0,0,0,1,2,3,3,3) 10080 2520 31 t0,1,5 Pentitruncated 8-simplex (tetane) (0,0,0,1,1,1,1,2,3) 21420 2520 32 t0,2,5 Penticantellated 8-simplex (turane) (0,0,0,1,1,1,2,2,3) 42840 5040 33 t1,2,5 Bisteritruncated 8-simplex (bictane) (0,0,0,1,1,1,2,3,3) 35280 5040 34 t0,3,5 Pentiruncinated 8-simplex (topene) (0,0,0,1,1,2,2,2,3) 37800 5040 35 t1,3,5 Bistericantellated 8-simplex (bocrane) (0,0,0,1,1,2,2,3,3) 52920 7560 36 t2,3,5 Triruncitruncated 8-simplex (toprane) (0,0,0,1,1,2,3,3,3) 27720 5040 37 t0,4,5 Pentistericated 8-simplex (tecane) (0,0,0,1,2,2,2,2,3) 13860 2520 38 t1,4,5 Bisteriruncinated 8-simplex (bacpane) (0,0,0,1,2,2,2,3,3) 30240 5040 39 t0,1,6 Hexitruncated 8-simplex (putene) (0,0,1,1,1,1,1,2,3) 12096 1512 40 t0,2,6 Hexicantellated 8-simplex (purene) (0,0,1,1,1,1,2,2,3) 34020 3780 41 t1,2,6 Bipentitruncated 8-simplex (bitotene) (0,0,1,1,1,1,2,3,3) 26460 3780 42 t0,3,6 Hexiruncinated 8-simplex (pupene) (0,0,1,1,1,2,2,2,3) 45360 5040 43 t1,3,6 Bipenticantellated 8-simplex (bitrene) (0,0,1,1,1,2,2,3,3) 60480 7560 44 t0,4,6 Hexistericated 8-simplex (pucane) (0,0,1,1,2,2,2,2,3) 30240 3780 45 t0,5,6 Hexipentellated 8-simplex (putane) (0,0,1,2,2,2,2,2,3) 9072 1512 46 t0,1,7 Heptitruncated 8-simplex (xotane) (0,1,1,1,1,1,1,2,3) 3276 504 47 t0,2,7 Hepticantellated 8-simplex (xorene)[3] (0,1,1,1,1,1,2,2,3) 12852 1512 48 t0,3,7 Heptiruncinated 8-simplex (xapane) (0,1,1,1,1,2,2,2,3) 23940 2520 49 t0,1,2,3 Runcicantitruncated 8-simplex (gapene) (0,0,0,0,0,1,2,3,4) 12096 3024 50 t0,1,2,4 Stericantitruncated 8-simplex (cograne) (0,0,0,0,1,1,2,3,4) 45360 7560 51 t0,1,3,4 Steriruncitruncated 8-simplex (coptane) (0,0,0,0,1,2,2,3,4) 34020 7560 52 t0,2,3,4 Steriruncicantellated 8-simplex (coprene) (0,0,0,0,1,2,3,3,4) 34020 7560 53 t1,2,3,4 Biruncicantitruncated 8-simplex (gabpene) (0,0,0,0,1,2,3,4,4) 30240 7560 54 t0,1,2,5 Penticantitruncated 8-simplex (tograne) (0,0,0,1,1,1,2,3,4) 70560 10080 55 t0,1,3,5 Pentiruncitruncated 8-simplex (taptane) (0,0,0,1,1,2,2,3,4) 98280 15120 56 t0,2,3,5 Pentiruncicantellated 8-simplex (taprene) (0,0,0,1,1,2,3,3,4) 90720 15120 57 t1,2,3,5 Bistericantitruncated 8-simplex (bocagrane) (0,0,0,1,1,2,3,4,4) 83160 15120 58 t0,1,4,5 Pentisteritruncated 8-simplex (tectane) (0,0,0,1,2,2,2,3,4) 50400 10080 59 t0,2,4,5 Pentistericantellated 8-simplex (tocrane) (0,0,0,1,2,2,3,3,4) 83160 15120 60 t1,2,4,5 Bisteriruncitruncated 8-simplex (bicpotane) (0,0,0,1,2,2,3,4,4) 68040 15120 61 t0,3,4,5 Pentisteriruncinated 8-simplex (tecpane) (0,0,0,1,2,3,3,3,4) 50400 10080 62 t1,3,4,5 Bisteriruncicantellated 8-simplex (bicprene) (0,0,0,1,2,3,3,4,4) 75600 15120 63 t2,3,4,5 Triruncicantitruncated 8-simplex (gatpeb) (0,0,0,1,2,3,4,4,4) 40320 10080 64 t0,1,2,6 Hexicantitruncated 8-simplex (pugrane) (0,0,1,1,1,1,2,3,4) 52920 7560 65 t0,1,3,6 Hexiruncitruncated 8-simplex (puptane) (0,0,1,1,1,2,2,3,4) 113400 15120 66 t0,2,3,6 Hexiruncicantellated 8-simplex (puprene) (0,0,1,1,1,2,3,3,4) 98280 15120 67 t1,2,3,6 Bipenticantitruncated 8-simplex (batograne) (0,0,1,1,1,2,3,4,4) 90720 15120 68 t0,1,4,6 Hexisteritruncated 8-simplex (puctane) (0,0,1,1,2,2,2,3,4) 105840 15120 69 t0,2,4,6 Hexistericantellated 8-simplex (pucrene) (0,0,1,1,2,2,3,3,4) 158760 22680 70 t1,2,4,6 Bipentiruncitruncated 8-simplex (batpitane) (0,0,1,1,2,2,3,4,4) 136080 22680 71 t0,3,4,6 Hexisteriruncinated 8-simplex (pocapine) (0,0,1,1,2,3,3,3,4) 90720 15120 72 t1,3,4,6 Bipentiruncicantellated 8-simplex (bitprop) (0,0,1,1,2,3,3,4,4) 136080 22680 73 t0,1,5,6 Hexipentitruncated 8-simplex (putatine) (0,0,1,2,2,2,2,3,4) 41580 7560 74 t0,2,5,6 Hexipenticantellated 8-simplex (putarene) (0,0,1,2,2,2,3,3,4) 98280 15120 75 t1,2,5,6 Bipentisteritruncated 8-simplex (batcotab) (0,0,1,2,2,2,3,4,4) 75600 15120 76 t0,3,5,6 Hexipentiruncinated 8-simplex (putapene) (0,0,1,2,2,3,3,3,4) 98280 15120 77 t0,4,5,6 Hexipentistericated 8-simplex (putacane) (0,0,1,2,3,3,3,3,4) 41580 7560 78 t0,1,2,7 Hepticantitruncated 8-simplex (xograne) (0,1,1,1,1,1,2,3,4) 18144 3024 79 t0,1,3,7 Heptiruncitruncated 8-simplex (xaptane) (0,1,1,1,1,2,2,3,4) 56700 7560 80 t0,2,3,7 Heptiruncicantellated 8-simplex (xeprane) (0,1,1,1,1,2,3,3,4) 45360 7560 81 t0,1,4,7 Heptisteritruncated 8-simplex (xactane) (0,1,1,1,2,2,2,3,4) 80640 10080 82 t0,2,4,7 Heptistericantellated 8-simplex (xacrene) (0,1,1,1,2,2,3,3,4) 113400 15120 83 t0,3,4,7 Heptisteriruncinated 8-simplex (xocapob) (0,1,1,1,2,3,3,3,4) 60480 10080 84 t0,1,5,7 Heptipentitruncated 8-simplex (xotatine) (0,1,1,2,2,2,2,3,4) 56700 7560 85 t0,2,5,7 Heptipenticantellated 8-simplex (xotrab) (0,1,1,2,2,2,3,3,4) 120960 15120 86 t0,1,6,7 Heptihexitruncated 8-simplex (xupatab) (0,1,2,2,2,2,2,3,4) 18144 3024 87 t0,1,2,3,4 Steriruncicantitruncated 8-simplex (gacene) (0,0,0,0,1,2,3,4,5) 