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.

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 , with v 7-polytope facets around each peak.

There are exactly three such convex regular 8-polytopes:
# - 8-simplex
# - 8-cube
# - 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 Topoplogy'', 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:

Selected regular and uniform 8-polytopes from each family include:
# Simplex family: A₈ ⁷-
#* 135 uniform 8-polytopes as permutations of rings in the group diagram, including one regular:
#*# - 8-simplex or ennea-9-tope or enneazetton -
# Hypercube/ orthoplex family: B₈ ,3⁶-
#* 255 uniform 8-polytopes as permutations of rings in the group diagram, including two regular ones:
#*# - 8-cube or ''octeract''-
#*# - 8-orthoplex or ''octacross'' -
# Demihypercube D₈ family: ^5,1,1-
#* 191 uniform 8-polytopes as permutations of rings in the group diagram, including:
#*# - 8-demicube or ''demiocteract'', 1₅₁ - ; also as h .
#*# - 8-orthoplex, 5₁₁ -
# E-polytope family E₈ family: ^4,1,1-
#* 255 uniform 8-polytopes as permutations of rings in the group diagram, including:
#*# - Thorold Gosset's semiregular 4₂₁,
#*# - the uniform 1₄₂, ,
#*# - 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.

{, class="wikitable collapsible collapsed"
!colspan=13, B₈ uniform polytopes
, -
!rowspan=2, #
!rowspan=2, Coxeter-Dynkin diagram
!rowspan=2, Schläfli
symbol
!rowspan=2, Name
!colspan=8, Element counts
, -
! 7, , 6, , 5, , 4, , 3, , 2, , 1, , 0
, - align=center BGCOLOR="#f0e0e0"
!1
, , , t₀{3⁶,4}, , 8-orthoplex
Diacosipentacontahexazetton (ek), , 256, , 1024, , 1792, , 1792, , 1120, , 448, , 112, , 16
, - align=center BGCOLOR="#f0e0e0"
!2
, , , t₁{3⁶,4}, , Rectified 8-orthoplex
Rectified diacosipentacontahexazetton (rek), , 272, , 3072, , 8960, , 12544, , 10080, , 4928, , 1344, , 112
, - align=center BGCOLOR="#f0e0e0"
!3
, , , t₂{3⁶,4}, , Birectified 8-orthoplex
Birectified diacosipentacontahexazetton (bark), , 272, , 3184, , 16128, , 34048, , 36960, , 22400, , 6720, , 448
, - align=center BGCOLOR="#f0e0e0"
!4
, , , t₃{3⁶,4}, , Trirectified 8-orthoplex
Trirectified diacosipentacontahexazetton (tark), , 272, , 3184, , 16576, , 48384, , 71680, , 53760, , 17920, , 1120
, - align=center BGCOLOR="#e0e0f0"
!5
, , , t₃{4,3⁶}, , Trirectified 8-cube
Trirectified octeract (tro), , 272, , 3184, , 16576, , 47712, , 80640, , 71680, , 26880, , 1792
, - align=center BGCOLOR="#e0e0f0"
!6
, , , t₂{4,3⁶}, , Birectified 8-cube
Birectified octeract (bro), , 272, , 3184, , 14784, , 36960, , 55552, , 50176, , 21504, , 1792
, - align=center BGCOLOR="#e0e0f0"
!7
, , , t₁{4,3⁶}, , Rectified 8-cube
Rectified octeract (recto), , 272, , 2160, , 7616, , 15456, , 19712, , 16128, , 7168, , 1024
, - align=center BGCOLOR="#e0e0f0"
!8
, , , t₀{4,3⁶}, , 8-cube
Octeract (octo), , 16, , 112, , 448, , 1120, , 1792, , 1792, , 1024, , 256
, - align=center BGCOLOR="#f0e0e0"
!9
, , , t_0,1{3⁶,4}, , Truncated 8-orthoplex
Truncated diacosipentacontahexazetton (tek), , , , , , , , , , , , , , 1456, , 224
, - align=center BGCOLOR="#f0e0e0"
!10
, , , t_0,2{3⁶,4}, , Cantellated 8-orthoplex
Small rhombated diacosipentacontahexazetton (srek), , , , , , , , , , , , , , 14784, , 1344
, - align=center BGCOLOR="#f0e0e0"
!11
, , , t_1,2{3⁶,4}, , Bitruncated 8-orthoplex
Bitruncated diacosipentacontahexazetton (batek), , , , , , , , , , , , , , 8064, , 1344
, - align=center BGCOLOR="#f0e0e0"
!12
, , , t_0,3{3⁶,4}, , Runcinated 8-orthoplex
Small prismated diacosipentacontahexazetton (spek), , , , , , , , , , , , , , 60480, , 4480
, - align=center BGCOLOR="#f0e0e0"
!13
, , , t_1,3{3⁶,4}, , Bicantellated 8-orthoplex
Small birhombated diacosipentacontahexazetton (sabork), , , , , , , , , , , , , , 67200, , 6720
, - align=center BGCOLOR="#f0e0e0"
!14
, , , t_2,3{3⁶,4}, , Tritruncated 8-orthoplex
Tritruncated diacosipentacontahexazetton (tatek), , , , , , , , , , , , , , 24640, , 4480
, - align=center BGCOLOR="#f0e0e0"
!15
, , , t_0,4{3⁶,4}, , Stericated 8-orthoplex
Small cellated diacosipentacontahexazetton (scak), , , , , , , , , , , , , , 125440, , 8960
, - align=center BGCOLOR="#f0e0e0"
!16
, , , t_1,4{3⁶,4}, , Biruncinated 8-orthoplex
Small biprismated diacosipentacontahexazetton (sabpek), , , , , , , , , , , , , , 215040, , 17920
, - align=center BGCOLOR="#f0e0e0"
!17
, , , t_2,4{3⁶,4}, , Tricantellated 8-orthoplex
Small trirhombated diacosipentacontahexazetton (satrek), , , , , , , , , , , , , , 161280, , 17920
, - align=center BGCOLOR="#e0f0e0"
!18
, , , t_3,4{4,3⁶}, , Quadritruncated 8-cube
Octeractidiacosipentacontahexazetton (oke), , , , , , , , , , , , , , 44800, , 8960
, - align=center BGCOLOR="#f0e0e0"
!19
, , , t_0,5{3⁶,4}, , Pentellated 8-orthoplex
Small terated diacosipentacontahexazetton (setek), , , , , , , , , , , , , , 134400, , 10752
, - align=center BGCOLOR="#f0e0e0"
!20
, , , t_1,5{3⁶,4}, , Bistericated 8-orthoplex
Small bicellated diacosipentacontahexazetton (sibcak), , , , , , , , , , , , , , 322560, , 26880
, - align=center BGCOLOR="#e0f0e0"
!21
, , , t_2,5{4,3⁶}, , Triruncinated 8-cube
Small triprismato-octeractidiacosipentacontahexazetton (sitpoke), , , , , , , , , , , , , , 376320, , 35840
, - align=center BGCOLOR="#e0e0f0"
!22
, , , t_2,4{4,3⁶}, , Tricantellated 8-cube
Small trirhombated octeract (satro), , , , , , , , , , , , , , 215040, , 26880
, - align=center BGCOLOR="#e0e0f0"
!23
, , , t_2,3{4,3⁶}, , Tritruncated 8-cube
Tritruncated octeract (tato), , , , , , , , , , , , , , 48384, , 10752
, - align=center BGCOLOR="#f0e0e0"
!24
, , , t_0,6{3⁶,4}, , Hexicated 8-orthoplex
Small petated diacosipentacontahexazetton (supek), , , , , , , , , , , , , , 64512, , 7168
, - align=center BGCOLOR="#e0f0e0"
!25
, , , t_1,6{4,3⁶}, , Bipentellated 8-cube
Small biteri-octeractidiacosipentacontahexazetton (sabtoke), , , , , , , , , , , , , , 215040, , 21504
, - align=center BGCOLOR="#e0e0f0"
!26
, , , t_1,5{4,3⁶}, , Bistericated 8-cube
Small bicellated octeract (sobco), , , , , , , , , , , , , , 358400, , 35840
, - align=center BGCOLOR="#e0e0f0"
!27