60480 15120 88 t0,1,2,3,5 Pentiruncicantitruncated 8-simplex (togapene) (0,0,0,1,1,2,3,4,5) 166320 30240 89 t0,1,2,4,5 Pentistericantitruncated 8-simplex (tecograne) (0,0,0,1,2,2,3,4,5) 136080 30240 90 t0,1,3,4,5 Pentisteriruncitruncated 8-simplex (tecpatane) (0,0,0,1,2,3,3,4,5) 136080 30240 91 t0,2,3,4,5 Pentisteriruncicantellated 8-simplex (ticprane) (0,0,0,1,2,3,4,4,5) 136080 30240 92 t1,2,3,4,5 Bisteriruncicantitruncated 8-simplex (gobcane) (0,0,0,1,2,3,4,5,5) 120960 30240 93 t0,1,2,3,6 Hexiruncicantitruncated 8-simplex (pogapene) (0,0,1,1,1,2,3,4,5) 181440 30240 94 t0,1,2,4,6 Hexistericantitruncated 8-simplex (pocagrane) (0,0,1,1,2,2,3,4,5) 272160 45360 95 t0,1,3,4,6 Hexisteriruncitruncated 8-simplex (pocpatine) (0,0,1,1,2,3,3,4,5) 249480 45360 96 t0,2,3,4,6 Hexisteriruncicantellated 8-simplex (pocpurene) (0,0,1,1,2,3,4,4,5) 249480 45360 97 t1,2,3,4,6 Bipentiruncicantitruncated 8-simplex (botagpane) (0,0,1,1,2,3,4,5,5) 226800 45360 98 t0,1,2,5,6 Hexipenticantitruncated 8-simplex (potagrene) (0,0,1,2,2,2,3,4,5) 151200 30240 99 t0,1,3,5,6 Hexipentiruncitruncated 8-simplex (potaptane) (0,0,1,2,2,3,3,4,5) 249480 45360 100 t0,2,3,5,6 Hexipentiruncicantellated 8-simplex (putaprene) (0,0,1,2,2,3,4,4,5) 226800 45360 101 t1,2,3,5,6 Bipentistericantitruncated 8-simplex (betcagrane) (0,0,1,2,2,3,4,5,5) 204120 45360 102 t0,1,4,5,6 Hexipentisteritruncated 8-simplex (putcatine) (0,0,1,2,3,3,3,4,5) 151200 30240 103 t0,2,4,5,6 Hexipentistericantellated 8-simplex (potacrane) (0,0,1,2,3,3,4,4,5) 249480 45360 104 t0,3,4,5,6 Hexipentisteriruncinated 8-simplex (potcapane) (0,0,1,2,3,4,4,4,5) 151200 30240 105 t0,1,2,3,7 Heptiruncicantitruncated 8-simplex (xigpane) (0,1,1,1,1,2,3,4,5) 83160 15120 106 t0,1,2,4,7 Heptistericantitruncated 8-simplex (xecagrane) (0,1,1,1,2,2,3,4,5) 196560 30240 107 t0,1,3,4,7 Heptisteriruncitruncated 8-simplex (xucaptane) (0,1,1,1,2,3,3,4,5) 166320 30240 108 t0,2,3,4,7 Heptisteriruncicantellated 8-simplex (xecaprane) (0,1,1,1,2,3,4,4,5) 166320 30240 109 t0,1,2,5,7 Heptipenticantitruncated 8-simplex (xotagrane) (0,1,1,2,2,2,3,4,5) 196560 30240 110 t0,1,3,5,7 Heptipentiruncitruncated 8-simplex (xitaptene) (0,1,1,2,2,3,3,4,5) 294840 45360 111 t0,2,3,5,7 Heptipentiruncicantellated 8-simplex (xataprane) (0,1,1,2,2,3,4,4,5) 272160 45360 112 t0,1,4,5,7 Heptipentisteritruncated 8-simplex (xotcatene) (0,1,1,2,3,3,3,4,5) 166320 30240 113 t0,1,2,6,7 Heptihexicantitruncated 8-simplex (xopugrane) (0,1,2,2,2,2,3,4,5) 83160 15120 114 t0,1,3,6,7 Heptihexiruncitruncated 8-simplex (xopupatane) (0,1,2,2,2,3,3,4,5) 196560 30240 115 t0,1,2,3,4,5 Pentisteriruncicantitruncated 8-simplex (gotane) (0,0,0,1,2,3,4,5,6) 241920 60480 116 t0,1,2,3,4,6 Hexisteriruncicantitruncated 8-simplex (pogacane) (0,0,1,1,2,3,4,5,6) 453600 90720 117 t0,1,2,3,5,6 Hexipentiruncicantitruncated 8-simplex (potegpane) (0,0,1,2,2,3,4,5,6) 408240 90720 118 t0,1,2,4,5,6 Hexipentistericantitruncated 8-simplex (potacagrane) (0,0,1,2,3,3,4,5,6) 408240 90720 119 t0,1,3,4,5,6 Hexipentisteriruncitruncated 8-simplex (poticaptine) (0,0,1,2,3,4,4,5,6) 408240 90720 120 t0,2,3,4,5,6 Hexipentisteriruncicantellated 8-simplex (poticoprane) (0,0,1,2,3,4,5,5,6) 408240 90720 121 t1,2,3,4,5,6 Bipentisteriruncicantitruncated 8-simplex (gobteb) (0,0,1,2,3,4,5,6,6) 362880 90720 122 t0,1,2,3,4,7 Heptisteriruncicantitruncated 8-simplex (xogacane) (0,1,1,1,2,3,4,5,6) 302400 60480 123 t0,1,2,3,5,7 Heptipentiruncicantitruncated 8-simplex (xotagapane) (0,1,1,2,2,3,4,5,6) 498960 90720 124 t0,1,2,4,5,7 Heptipentistericantitruncated 8-simplex (xotcagrane) (0,1,1,2,3,3,4,5,6) 453600 90720 125 t0,1,3,4,5,7 Heptipentisteriruncitruncated 8-simplex (xotacaptane) (0,1,1,2,3,4,4,5,6) 453600 90720 126 t0,2,3,4,5,7 Heptipentisteriruncicantellated 8-simplex (xotacaparb) (0,1,1,2,3,4,5,5,6) 453600 90720 127 t0,1,2,3,6,7 Heptihexiruncicantitruncated 8-simplex (xupogapene) (0,1,2,2,2,3,4,5,6) 302400 60480 128 t0,1,2,4,6,7 Heptihexistericantitruncated 8-simplex (xupcagrene) (0,1,2,2,3,3,4,5,6) 498960 90720 129 t0,1,3,4,6,7 Heptihexisteriruncitruncated 8-simplex (xupacputob) (0,1,2,2,3,4,4,5,6) 453600 90720 130 t0,1,2,5,6,7 Heptihexipenticantitruncated 8-simplex (xuptagrab) (0,1,2,3,3,3,4,5,6) 302400 60480 131 t0,1,2,3,4,5,6 Hexipentisteriruncicantitruncated 8-simplex (gupane) (0,0,1,2,3,4,5,6,7) 725760 181440 132 t0,1,2,3,4,5,7 Heptipentisteriruncicantitruncated 8-simplex (xogtane) (0,1,1,2,3,4,5,6,7) 816480 181440 133 t0,1,2,3,4,6,7 Heptihexisteriruncicantitruncated 8-simplex (xupogacane) (0,1,2,2,3,4,5,6,7) 816480 181440 134 t0,1,2,3,5,6,7 Heptihexipentiruncicantitruncated 8-simplex (xuptagapene) (0,1,2,3,3,4,5,6,7) 816480 181440 135 t0,1,2,3,4,5,6,7 Omnitruncated 8-simplex (goxeb) (0,1,2,3,4,5,6,7,8) 1451520 362880

### The B8 family

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

See also a [list of B8 polytopes](/source/List_of_B8_polytopes) for symmetric [Coxeter plane](/source/Coxeter_plane) graphs of these polytopes.

B8 uniform polytopes # Coxeter-Dynkin diagram Schläfli symbol Name Element counts 7 6 5 4 3 2 1 0 1 t0{36,4} 8-orthoplex Diacosipentacontahexazetton (ek) 256 1024 1792 1792 1120 448 112 16 2 t1{36,4} Rectified 8-orthoplex Rectified diacosipentacontahexazetton (rek) 272 3072 8960 12544 10080 4928 1344 112 3 t2{36,4} Birectified 8-orthoplex Birectified diacosipentacontahexazetton (bark) 272 3184 16128 34048 36960 22400 6720 448 4 t3{36,4} Trirectified 8-orthoplex Trirectified diacosipentacontahexazetton (tark) 272 3184 16576 48384 71680 53760 17920 1120 5 t3{4,36} Trirectified 8-cube Trirectified octeract (tro) 272 3184 16576 47712 80640 71680 26880 1792 6 t2{4,36} Birectified 8-cube Birectified octeract (bro) 272 3184 14784 36960 55552 50176 21504 1792 7 t1{4,36} Rectified 8-cube Rectified octeract (recto) 272 2160 7616 15456 19712 16128 7168 1024 8 t0{4,36} 8-cube Octeract (octo) 16 112 448 1120 1792 1792 1024 256 9 t0,1{36,4} Truncated 8-orthoplex Truncated diacosipentacontahexazetton (tek) 1456 224 10 t0,2{36,4} Cantellated 8-orthoplex[4] Small rhombated diacosipentacontahexazetton (srek) 14784 1344 11 t1,2{36,4} Bitruncated 8-orthoplex Bitruncated diacosipentacontahexazetton (batek) 8064 1344 12 t0,3{36,4} Runcinated 8-orthoplex Small prismated diacosipentacontahexazetton (spek) 60480 4480 13 t1,3{36,4} Bicantellated 8-orthoplex Small birhombated diacosipentacontahexazetton (sabork) 67200 6720 14 t2,3{36,4} Tritruncated 8-orthoplex Tritruncated diacosipentacontahexazetton (tatek) 24640 4480 15 t0,4{36,4} Stericated 8-orthoplex Small cellated diacosipentacontahexazetton (scak) 125440 8960 16 t1,4{36,4} Biruncinated 8-orthoplex Small biprismated diacosipentacontahexazetton (sabpek) 215040 17920 17 t2,4{36,4} Tricantellated 8-orthoplex Small trirhombated diacosipentacontahexazetton (satrek) 161280 17920 18 t3,4{4,36} Quadritruncated 8-cube Octeractidiacosipentacontahexazetton (oke) 44800 8960 19 t0,5{36,4} Pentellated 8-orthoplex Small terated diacosipentacontahexazetton (setek) 134400 10752 20 t1,5{36,4} Bistericated 8-orthoplex Small bicellated diacosipentacontahexazetton (sibcak) 322560 26880 21 t2,5{4,36} Triruncinated 8-cube Small triprismato-octeractidiacosipentacontahexazetton (sitpoke) 376320 35840 22 t2,4{4,36} Tricantellated 8-cube Small trirhombated octeract (satro) 215040 26880 23 t2,3{4,36} Tritruncated 8-cube Tritruncated octeract (tato) 48384 10752 24 t0,6{36,4} Hexicated 8-orthoplex Small petated diacosipentacontahexazetton (supek) 64512 7168 25 t1,6{4,36} Bipentellated 8-cube Small biteri-octeractidiacosipentacontahexazetton (sabtoke) 215040 21504 26 t1,5{4,36} Bistericated 8-cube Small bicellated octeract (sobco) 358400 35840 27 t1,4{4,36} Biruncinated 8-cube Small biprismated octeract (sabepo) 322560 35840 28 t1,3{4,36} Bicantellated 8-cube Small birhombated octeract (subro) 150528 21504 29 t1,2{4,36} Bitruncated 8-cube Bitruncated octeract (bato) 28672 7168 30 t0,7{4,36} Heptellated 8-cube Small exi-octeractidiacosipentacontahexazetton (saxoke) 14336 2048 31 t0,6{4,36} Hexicated 8-cube Small petated octeract (supo) 64512 7168 32 t0,5{4,36} Pentellated 8-cube Small terated octeract (soto) 143360 14336 33 t0,4{4,36} Stericated 8-cube Small cellated octeract (soco) 179200 17920 34 t0,3{4,36} Runcinated 8-cube Small prismated octeract (sopo) 129024 14336 35 t0,2{4,36} Cantellated 8-cube Small rhombated octeract (soro) 50176 7168 36 t0,1{4,36} Truncated 8-cube Truncated octeract (tocto) 8192 2048 37 t0,1,2{36,4} Cantitruncated 8-orthoplex Great rhombated diacosipentacontahexazetton 16128 2688 38 t0,1,3{36,4} Runcitruncated 8-orthoplex Prismatotruncated diacosipentacontahexazetton 127680 13440 39 t0,2,3{36,4} Runcicantellated 8-orthoplex Prismatorhombated diacosipentacontahexazetton 80640 13440 40 t1,2,3{36,4} Bicantitruncated 8-orthoplex Great birhombated diacosipentacontahexazetton 73920 13440 41 t0,1,4{36,4} Steritruncated 8-orthoplex Cellitruncated diacosipentacontahexazetton 394240 35840 42 t0,2,4{36,4} Stericantellated 8-orthoplex Cellirhombated diacosipentacontahexazetton 483840 53760 43 t1,2,4{36,4} Biruncitruncated 8-orthoplex Biprismatotruncated diacosipentacontahexazetton 430080 53760 44 t0,3,4{36,4} Steriruncinated 8-orthoplex Celliprismated diacosipentacontahexazetton 215040 35840 45 t1,3,4{36,4} Biruncicantellated 8-orthoplex Biprismatorhombated diacosipentacontahexazetton 322560 53760 46 t2,3,4{36,4} Tricantitruncated 8-orthoplex Great trirhombated diacosipentacontahexazetton 179200 35840 47 t0,1,5{36,4} Pentitruncated 8-orthoplex Teritruncated diacosipentacontahexazetton 564480 53760 48 t0,2,5{36,4} Penticantellated 8-orthoplex Terirhombated diacosipentacontahexazetton 1075200 107520 49 t1,2,5{36,4} Bisteritruncated 8-orthoplex Bicellitruncated diacosipentacontahexazetton 913920 107520 50 t0,3,5{36,4} Pentiruncinated 8-orthoplex Teriprismated diacosipentacontahexazetton 913920 107520 51 t1,3,5{36,4} Bistericantellated 8-orthoplex Bicellirhombated diacosipentacontahexazetton 1290240 161280 52 t2,3,5{36,4} Triruncitruncated 8-orthoplex Triprismatotruncated diacosipentacontahexazetton 698880 107520 53 t0,4,5{36,4} Pentistericated 8-orthoplex Tericellated diacosipentacontahexazetton 322560 53760 54 t1,4,5{36,4} Bisteriruncinated 8-orthoplex Bicelliprismated diacosipentacontahexazetton 698880 107520 55 t2,3,5{4,36} Triruncitruncated 8-cube Triprismatotruncated octeract 645120 107520 56 t2,3,4{4,36} Tricantitruncated 8-cube Great trirhombated octeract 241920 53760 57 t0,1,6{36,4} Hexitruncated 8-orthoplex Petitruncated diacosipentacontahexazetton 344064 43008 58 t0,2,6{36,4} Hexicantellated 8-orthoplex Petirhombated diacosipentacontahexazetton 967680 107520 59 t1,2,6{36,4} Bipentitruncated 8-orthoplex Biteritruncated diacosipentacontahexazetton 752640 107520 60 t0,3,6{36,4} Hexiruncinated 8-orthoplex Petiprismated diacosipentacontahexazetton 1290240 143360 61 t1,3,6{36,4} Bipenticantellated 8-orthoplex Biterirhombated diacosipentacontahexazetton 1720320 215040 62 t1,4,5{4,36} Bisteriruncinated 8-cube Bicelliprismated octeract 860160 143360 63 t0,4,6{36,4} Hexistericated 8-orthoplex Peticellated diacosipentacontahexazetton 860160 107520 64 t1,3,6{4,36} Bipenticantellated 8-cube Biterirhombated octeract 1720320 215040 65 t1,3,5{4,36} Bistericantellated 8-cube Bicellirhombated octeract 1505280 215040 66 t1,3,4{4,36} Biruncicantellated 8-cube Biprismatorhombated octeract 537600 107520 67 t0,5,6{36,4} Hexipentellated 8-orthoplex Petiterated diacosipentacontahexazetton 