, , , t_1,4{4,3⁶}, , Biruncinated 8-cube
Small biprismated octeract (sabepo), , , , , , , , , , , , , , 322560, , 35840
, - align=center BGCOLOR="#e0e0f0"
!28
, , , t_1,3{4,3⁶}, , Bicantellated 8-cube
Small birhombated octeract (subro), , , , , , , , , , , , , , 150528, , 21504
, - align=center BGCOLOR="#e0e0f0"
!29
, , , t_1,2{4,3⁶}, , Bitruncated 8-cube
Bitruncated octeract (bato), , , , , , , , , , , , , , 28672, , 7168
, - align=center BGCOLOR="#e0f0e0"
!30
, , , t_0,7{4,3⁶}, , Heptellated 8-cube
Small exi-octeractidiacosipentacontahexazetton (saxoke), , , , , , , , , , , , , , 14336, , 2048
, - align=center BGCOLOR="#e0e0f0"
!31
, , , t_0,6{4,3⁶}, , Hexicated 8-cube
Small petated octeract (supo), , , , , , , , , , , , , , 64512, , 7168
, - align=center BGCOLOR="#e0e0f0"
!32
, , , t_0,5{4,3⁶}, , Pentellated 8-cube
Small terated octeract (soto), , , , , , , , , , , , , , 143360, , 14336
, - align=center BGCOLOR="#e0e0f0"
!33
, , , t_0,4{4,3⁶}, , Stericated 8-cube
Small cellated octeract (soco), , , , , , , , , , , , , , 179200, , 17920
, - align=center BGCOLOR="#e0e0f0"
!34
, , , t_0,3{4,3⁶}, , Runcinated 8-cube
Small prismated octeract (sopo), , , , , , , , , , , , , , 129024, , 14336
, - align=center BGCOLOR="#e0e0f0"
!35
, , , t_0,2{4,3⁶}, , Cantellated 8-cube
Small rhombated octeract (soro), , , , , , , , , , , , , , 50176, , 7168
, - align=center BGCOLOR="#e0e0f0"
!36
, , , t_0,1{4,3⁶}, , Truncated 8-cube
Truncated octeract (tocto), , , , , , , , , , , , , , 8192, , 2048
, - align=center BGCOLOR="#f0e0e0"
!37
, , , t_0,1,2{3⁶,4}, , Cantitruncated 8-orthoplex
Great rhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 16128, , 2688
, - align=center BGCOLOR="#f0e0e0"
!38
, , , t_0,1,3{3⁶,4}, , Runcitruncated 8-orthoplex
Prismatotruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 127680, , 13440
, - align=center BGCOLOR="#f0e0e0"
!39
, , , t_0,2,3{3⁶,4}, , Runcicantellated 8-orthoplex
Prismatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 80640, , 13440
, - align=center BGCOLOR="#f0e0e0"
!40
, , , t_1,2,3{3⁶,4}, , Bicantitruncated 8-orthoplex
Great birhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 73920, , 13440
, - align=center BGCOLOR="#f0e0e0"
!41
, , , t_0,1,4{3⁶,4}, , Steritruncated 8-orthoplex
Cellitruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 394240, , 35840
, - align=center BGCOLOR="#f0e0e0"
!42
, , , t_0,2,4{3⁶,4}, , Stericantellated 8-orthoplex
Cellirhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 483840, , 53760
, - align=center BGCOLOR="#f0e0e0"
!43
, , , t_1,2,4{3⁶,4}, , Biruncitruncated 8-orthoplex
Biprismatotruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 430080, , 53760
, - align=center BGCOLOR="#f0e0e0"
!44
, , , t_0,3,4{3⁶,4}, , Steriruncinated 8-orthoplex
Celliprismated diacosipentacontahexazetton, , , , , , , , , , , , , , 215040, , 35840
, - align=center BGCOLOR="#f0e0e0"
!45
, , , t_1,3,4{3⁶,4}, , Biruncicantellated 8-orthoplex
Biprismatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 322560, , 53760