258048 43008 68 t1,2,6{4,36} Bipentitruncated 8-cube Biteritruncated octeract 752640 107520 69 t1,2,5{4,36} Bisteritruncated 8-cube Bicellitruncated octeract 1003520 143360 70 t1,2,4{4,36} Biruncitruncated 8-cube Biprismatotruncated octeract 645120 107520 71 t1,2,3{4,36} Bicantitruncated 8-cube Great birhombated octeract 172032 43008 72 t0,1,7{36,4} Heptitruncated 8-orthoplex Exitruncated diacosipentacontahexazetton 93184 14336 73 t0,2,7{36,4} Hepticantellated 8-orthoplex Exirhombated diacosipentacontahexazetton 365568 43008 74 t0,5,6{4,36} Hexipentellated 8-cube Petiterated octeract 258048 43008 75 t0,3,7{36,4} Heptiruncinated 8-orthoplex Exiprismated diacosipentacontahexazetton 680960 71680 76 t0,4,6{4,36} Hexistericated 8-cube Peticellated octeract 860160 107520 77 t0,4,5{4,36} Pentistericated 8-cube Tericellated octeract 394240 71680 78 t0,3,7{4,36} Heptiruncinated 8-cube Exiprismated octeract 680960 71680 79 t0,3,6{4,36} Hexiruncinated 8-cube Petiprismated octeract 1290240 143360 80 t0,3,5{4,36} Pentiruncinated 8-cube Teriprismated octeract 1075200 143360 81 t0,3,4{4,36} Steriruncinated 8-cube Celliprismated octeract 358400 71680 82 t0,2,7{4,36} Hepticantellated 8-cube Exirhombated octeract 365568 43008 83 t0,2,6{4,36} Hexicantellated 8-cube Petirhombated octeract 967680 107520 84 t0,2,5{4,36} Penticantellated 8-cube Terirhombated octeract 1218560 143360 85 t0,2,4{4,36} Stericantellated 8-cube Cellirhombated octeract 752640 107520 86 t0,2,3{4,36} Runcicantellated 8-cube Prismatorhombated octeract 193536 43008 87 t0,1,7{4,36} Heptitruncated 8-cube Exitruncated octeract 93184 14336 88 t0,1,6{4,36} Hexitruncated 8-cube Petitruncated octeract 344064 43008 89 t0,1,5{4,36} Pentitruncated 8-cube Teritruncated octeract 609280 71680 90 t0,1,4{4,36} Steritruncated 8-cube Cellitruncated octeract 573440 71680 91 t0,1,3{4,36} Runcitruncated 8-cube Prismatotruncated octeract 279552 43008 92 t0,1,2{4,36} Cantitruncated 8-cube Great rhombated octeract 57344 14336 93 t0,1,2,3{36,4} Runcicantitruncated 8-orthoplex Great prismated diacosipentacontahexazetton 147840 26880 94 t0,1,2,4{36,4} Stericantitruncated 8-orthoplex Celligreatorhombated diacosipentacontahexazetton 860160 107520 95 t0,1,3,4{36,4} Steriruncitruncated 8-orthoplex Celliprismatotruncated diacosipentacontahexazetton 591360 107520 96 t0,2,3,4{36,4} Steriruncicantellated 8-orthoplex Celliprismatorhombated diacosipentacontahexazetton 591360 107520 97 t1,2,3,4{36,4} Biruncicantitruncated 8-orthoplex Great biprismated diacosipentacontahexazetton 537600 107520 98 t0,1,2,5{36,4} Penticantitruncated 8-orthoplex Terigreatorhombated diacosipentacontahexazetton 1827840 215040 99 t0,1,3,5{36,4} Pentiruncitruncated 8-orthoplex Teriprismatotruncated diacosipentacontahexazetton 2419200 322560 100 t0,2,3,5{36,4} Pentiruncicantellated 8-orthoplex Teriprismatorhombated diacosipentacontahexazetton 2257920 322560 101 t1,2,3,5{36,4} Bistericantitruncated 8-orthoplex Bicelligreatorhombated diacosipentacontahexazetton 2096640 322560 102 t0,1,4,5{36,4} Pentisteritruncated 8-orthoplex Tericellitruncated diacosipentacontahexazetton 1182720 215040 103 t0,2,4,5{36,4} Pentistericantellated 8-orthoplex Tericellirhombated diacosipentacontahexazetton 1935360 322560 104 t1,2,4,5{36,4} Bisteriruncitruncated 8-orthoplex Bicelliprismatotruncated diacosipentacontahexazetton 1612800 322560 105 t0,3,4,5{36,4} Pentisteriruncinated 8-orthoplex Tericelliprismated diacosipentacontahexazetton 1182720 215040 106 t1,3,4,5{36,4} Bisteriruncicantellated 8-orthoplex Bicelliprismatorhombated diacosipentacontahexazetton 1774080 322560 107 t2,3,4,5{4,36} Triruncicantitruncated 8-cube Great triprismato-octeractidiacosipentacontahexazetton 967680 215040 108 t0,1,2,6{36,4} Hexicantitruncated 8-orthoplex Petigreatorhombated diacosipentacontahexazetton 1505280 215040 109 t0,1,3,6{36,4} Hexiruncitruncated 8-orthoplex Petiprismatotruncated diacosipentacontahexazetton 3225600 430080 110 t0,2,3,6{36,4} Hexiruncicantellated 8-orthoplex Petiprismatorhombated diacosipentacontahexazetton 2795520 430080 111 t1,2,3,6{36,4} Bipenticantitruncated 8-orthoplex Biterigreatorhombated diacosipentacontahexazetton 2580480 430080 112 t0,1,4,6{36,4} Hexisteritruncated 8-orthoplex Peticellitruncated diacosipentacontahexazetton 3010560 430080 113 t0,2,4,6{36,4} Hexistericantellated 8-orthoplex Peticellirhombated diacosipentacontahexazetton 4515840 645120 114 t1,2,4,6{36,4} Bipentiruncitruncated 8-orthoplex Biteriprismatotruncated diacosipentacontahexazetton 3870720 645120 115 t0,3,4,6{36,4} Hexisteriruncinated 8-orthoplex Peticelliprismated diacosipentacontahexazetton 2580480 430080 116 t1,3,4,6{4,36} Bipentiruncicantellated 8-cube Biteriprismatorhombi-octeractidiacosipentacontahexazetton 3870720 645120 117 t1,3,4,5{4,36} Bisteriruncicantellated 8-cube Bicelliprismatorhombated octeract 2150400 430080 118 t0,1,5,6{36,4} Hexipentitruncated 8-orthoplex Petiteritruncated diacosipentacontahexazetton 1182720 215040 119 t0,2,5,6{36,4} Hexipenticantellated 8-orthoplex Petiterirhombated diacosipentacontahexazetton 2795520 430080 120 t1,2,5,6{4,36} Bipentisteritruncated 8-cube Bitericellitrunki-octeractidiacosipentacontahexazetton 2150400 430080 121 t0,3,5,6{36,4} Hexipentiruncinated 8-orthoplex Petiteriprismated diacosipentacontahexazetton 2795520 430080 122 t1,2,4,6{4,36} Bipentiruncitruncated 8-cube Biteriprismatotruncated octeract 3870720 645120 123 t1,2,4,5{4,36} Bisteriruncitruncated 8-cube Bicelliprismatotruncated octeract 1935360 430080 124 t0,4,5,6{36,4} Hexipentistericated 8-orthoplex Petitericellated diacosipentacontahexazetton 1182720 215040 125 t1,2,3,6{4,36} Bipenticantitruncated 8-cube Biterigreatorhombated octeract 2580480 430080 126 t1,2,3,5{4,36} Bistericantitruncated 8-cube Bicelligreatorhombated octeract 2365440 430080 127 t1,2,3,4{4,36} Biruncicantitruncated 8-cube Great biprismated octeract 860160 215040 128 t0,1,2,7{36,4} Hepticantitruncated 8-orthoplex Exigreatorhombated diacosipentacontahexazetton 516096 86016 129 t0,1,3,7{36,4} Heptiruncitruncated 8-orthoplex Exiprismatotruncated diacosipentacontahexazetton 1612800 215040 130 t0,2,3,7{36,4} Heptiruncicantellated 8-orthoplex Exiprismatorhombated diacosipentacontahexazetton 1290240 215040 131 t0,4,5,6{4,36} Hexipentistericated 8-cube Petitericellated octeract 1182720 215040 132 t0,1,4,7{36,4} Heptisteritruncated 8-orthoplex Exicellitruncated diacosipentacontahexazetton 2293760 286720 133 t0,2,4,7{36,4} Heptistericantellated 8-orthoplex Exicellirhombated diacosipentacontahexazetton 3225600 430080 134 t0,3,5,6{4,36} Hexipentiruncinated 8-cube Petiteriprismated octeract 2795520 430080 135 t0,3,4,7{4,36} Heptisteriruncinated 8-cube Exicelliprismato-octeractidiacosipentacontahexazetton 1720320 286720 136 t0,3,4,6{4,36} Hexisteriruncinated 8-cube Peticelliprismated octeract 2580480 430080 137 t0,3,4,5{4,36} Pentisteriruncinated 8-cube Tericelliprismated octeract 1433600 286720 138 t0,1,5,7{36,4} Heptipentitruncated 8-orthoplex Exiteritruncated diacosipentacontahexazetton 1612800 215040 139 t0,2,5,7{4,36} Heptipenticantellated 8-cube Exiterirhombi-octeractidiacosipentacontahexazetton 3440640 430080 140 t0,2,5,6{4,36} Hexipenticantellated 8-cube Petiterirhombated octeract 2795520 430080 141 t0,2,4,7{4,36} Heptistericantellated 8-cube Exicellirhombated octeract 3225600 430080 142 t0,2,4,6{4,36} Hexistericantellated 8-cube Peticellirhombated octeract 4515840 645120 143 t0,2,4,5{4,36} Pentistericantellated 8-cube Tericellirhombated octeract 2365440 430080 144 t0,2,3,7{4,36} Heptiruncicantellated 8-cube Exiprismatorhombated octeract 1290240 215040 145 t0,2,3,6{4,36} Hexiruncicantellated 8-cube Petiprismatorhombated octeract 2795520 430080 146 t0,2,3,5{4,36} Pentiruncicantellated 8-cube Teriprismatorhombated octeract 2580480 430080 147 t0,2,3,4{4,36} Steriruncicantellated 8-cube Celliprismatorhombated octeract 967680 215040 148 t0,1,6,7{4,36} Heptihexitruncated 8-cube Exipetitrunki-octeractidiacosipentacontahexazetton 516096 86016 149 t0,1,5,7{4,36} Heptipentitruncated 8-cube Exiteritruncated octeract 1612800 215040 150 t0,1,5,6{4,36} Hexipentitruncated 8-cube Petiteritruncated octeract 1182720 215040 151 t0,1,4,7{4,36} Heptisteritruncated 8-cube Exicellitruncated octeract 2293760 286720 152 t0,1,4,6{4,36} Hexisteritruncated 8-cube Peticellitruncated octeract 3010560 430080 153 t0,1,4,5{4,36} Pentisteritruncated 8-cube Tericellitruncated octeract 1433600 286720 154 t0,1,3,7{4,36} Heptiruncitruncated 8-cube Exiprismatotruncated octeract 1612800 215040 155 t0,1,3,6{4,36} Hexiruncitruncated 8-cube Petiprismatotruncated octeract 3225600 430080 156 t0,1,3,5{4,36} Pentiruncitruncated 8-cube Teriprismatotruncated octeract 2795520 430080 157 t0,1,3,4{4,36} Steriruncitruncated 8-cube Celliprismatotruncated octeract 967680 215040 158 t0,1,2,7{4,36} Hepticantitruncated 8-cube Exigreatorhombated octeract 516096 86016 159 t0,1,2,6{4,36} Hexicantitruncated 8-cube Petigreatorhombated octeract 1505280 215040 160 t0,1,2,5{4,36} Penticantitruncated 8-cube Terigreatorhombated octeract 2007040 286720 161 t0,1,2,4{4,36} Stericantitruncated 8-cube Celligreatorhombated octeract 1290240 215040 162 t0,1,2,3{4,36} Runcicantitruncated 8-cube Great prismated octeract 344064 86016 163 t0,1,2,3,4{36,4} Steriruncicantitruncated 8-orthoplex Great cellated diacosipentacontahexazetton 1075200 215040 164 t0,1,2,3,5{36,4} Pentiruncicantitruncated 8-orthoplex Terigreatoprismated diacosipentacontahexazetton 4193280 645120 165 t0,1,2,4,5{36,4} Pentistericantitruncated 8-orthoplex Tericelligreatorhombated diacosipentacontahexazetton 3225600 645120 166 t0,1,3,4,5{36,4} Pentisteriruncitruncated 8-orthoplex Tericelliprismatotruncated diacosipentacontahexazetton 3225600 645120 167 t0,2,3,4,5{36,4} Pentisteriruncicantellated 8-orthoplex Tericelliprismatorhombated diacosipentacontahexazetton 3225600 645120 168 t1,2,3,4,5{36,4} Bisteriruncicantitruncated 8-orthoplex Great bicellated diacosipentacontahexazetton 2903040 645120 169 t0,1,2,3,6{36,4} Hexiruncicantitruncated 8-orthoplex Petigreatoprismated diacosipentacontahexazetton 5160960 860160 170 t0,1,2,4,6{36,4} Hexistericantitruncated 8-orthoplex Peticelligreatorhombated diacosipentacontahexazetton 7741440 1290240 171 t0,1,3,4,6{36,4} Hexisteriruncitruncated 8-orthoplex Peticelliprismatotruncated diacosipentacontahexazetton 7096320 1290240 172 t0,2,3,4,6{36,4} Hexisteriruncicantellated 8-orthoplex Peticelliprismatorhombated diacosipentacontahexazetton 7096320 1290240 173 t1,2,3,4,6{36,4} Bipentiruncicantitruncated 8-orthoplex Biterigreatoprismated diacosipentacontahexazetton 6451200 1290240 174 t0,1,2,5,6{36,4} Hexipenticantitruncated 8-orthoplex Petiterigreatorhombated diacosipentacontahexazetton 4300800 860160 175 t0,1,3,5,6{36,4} Hexipentiruncitruncated 8-orthoplex Petiteriprismatotruncated diacosipentacontahexazetton 7096320 1290240 176 t0,2,3,5,6{36,4} Hexipentiruncicantellated 8-orthoplex Petiteriprismatorhombated diacosipentacontahexazetton 6451200 1290240 177 t1,2,3,5,6{36,4} Bipentistericantitruncated 8-orthoplex Bitericelligreatorhombated diacosipentacontahexazetton 5806080 1290240 178 t0,1,4,5,6{36,4} Hexipentisteritruncated 8-orthoplex Petitericellitruncated diacosipentacontahexazetton 4300800 860160 179 t0,2,4,5,6{36,4} Hexipentistericantellated 8-orthoplex Petitericellirhombated diacosipentacontahexazetton 7096320 1290240 180 t1,2,3,5,6{4,36} Bipentistericantitruncated 8-cube Bitericelligreatorhombated octeract 5806080 1290240 181 t0,3,4,5,6{36,4} Hexipentisteriruncinated 8-orthoplex Petitericelliprismated diacosipentacontahexazetton 4300800 860160 182 t1,2,3,4,6{4,36} Bipentiruncicantitruncated 8-cube Biterigreatoprismated octeract 6451200 1290240 183 t1,2,3,4,5{4,36} Bisteriruncicantitruncated 8-cube Great bicellated octeract 3440640 860160 184 t0,1,2,3,7{36,4} Heptiruncicantitruncated 8-orthoplex Exigreatoprismated diacosipentacontahexazetton 2365440 430080 185 t0,1,2,4,7{36,4} Heptistericantitruncated 8-orthoplex Exicelligreatorhombated diacosipentacontahexazetton 5591040 860160 186 t0,1,3,4,7{36,4} Heptisteriruncitruncated 8-orthoplex Exicelliprismatotruncated diacosipentacontahexazetton 4730880 860160 187 t0,2,3,4,7{36,4} Heptisteriruncicantellated 8-orthoplex Exicelliprismatorhombated diacosipentacontahexazetton 4730880 860160 188 t0,3,4,5,6{4,36} Hexipentisteriruncinated 8-cube Petitericelliprismated octeract 4300800 860160 189 t0,1,2,5,7{36,4} Heptipenticantitruncated 8-orthoplex Exiterigreatorhombated diacosipentacontahexazetton 5591040 860160 190 t0,1,3,5,7{36,4} Heptipentiruncitruncated 8-orthoplex Exiteriprismatotruncated diacosipentacontahexazetton 8386560 1290240 191 t0,2,3,5,7{36,4} Heptipentiruncicantellated 8-orthoplex Exiteriprismatorhombated diacosipentacontahexazetton 7741440 1290240 192 t0,2,4,5,6{4,36} Hexipentistericantellated 8-cube Petitericellirhombated octeract 7096320 1290240 193 t0,1,4,5,7{36,4} Heptipentisteritruncated 8-orthoplex Exitericellitruncated diacosipentacontahexazetton 4730880 860160 194 t0,2,3,5,7{4,36} Heptipentiruncicantellated 8-cube Exiteriprismatorhombated octeract 7741440 1290240 195 t0,2,3,5,6{4,36} Hexipentiruncicantellated 8-cube Petiteriprismatorhombated octeract 6451200 1290240 196 t0,2,3,4,7{4,36} Heptisteriruncicantellated 8-cube Exicelliprismatorhombated octeract 4730880 860160 197 t0,2,3,4,6{4,36} Hexisteriruncicantellated 8-cube Peticelliprismatorhombated octeract 7096320 1290240 198 