, - align=center BGCOLOR="#f0e0e0"
!46
, , , t_2,3,4{3⁶,4}, , Tricantitruncated 8-orthoplex
Great trirhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 179200, , 35840
, - align=center BGCOLOR="#f0e0e0"
!47
, , , t_0,1,5{3⁶,4}, , Pentitruncated 8-orthoplex
Teritruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 564480, , 53760
, - align=center BGCOLOR="#f0e0e0"
!48
, , , t_0,2,5{3⁶,4}, , Penticantellated 8-orthoplex
Terirhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 1075200, , 107520
, - align=center BGCOLOR="#f0e0e0"
!49
, , , t_1,2,5{3⁶,4}, , Bisteritruncated 8-orthoplex
Bicellitruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 913920, , 107520
, - align=center BGCOLOR="#f0e0e0"
!50
, , , t_0,3,5{3⁶,4}, , Pentiruncinated 8-orthoplex
Teriprismated diacosipentacontahexazetton, , , , , , , , , , , , , , 913920, , 107520
, - align=center BGCOLOR="#f0e0e0"
!51
, , , t_1,3,5{3⁶,4}, , Bistericantellated 8-orthoplex
Bicellirhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 1290240, , 161280
, - align=center BGCOLOR="#f0e0e0"
!52
, , , t_2,3,5{3⁶,4}, , Triruncitruncated 8-orthoplex
Triprismatotruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 698880, , 107520
, - align=center BGCOLOR="#f0e0e0"
!53
, , , t_0,4,5{3⁶,4}, , Pentistericated 8-orthoplex
Tericellated diacosipentacontahexazetton, , , , , , , , , , , , , , 322560, , 53760
, - align=center BGCOLOR="#f0e0e0"
!54
, , , t_1,4,5{3⁶,4}, , Bisteriruncinated 8-orthoplex
Bicelliprismated diacosipentacontahexazetton, , , , , , , , , , , , , , 698880, , 107520
, - align=center BGCOLOR="#e0e0f0"
!55
, , , t_2,3,5{4,3⁶}, , Triruncitruncated 8-cube
Triprismatotruncated octeract, , , , , , , , , , , , , , 645120, , 107520
, - align=center BGCOLOR="#e0e0f0"
!56
, , , t_2,3,4{4,3⁶}, , Tricantitruncated 8-cube
Great trirhombated octeract, , , , , , , , , , , , , , 241920, , 53760
, - align=center BGCOLOR="#f0e0e0"
!57
, , , t_0,1,6{3⁶,4}, , Hexitruncated 8-orthoplex
Petitruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 344064, , 43008
, - align=center BGCOLOR="#f0e0e0"
!58
, , , t_0,2,6{3⁶,4}, , Hexicantellated 8-orthoplex
Petirhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 967680, , 107520
, - align=center BGCOLOR="#f0e0e0"
!59
, , , t_1,2,6{3⁶,4}, , Bipentitruncated 8-orthoplex
Biteritruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 752640, , 107520
, - align=center BGCOLOR="#f0e0e0"
!60
, , , t_0,3,6{3⁶,4}, , Hexiruncinated 8-orthoplex
Petiprismated diacosipentacontahexazetton, , , , , , , , , , , , , , 1290240, , 143360
, - align=center BGCOLOR="#f0e0e0"
!61
, , , t_1,3,6{3⁶,4}, , Bipenticantellated 8-orthoplex
Biterirhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 1720320, , 215040
, - align=center BGCOLOR="#e0e0f0"
!62
, , , t_1,4,5{4,3⁶}, , Bisteriruncinated 8-cube
Bicelliprismated octeract, , , , , , , , , , , , , , 860160, , 143360
, - align=center BGCOLOR="#f0e0e0"
!63
, , , t_0,4,6{3⁶,4}, , Hexistericated 8-orthoplex
Peticellated diacosipentacontahexazetton, , , , , , , , , , , , , , 860160, , 107520
, - align=center BGCOLOR="#e0e0f0"
!64
, , , t_1,3,6{4,3⁶}, , Bipenticantellated 8-cube
Biterirhombated octeract, , , , , , , , , , , , , , 1720320, , 215040
, - align=center BGCOLOR="#e0e0f0"
!65
, , , t_1,3,5{4,3⁶}, , Bistericantellated 8-cube
Bicellirhombated octeract, , , , , , , , , , , , , , 1505280, , 215040
, - align=center BGCOLOR="#e0e0f0"
!66
, , , t_1,3,4{4,3⁶}, , Biruncicantellated 8-cube
Biprismatorhombated octeract, , , , , , , , , , , , , , 537600, , 107520
, - align=center BGCOLOR="#f0e0e0"
!67
, , , t_0,5,6{3⁶,4}, , Hexipentellated 8-orthoplex
Petiterated diacosipentacontahexazetton, , , , , , , , , , , , , , 258048, , 43008
, - align=center BGCOLOR="#e0e0f0"
!68
, , , t_1,2,6{4,3⁶}, , Bipentitruncated 8-cube
Biteritruncated octeract, , , , , , , , , , , , , , 752640, , 107520
, - align=center BGCOLOR="#e0e0f0"
!69
, , , t_1,2,5{4,3⁶}, , Bisteritruncated 8-cube
Bicellitruncated octeract, , , , , , , , , , , , , , 1003520, , 143360
, - align=center BGCOLOR="#e0e0f0"
!70
, , , t_1,2,4{4,3⁶}, , Biruncitruncated 8-cube
Biprismatotruncated octeract, , , , , , , , , , , , , , 645120, , 107520
, - align=center BGCOLOR="#e0e0f0"
!71
, , , t_1,2,3{4,3⁶}, , Bicantitruncated 8-cube
Great birhombated octeract, , , , , , , , , , , , , , 172032, , 43008
, - align=center BGCOLOR="#f0e0e0"
!72
, , , t_0,1,7{3⁶,4}, , Heptitruncated 8-orthoplex
Exitruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 93184, , 14336
, - align=center BGCOLOR="#f0e0e0"
!73
, , , t_0,2,7{3⁶,4}, , Hepticantellated 8-orthoplex
Exirhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 365568, , 43008
, - align=center BGCOLOR="#e0e0f0"
!74
, , , t_0,5,6{4,3⁶}, , Hexipentellated 8-cube
Petiterated octeract, , , , , , , , , , , , , , 258048, , 43008
, - align=center BGCOLOR="#f0e0e0"
!75
, , , t_0,3,7{3⁶,4}, , Heptiruncinated 8-orthoplex
Exiprismated diacosipentacontahexazetton, , , , , , , , , , , , , , 680960, , 71680
, - align=center BGCOLOR="#e0e0f0"
!76
, , , t_0,4,6{4,3⁶}, , Hexistericated 8-cube
Peticellated octeract, , , , , , , , , , , , , , 860160, , 107520
, - align=center BGCOLOR="#e0e0f0"
!77
, , , t_0,4,5{4,3⁶}, , Pentistericated 8-cube
Tericellated octeract, , , , , , , , , , , , , , 394240, , 71680
, - align=center BGCOLOR="#e0e0f0"
!78
, , , t_0,3,7{4,3⁶}, , Heptiruncinated 8-cube
Exiprismated octeract, , , , , , , , , , , , , , 680960, , 71680
, - align=center BGCOLOR="#e0e0f0"
!79
, , , t_0,3,6{4,3⁶}, , Hexiruncinated 8-cube
Petiprismated octeract, , , , , , , , , , , , , , 1290240, , 143360
, - align=center BGCOLOR="#e0e0f0"
!80
, , , t_0,3,5{4,3⁶}, , Pentiruncinated 8-cube
Teriprismated octeract, , , , , , , , , , , , , , 1075200, , 143360
, - align=center BGCOLOR="#e0e0f0"
!81
, , , t_0,3,4{4,3⁶}, , Steriruncinated 8-cube
Celliprismated octeract, , , , , , , , , , , , , , 358400, , 71680
, - align=center BGCOLOR="#e0e0f0"
!82
, , , t_0,2,7{4,3⁶}, , Hepticantellated 8-cube
Exirhombated octeract, , , , , , , , , , , , , , 365568, , 43008
, - align=center BGCOLOR="#e0e0f0"