t0,2,3,4,5{4,36} Pentisteriruncicantellated 8-cube Tericelliprismatorhombated octeract 3870720 860160 199 t0,1,2,6,7{36,4} Heptihexicantitruncated 8-orthoplex Exipetigreatorhombated diacosipentacontahexazetton 2365440 430080 200 t0,1,3,6,7{36,4} Heptihexiruncitruncated 8-orthoplex Exipetiprismatotruncated diacosipentacontahexazetton 5591040 860160 201 t0,1,4,5,7{4,36} Heptipentisteritruncated 8-cube Exitericellitruncated octeract 4730880 860160 202 t0,1,4,5,6{4,36} Hexipentisteritruncated 8-cube Petitericellitruncated octeract 4300800 860160 203 t0,1,3,6,7{4,36} Heptihexiruncitruncated 8-cube Exipetiprismatotruncated octeract 5591040 860160 204 t0,1,3,5,7{4,36} Heptipentiruncitruncated 8-cube Exiteriprismatotruncated octeract 8386560 1290240 205 t0,1,3,5,6{4,36} Hexipentiruncitruncated 8-cube Petiteriprismatotruncated octeract 7096320 1290240 206 t0,1,3,4,7{4,36} Heptisteriruncitruncated 8-cube Exicelliprismatotruncated octeract 4730880 860160 207 t0,1,3,4,6{4,36} Hexisteriruncitruncated 8-cube Peticelliprismatotruncated octeract 7096320 1290240 208 t0,1,3,4,5{4,36} Pentisteriruncitruncated 8-cube Tericelliprismatotruncated octeract 3870720 860160 209 t0,1,2,6,7{4,36} Heptihexicantitruncated 8-cube Exipetigreatorhombated octeract 2365440 430080 210 t0,1,2,5,7{4,36} Heptipenticantitruncated 8-cube Exiterigreatorhombated octeract 5591040 860160 211 t0,1,2,5,6{4,36} Hexipenticantitruncated 8-cube Petiterigreatorhombated octeract 4300800 860160 212 t0,1,2,4,7{4,36} Heptistericantitruncated 8-cube Exicelligreatorhombated octeract 5591040 860160 213 t0,1,2,4,6{4,36} Hexistericantitruncated 8-cube Peticelligreatorhombated octeract 7741440 1290240 214 t0,1,2,4,5{4,36} Pentistericantitruncated 8-cube Tericelligreatorhombated octeract 3870720 860160 215 t0,1,2,3,7{4,36} Heptiruncicantitruncated 8-cube Exigreatoprismated octeract 2365440 430080 216 t0,1,2,3,6{4,36} Hexiruncicantitruncated 8-cube Petigreatoprismated octeract 5160960 860160 217 t0,1,2,3,5{4,36} Pentiruncicantitruncated 8-cube Terigreatoprismated octeract 4730880 860160 218 t0,1,2,3,4{4,36} Steriruncicantitruncated 8-cube Great cellated octeract 1720320 430080 219 t0,1,2,3,4,5{36,4} Pentisteriruncicantitruncated 8-orthoplex Great terated diacosipentacontahexazetton 5806080 1290240 220 t0,1,2,3,4,6{36,4} Hexisteriruncicantitruncated 8-orthoplex Petigreatocellated diacosipentacontahexazetton 12902400 2580480 221 t0,1,2,3,5,6{36,4} Hexipentiruncicantitruncated 8-orthoplex Petiterigreatoprismated diacosipentacontahexazetton 11612160 2580480 222 t0,1,2,4,5,6{36,4} Hexipentistericantitruncated 8-orthoplex Petitericelligreatorhombated diacosipentacontahexazetton 11612160 2580480 223 t0,1,3,4,5,6{36,4} Hexipentisteriruncitruncated 8-orthoplex Petitericelliprismatotruncated diacosipentacontahexazetton 11612160 2580480 224 t0,2,3,4,5,6{36,4} Hexipentisteriruncicantellated 8-orthoplex Petitericelliprismatorhombated diacosipentacontahexazetton 11612160 2580480 225 t1,2,3,4,5,6{4,36} Bipentisteriruncicantitruncated 8-cube Great biteri-octeractidiacosipentacontahexazetton 10321920 2580480 226 t0,1,2,3,4,7{36,4} Heptisteriruncicantitruncated 8-orthoplex Exigreatocellated diacosipentacontahexazetton 8601600 1720320 227 t0,1,2,3,5,7{36,4} Heptipentiruncicantitruncated 8-orthoplex Exiterigreatoprismated diacosipentacontahexazetton 14192640 2580480 228 t0,1,2,4,5,7{36,4} Heptipentistericantitruncated 8-orthoplex Exitericelligreatorhombated diacosipentacontahexazetton 12902400 2580480 229 t0,1,3,4,5,7{36,4} Heptipentisteriruncitruncated 8-orthoplex Exitericelliprismatotruncated diacosipentacontahexazetton 12902400 2580480 230 t0,2,3,4,5,7{4,36} Heptipentisteriruncicantellated 8-cube Exitericelliprismatorhombi-octeractidiacosipentacontahexazetton 12902400 2580480 231 t0,2,3,4,5,6{4,36} Hexipentisteriruncicantellated 8-cube Petitericelliprismatorhombated octeract 11612160 2580480 232 t0,1,2,3,6,7{36,4} Heptihexiruncicantitruncated 8-orthoplex Exipetigreatoprismated diacosipentacontahexazetton 8601600 1720320 233 t0,1,2,4,6,7{36,4} Heptihexistericantitruncated 8-orthoplex Exipeticelligreatorhombated diacosipentacontahexazetton 14192640 2580480 234 t0,1,3,4,6,7{4,36} Heptihexisteriruncitruncated 8-cube Exipeticelliprismatotrunki-octeractidiacosipentacontahexazetton 12902400 2580480 235 t0,1,3,4,5,7{4,36} Heptipentisteriruncitruncated 8-cube Exitericelliprismatotruncated octeract 12902400 2580480 236 t0,1,3,4,5,6{4,36} Hexipentisteriruncitruncated 8-cube Petitericelliprismatotruncated octeract 11612160 2580480 237 t0,1,2,5,6,7{4,36} Heptihexipenticantitruncated 8-cube Exipetiterigreatorhombi-octeractidiacosipentacontahexazetton 8601600 1720320 238 t0,1,2,4,6,7{4,36} Heptihexistericantitruncated 8-cube Exipeticelligreatorhombated octeract 14192640 2580480 239 t0,1,2,4,5,7{4,36} Heptipentistericantitruncated 8-cube Exitericelligreatorhombated octeract 12902400 2580480 240 t0,1,2,4,5,6{4,36} Hexipentistericantitruncated 8-cube Petitericelligreatorhombated octeract 11612160 2580480 241 t0,1,2,3,6,7{4,36} Heptihexiruncicantitruncated 8-cube Exipetigreatoprismated octeract 8601600 1720320 242 t0,1,2,3,5,7{4,36} Heptipentiruncicantitruncated 8-cube Exiterigreatoprismated octeract 14192640 2580480 243 t0,1,2,3,5,6{4,36} Hexipentiruncicantitruncated 8-cube Petiterigreatoprismated octeract 11612160 2580480 244 t0,1,2,3,4,7{4,36} Heptisteriruncicantitruncated 8-cube Exigreatocellated octeract 8601600 1720320 245 t0,1,2,3,4,6{4,36} Hexisteriruncicantitruncated 8-cube Petigreatocellated octeract 12902400 2580480 246 t0,1,2,3,4,5{4,36} Pentisteriruncicantitruncated 8-cube Great terated octeract 6881280 1720320 247 t0,1,2,3,4,5,6{36,4} Hexipentisteriruncicantitruncated 8-orthoplex Great petated diacosipentacontahexazetton 20643840 5160960 248 t0,1,2,3,4,5,7{36,4} Heptipentisteriruncicantitruncated 8-orthoplex Exigreatoterated diacosipentacontahexazetton 23224320 5160960 249 t0,1,2,3,4,6,7{36,4} Heptihexisteriruncicantitruncated 8-orthoplex Exipetigreatocellated diacosipentacontahexazetton 23224320 5160960 250 t0,1,2,3,5,6,7{36,4} Heptihexipentiruncicantitruncated 8-orthoplex Exipetiterigreatoprismated diacosipentacontahexazetton 23224320 5160960 251 t0,1,2,3,5,6,7{4,36} Heptihexipentiruncicantitruncated 8-cube Exipetiterigreatoprismated octeract 23224320 5160960 252 t0,1,2,3,4,6,7{4,36} Heptihexisteriruncicantitruncated 8-cube Exipetigreatocellated octeract 23224320 5160960 253 t0,1,2,3,4,5,7{4,36} Heptipentisteriruncicantitruncated 8-cube Exigreatoterated octeract 23224320 5160960 254 t0,1,2,3,4,5,6{4,36} Hexipentisteriruncicantitruncated 8-cube Great petated octeract 20643840 5160960 255 t0,1,2,3,4,5,6,7{4,36} Omnitruncated 8-cube Great exi-octeractidiacosipentacontahexazetton 41287680 10321920

### The D8 family

The D8 family has symmetry of order 5,160,960 (8 [factorial](/source/Factorial) × 27).

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

See [list of D8 polytopes](/source/List_of_D8_polytopes) for Coxeter plane graphs of these polytopes.

D8 uniform polytopes # Coxeter-Dynkin diagram Name (acronym)[2] Schläfli symbol Base point (Alternately signed) Element counts Circumrad 7 6 5 4 3 2 1 0 1 = 8-demicube (hocto) h{4,3,3,3,3,3,3} (1,1,1,1,1,1,1,1) 144 1136 4032 8288 10752 7168 1792 128 1.0000000 2 = Cantic 8-cube (thocto) h2{4,3,3,3,3,3,3} (1,1,3,3,3,3,3,3) 23296 3584 2.6457512 3 = Runcic 8-cube (sreho) h3{4,3,3,3,3,3,3} (1,1,1,3,3,3,3,3) 64512 7168 2.4494896 4 = Steric 8-cube (sapho) h4{4,3,3,3,3,3,3} (1,1,1,1,3,3,3,3) 98560 8960 2.2360678 5 = Pentic 8-cube (sacho) h5{4,3,3,3,3,3,3} (1,1,1,1,1,3,3,3) 89600 7168 1.9999999 6 = Hexic 8-cube (sotho)[5] h6{4,3,3,3,3,3,3} (1,1,1,1,1,1,3,3) 48384 3584 1.7320508 7 = Heptic 8-cube (spuho)[6] h7{4,3,3,3,3,3,3} (1,1,1,1,1,1,1,3) 14336 1024 1.4142135 8 = Runcicantic 8-cube (garho) h2,3{4,3,3,3,3,3,3} (1,1,3,5,5,5,5,5) 86016 21504 4.1231055 9 = Stericantic 8-cube (petho) h2,4{4,3,3,3,3,3,3} (1,1,3,3,5,5,5,5) 349440 53760 3.8729835 10 = Steriruncic 8-cube (preho) h3,4{4,3,3,3,3,3,3} (1,1,1,3,5,5,5,5) 179200 35840 3.7416575 11 = Penticantic 8-cube (catho) h2,5{4,3,3,3,3,3,3} (1,1,3,3,3,5,5,5) 573440 71680 3.6055512 12 = Pentiruncic 8-cube (craho) h3,5{4,3,3,3,3,3,3} (1,1,1,3,3,5,5,5) 537600 71680 3.4641016 13 = Pentisteric 8-cube (cepho) h4,5{4,3,3,3,3,3,3} (1,1,1,1,3,5,5,5) 232960 35840 3.3166249 14 = Hexicantic 8-cube (totho) h2,6{4,3,3,3,3,3,3} (1,1,3,3,3,3,5,5) 456960 53760 3.3166249 15 = Hexiruncic 8-cube (tarho) h3,6{4,3,3,3,3,3,3} (1,1,1,3,3,3,5,5) 645120 71680 3.1622777 16 = Hexisteric 8-cube (tupho) h4,6{4,3,3,3,3,3,3} (1,1,1,1,3,3,5,5) 483840 53760 3 17 = Hexipentic 8-cube (tucho) h5,6{4,3,3,3,3,3,3} (1,1,1,1,1,3,5,5) 182784 21504 2.8284271 18 = Hepticantic 8-cube (putho) h2,7{4,3,3,3,3,3,3} (1,1,3,3,3,3,3,5) 172032 21504 3 19 = Heptiruncic 8-cube (pruho) h3,7{4,3,3,3,3,3,3} (1,1,1,3,3,3,3,5) 340480 35840 2.8284271 20 = Heptisteric 8-cube (pupaho) h4,7{4,3,3,3,3,3,3} (1,1,1,1,3,3,3,5) 376320 35840 2.6457512 21 = Heptipentic 8-cube (pucho) h5,7{4,3,3,3,3,3,3} (1,1,1,1,1,3,3,5) 236544 21504 2.4494898 22 = Heptihexic 8-cube (puteho) h6,7{4,3,3,3,3,3,3} (1,1,1,1,1,1,3,5) 78848 7168 2.236068 23 = Steriruncicantic 8-cube (gapho) h2,3,4{4,36} (1,1,3,5,7,7,7,7) 430080 107520 5.3851647 24 = Pentiruncicantic 8-cube (cagreho) h2,3,5{4,36} (1,1,3,5,5,7,7,7) 1182720 215040 5.0990195 25 = Pentistericantic 8-cube (copatho) h2,4,5{4,36} (1,1,3,3,5,7,7,7) 1075200 215040 4.8989797 26 = Pentisteriruncic 8-cube (cepraho) h3,4,5{4,36} (1,1,1,3,5,7,7,7) 716800 143360 4.7958317 27 = Hexiruncicantic 8-cube (tugreho) h2,3,6{4,36} (1,1,3,5,5,5,7,7) 1290240 215040 4.7958317 28 = Hexistericantic 8-cube (tupetho) h2,4,6{4,36} (1,1,3,3,5,5,7,7) 2096640 322560 4.5825758 29 = Hexisteriruncic 8-cube (topreho) h3,4,6{4,36} (1,1,1,3,5,5,7,7) 1290240 215040 4.472136 30 = Hexipenticantic 8-cube (tucatho) h2,5,6{4,36} (1,1,3,3,3,5,7,7) 1290240 215040 4.3588991 31 = Hexipentiruncic 8-cube (tucreho) h3,5,6{4,36} (1,1,1,3,3,5,7,7) 1397760 215040 4.2426405 32 = Hexipentisteric 8-cube (tocapho) h4,5,6{4,36} (1,1,1,1,3,5,7,7) 698880 107520 4.1231055 33 = Heptiruncicantic 8-cube (pugerho) h2,3,7{4,36} (1,1,3,5,5,5,5,7) 591360 107520 4.472136 34 = Heptistericantic 8-cube (pupatho) h2,4,7{4,36} (1,1,3,3,5,5,5,7) 1505280 215040 4.2426405 35 = Heptisteriruncic 8-cube (pupraho) h3,4,7{4,36} (1,1,1,3,5,5,5,7) 860160 143360 4.1231055 36 = Heptipenticantic 8-cube (pucatho) h2,5,7{4,36} (1,1,3,3,3,5,5,7) 1612800 215040 4 37 = Heptipentiruncic 8-cube (pucarho) h3,5,7{4,36} (1,1,1,3,3,5,5,7) 1612800 215040 3.8729835 38 = Heptipentisteric 8-cube (pucapho) h4,5,7{4,36} (1,1,1,1,3,5,5,7) 752640 107520 3.7416575 39 = Heptihexicantic 8-cube (putetho) h2,6,7{4,36} (1,1,3,3,3,3,5,7) 752640 107520 3.7416575 40 = Heptihexiruncic 8-cube (putarho) h3,6,7{4,36} (1,1,1,3,3,3,5,7) 1146880 143360 3.6055512 41 = Heptihexisteric 8-cube (putopho) h4,6,7{4,36} (1,1,1,1,3,3,5,7) 913920 107520 3.4641016 42 = Heptihexipentic 8-cube (potecho) h5,6,7{4,36} (1,1,1,1,1,3,5,7) 365568 43008 3.3166249 43 = Pentisteriruncicantic 8-cube (gacho) h2,3,4,5{4,36} (1,1,3,5,7,9,9,9) 1720320 430080 6.4031243 44 = Hexisteriruncicantic 8-cube (tugepho) h2,3,4,6{4,36} (1,1,3,5,7,7,9,9) 3225600 645120 6.0827627 45 = Hexipentiruncicantic 8-cube (tucagreho) h2,3,5,6{4,36} (1,1,3,5,5,7,9,9) 2903040 645120 5.8309517 46 = Hexipentistericantic 8-cube (tucpetho) h2,4,5,6{4,36} (1,1,3,3,5,7,9,9) 3225600 645120 5.6568542 47 = Hexipentisteriruncic 8-cube (tocparho) h3,4,5,6{4,36} (1,1,1,3,5,7,9,9) 2150400 430080 5.5677648 48 = Heptisteriruncicantic 8-cube (pugapho) h2,3,4,7{4,36} (1,1,3,5,7,7,7,9) 2150400 430080 5.7445626 49 = Heptipentiruncicantic 8-cube (pucgreho) h2,3,5,7{4,36} (1,1,3,5,5,7,7,9) 3548160 645120 5.4772258 50 = Heptipentistericantic 8-cube (pocpatho) h2,4,5,7{4,36} (1,1,3,3,5,7,7,9) 3548160 645120 5.291503 51 = Heptipentisteriruncic 8-cube (pocpreho) h3,4,5,7{4,36} (1,1,1,3,5,7,7,9) 2365440 430080 5.1961527 52 = Heptihexiruncicantic 8-cube (putagreho) h2,3,6,7{4,36} (1,1,3,5,5,5,7,9) 2150400 430080 5.1961527 53 = Heptihexistericantic 8-cube (putapatho) h2,4,6,7{4,36} (1,1,3,3,5,5,7,9) 3870720 645120 5 54 = Heptihexisteriruncic 8-cube (poteparho) h3,4,6,7{4,36} (1,1,1,3,5,5,7,9) 2365440 430080 4.8989797 55 = Heptihexipenticantic 8-cube (potacotho) h2,5,6,7{4,36} (1,1,3,3,3,5,7,9) 2580480 430080 4.7958317 56 = Heptihexipentiruncic 8-cube (potcarho) h3,5,6,7{4,36} (1,1,1,3,3,5,7,9) 2795520 430080 4.6904159 57 = Heptihexipentisteric 8-cube (potcapho) h4,5,6,7{4,36} (1,1,1,1,3,5,7,9) 1397760 215040 4.5825758 58 = Hexipentisteriruncicantic 8-cube (gotho) h2,3,4,5,6{4,36} (1,1,3,5,7,9,11,11) 5160960 1290240 7.1414285 59 = Heptipentisteriruncicantic 8-cube (pugecho) h2,3,4,5,7{4,36} (1,1,3,5,7,9,9,11) 5806080 1290240 6.78233 60 = Heptihexisteriruncicantic 8-cube (potegapho) h2,3,4,6,7{4,36} (1,1,3,5,7,7,9,11) 5806080 1290240 6.480741 61 = Heptihexipentiruncicantic 8-cube (potcograho) h2,3,5,6,7{4,36} (1,1,3,5,5,7,9,11) 5806080 1290240 6.244998 62 = Heptihexipentistericantic 8-cube (potcupetho) h2,4,5,6,7{4,36} (1,1,3,3,5,7,9,11) 6451200 1290240 6.0827627 63 = Heptihexipentisteriruncic 8-cube (potcoparho) h3,4,5,6,7{4,36} (1,1,1,3,5,7,9,11) 4300800 860160 6.0000000 64 = Heptihexipentisteriruncicantic 8-cube (gupho) h2,3,4,5,6,7{4,36} (1,1,3,5,7,9,11,13) 2580480 10321920 7.5498347

### 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](/source/Coxeter-Dynkin_diagram) 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](/source/List_of_E8_polytopes) for Coxeter plane graphs of this family.

E8 uniform polytopes # Coxeter-Dynkin diagram Names Element counts 7-faces 6-faces 5-faces 4-faces Cells Faces Edges Vertices 1 421 (fy) 19440 207360 483840 483840 241920 60480 6720 240 2 Truncated 421 (tiffy) 188160 13440 3 Rectified 421 (riffy) 19680 375840 1935360 3386880 2661120 1028160 181440 6720 4 Birectified 421 (borfy) 19680 382560 2600640 7741440 9918720 5806080 1451520 60480 5 Trirectified 421 (torfy) 19680 382560 2661120 9313920 16934400 14515200 4838400 241920 6 Rectified 142 (buffy) 19680 382560 2661120 9072000 16934400 16934400 7257600 483840 7 Rectified 241 (robay) 19680 313440 1693440 4717440 7257600 5322240 1451520 69120 8 241 (bay) 17520 144960 544320 1209600 1209600 483840 69120 2160 9 Truncated 241 138240 10 142 (bif) 2400 106080 725760 2298240 3628800 2419200 483840 17280 11 Truncated 142 967680 12 Omnitruncated 421 696729600

## 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.

There are five fundamental affine [Coxeter groups](/source/Coxeter_groups) that generate regular and uniform tessellations in 7-space:

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

Regular and uniform tessellations include:

- A ~ 7 {\displaystyle {\tilde {A}}_{7}} 29 uniquely ringed forms, including: - [7-simplex honeycomb](/source/7-simplex_honeycomb): {3[8]}

- C ~ 7 {\displaystyle {\tilde {C}}_{7}} 135 uniquely ringed forms, including: - Regular [7-cube honeycomb](/source/7-cube_honeycomb): {4,34,4} = {4,34,31,1}, =

- B ~ 7 {\displaystyle {\tilde {B}}_{7}} 191 uniquely ringed forms, 127 shared with C ~ 7 {\displaystyle {\tilde {C}}_{7}} , and 64 new, including: - [7-demicube honeycomb](/source/7-demicube_honeycomb): h{4,34,4} = {31,1,34,4}, =

- D ~ 7 {\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](/source/Quarter_7-cubic_honeycomb). - , , , , , , , , ,

- E ~ 7 {\displaystyle {\tilde {E}}_{7}} 143 uniquely ringed forms, including: - [133 honeycomb](/source/1_33_honeycomb): {3,33,3}, - [331 honeycomb](/source/3_31_honeycomb): {3,3,3,33,1},

### 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](/source/Vertex_figure). However, there are [4 paracompact hyperbolic Coxeter groups](/source/Coxeter-Dynkin_diagram#Rank_4_to_10) of rank 8, each generating uniform honeycombs in 7-space as permutations of rings of the Coxeter diagrams.

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

## References

1. ^ [***a***](#cite_ref-richeson_1-0) [***b***](#cite_ref-richeson_1-1) [***c***](#cite_ref-richeson_1-2) Richeson, D.; *Euler's Gem: The Polyhedron Formula and the Birth of Topology*, Princeton, 2008.

1. ^ [***a***](#cite_ref-FOOTNOTEKlitzing_2-0) [***b***](#cite_ref-FOOTNOTEKlitzing_2-1) [Klitzing](#CITEREFKlitzing).

1. **[^](#cite_ref-FOOTNOTEKlitzing[httpsbendwavyorgklitzingincmatsxorenehtm_(x3o3x3o3o3o3o3x3_-_xorene)]_3-0)** [Klitzing](#CITEREFKlitzing), [(x3o3x3o3o3o3o3x3 - xorene)](https://bendwavy.org/klitzing/incmats/xorene.htm).

1. **[^](#cite_ref-FOOTNOTEKlitzing[httpsbendwavyorgklitzingincmatssrekhtm_x3o3x3o3o3o3o4o_-_srek]_4-0)** [Klitzing](#CITEREFKlitzing), [x3o3x3o3o3o3o4o - srek](https://bendwavy.org/klitzing/incmats/srek.htm).

1. **[^](#cite_ref-FOOTNOTEKlitzing[httpsbendwavyorgklitzingincmatssothohtm_(x3o3o_*b3o3o3o3x3o_-_sotho)]_5-0)** [Klitzing](#CITEREFKlitzing), [(x3o3o *b3o3o3o3x3o - sotho)](https://bendwavy.org/klitzing/incmats/sotho.htm).

1. **[^](#cite_ref-FOOTNOTEKlitzing[httpsbendwavyorgklitzingincmatsspuhohtm_(x3o3o_*b3o3o3o3o3x_-_spuho)]_6-0)** [Klitzing](#CITEREFKlitzing), [(x3o3o *b3o3o3o3o3x - spuho)](https://bendwavy.org/klitzing/incmats/spuho.htm).

- [T. Gosset](/source/Thorold_Gosset): *On the Regular and Semi-Regular Figures in Space of n Dimensions*, [Messenger of Mathematics](/source/Messenger_of_Mathematics), Macmillan, 1900

- [A. Boole Stott](/source/Alicia_Boole_Stott) (1910). ["Geometrical deduction of semiregular from regular polytopes and space fillings"](https://web.archive.org/web/20250429000816/https://dwc.knaw.nl/DL/publications/PU00011492.pdf) (PDF). *Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam*. **XI** (1). Amsterdam: Johannes Müller. Archived from [the original](https://dwc.knaw.nl/DL/publications/PU00011492.pdf) (PDF) on 29 April 2025.

- [H.S.M. Coxeter](/source/Harold_Scott_MacDonald_Coxeter): - H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: *Uniform Polyhedra*, Philosophical Transactions of the Royal Society of London, 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 Ivić Weiss, Wiley-Interscience Publication, 1995, [wiley.com](https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036), [ISBN](/source/ISBN_(identifier)) [978-0-471-01003-6](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-01003-6) - (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](/source/Norman_Johnson_(mathematician)): *The Theory of Uniform Polytopes and Honeycombs*, Ph.D. Dissertation, University of Toronto, 1966

- Klitzing, Richard. ["8D uniform polytopes (polyzetta) with acronyms"](https://bendwavy.org/klitzing/dimensions/polyzetta.htm).

## External links

- [Polytope names](http://www.steelpillow.com/polyhedra/ditela.html)

- [Polytopes of Various Dimensions](http://www.polytope.net/hedrondude/topes.htm)

- [Multi-dimensional Glossary](http://tetraspace.alkaline.org/glossary.htm)

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10 Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn Regular polygon Triangle Square p-gon Hexagon Pentagon Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron Uniform polychoron Pentachoron 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221 Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321 Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421 Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations

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Adapted from the Wikipedia article [Uniform 8-polytope](https://en.wikipedia.org/wiki/Uniform_8-polytope) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Uniform_8-polytope?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.