!83
, , , t_0,2,6{4,3⁶}, , Hexicantellated 8-cube
Petirhombated octeract, , , , , , , , , , , , , , 967680, , 107520
, - align=center BGCOLOR="#e0e0f0"
!84
, , , t_0,2,5{4,3⁶}, , Penticantellated 8-cube
Terirhombated octeract, , , , , , , , , , , , , , 1218560, , 143360
, - align=center BGCOLOR="#e0e0f0"
!85
, , , t_0,2,4{4,3⁶}, , Stericantellated 8-cube
Cellirhombated octeract, , , , , , , , , , , , , , 752640, , 107520
, - align=center BGCOLOR="#e0e0f0"
!86
, , , t_0,2,3{4,3⁶}, , Runcicantellated 8-cube
Prismatorhombated octeract, , , , , , , , , , , , , , 193536, , 43008
, - align=center BGCOLOR="#e0e0f0"
!87
, , , t_0,1,7{4,3⁶}, , Heptitruncated 8-cube
Exitruncated octeract, , , , , , , , , , , , , , 93184, , 14336
, - align=center BGCOLOR="#e0e0f0"
!88
, , , t_0,1,6{4,3⁶}, , Hexitruncated 8-cube
Petitruncated octeract, , , , , , , , , , , , , , 344064, , 43008
, - align=center BGCOLOR="#e0e0f0"
!89
, , , t_0,1,5{4,3⁶}, , Pentitruncated 8-cube
Teritruncated octeract, , , , , , , , , , , , , , 609280, , 71680
, - align=center BGCOLOR="#e0e0f0"
!90
, , , t_0,1,4{4,3⁶}, , Steritruncated 8-cube
Cellitruncated octeract, , , , , , , , , , , , , , 573440, , 71680
, - align=center BGCOLOR="#e0e0f0"
!91
, , , t_0,1,3{4,3⁶}, , Runcitruncated 8-cube
Prismatotruncated octeract, , , , , , , , , , , , , , 279552, , 43008
, - align=center BGCOLOR="#e0e0f0"
!92
, , , t_0,1,2{4,3⁶}, , Cantitruncated 8-cube
Great rhombated octeract, , , , , , , , , , , , , , 57344, , 14336
, - align=center BGCOLOR="#f0e0e0"
!93
, , , t_0,1,2,3{3⁶,4}, , Runcicantitruncated 8-orthoplex
Great prismated diacosipentacontahexazetton, , , , , , , , , , , , , , 147840, , 26880
, - align=center BGCOLOR="#f0e0e0"
!94
, , , t_0,1,2,4{3⁶,4}, , Stericantitruncated 8-orthoplex
Celligreatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 860160, , 107520
, - align=center BGCOLOR="#f0e0e0"
!95
, , , t_0,1,3,4{3⁶,4}, , Steriruncitruncated 8-orthoplex
Celliprismatotruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 591360, , 107520
, - align=center BGCOLOR="#f0e0e0"
!96
, , , t_0,2,3,4{3⁶,4}, , Steriruncicantellated 8-orthoplex
Celliprismatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 591360, , 107520
, - align=center BGCOLOR="#f0e0e0"
!97
, , , t_1,2,3,4{3⁶,4}, , Biruncicantitruncated 8-orthoplex
Great biprismated diacosipentacontahexazetton, , , , , , , , , , , , , , 537600, , 107520
, - align=center BGCOLOR="#f0e0e0"
!98
, , , t_0,1,2,5{3⁶,4}, , Penticantitruncated 8-orthoplex
Terigreatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 1827840, , 215040
, - align=center BGCOLOR="#f0e0e0"
!99
, , , t_0,1,3,5{3⁶,4}, , Pentiruncitruncated 8-orthoplex
Teriprismatotruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 2419200, , 322560
, - align=center BGCOLOR="#f0e0e0"
!100
, , , t_0,2,3,5{3⁶,4}, , Pentiruncicantellated 8-orthoplex
Teriprismatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 2257920, , 322560
, - align=center BGCOLOR="#f0e0e0"
!101
, , , t_1,2,3,5{3⁶,4}, , Bistericantitruncated 8-orthoplex
Bicelligreatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 2096640, , 322560
, - align=center BGCOLOR="#f0e0e0"
!102
, , , t_0,1,4,5{3⁶,4}, , Pentisteritruncated 8-orthoplex
Tericellitruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 1182720, , 215040
, - align=center BGCOLOR="#f0e0e0"
!103
, , , t_0,2,4,5{3⁶,4}, , Pentistericantellated 8-orthoplex
Tericellirhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 1935360, , 322560
, - align=center BGCOLOR="#f0e0e0"
!104
, , , t_1,2,4,5{3⁶,4}, , Bisteriruncitruncated 8-orthoplex
Bicelliprismatotruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 1612800, , 322560
, - align=center BGCOLOR="#f0e0e0"
!105
, , , t_0,3,4,5{3⁶,4}, , Pentisteriruncinated 8-orthoplex
Tericelliprismated diacosipentacontahexazetton, , , , , , , , , , , , , , 1182720, , 215040
, - align=center BGCOLOR="#f0e0e0"
!106
, , , t_1,3,4,5{3⁶,4}, , Bisteriruncicantellated 8-orthoplex
Bicelliprismatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 1774080, , 322560
, - align=center BGCOLOR="#e0f0e0"
!107
, , , t_2,3,4,5{4,3⁶}, , Triruncicantitruncated 8-cube
Great triprismato-octeractidiacosipentacontahexazetton, , , , , , , , , , , , , , 967680, , 215040
, - align=center BGCOLOR="#f0e0e0"
!108
, , , t_0,1,2,6{3⁶,4}, , Hexicantitruncated 8-orthoplex
Petigreatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 1505280, , 215040
, - align=center BGCOLOR="#f0e0e0"
!109
, , , t_0,1,3,6{3⁶,4}, , Hexiruncitruncated 8-orthoplex
Petiprismatotruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 3225600, , 430080
, - align=center BGCOLOR="#f0e0e0"
!110
, , , t_0,2,3,6{3⁶,4}, , Hexiruncicantellated 8-orthoplex
Petiprismatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 2795520, , 430080
, - align=center BGCOLOR="#f0e0e0"
!111
, , , t_1,2,3,6{3⁶,4}, , Bipenticantitruncated 8-orthoplex
Biterigreatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 2580480, , 430080
, - align=center BGCOLOR="#f0e0e0"
!112
, , , t_0,1,4,6{3⁶,4}, , Hexisteritruncated 8-orthoplex
Peticellitruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 3010560, , 430080
, - align=center BGCOLOR="#f0e0e0"
!113
, , , t_0,2,4,6{3⁶,4}, , Hexistericantellated 8-orthoplex
Peticellirhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 4515840, , 645120
, - align=center BGCOLOR="#f0e0e0"
!114
, , , t_1,2,4,6{3⁶,4}, , Bipentiruncitruncated 8-orthoplex
Biteriprismatotruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 3870720, , 645120
, - align=center BGCOLOR="#f0e0e0"
!115
, , , t_0,3,4,6{3⁶,4}, , Hexisteriruncinated 8-orthoplex
Peticelliprismated diacosipentacontahexazetton, , , , , , , , , , , , , , 2580480, , 430080
, - align=center BGCOLOR="#e0f0e0"
!116
, , , t_1,3,4,6{4,3⁶}, , Bipentiruncicantellated 8-cube
Biteriprismatorhombi-octeractidiacosipentacontahexazetton, , , , , , , , , , , , , , 3870720, , 645120
, - align=center BGCOLOR="#e0e0f0"
!117
, , , t_1,3,4,5{4,3⁶}, , Bisteriruncicantellated 8-cube
Bicelliprismatorhombated octeract, , , , , , , , , , , , , , 2150400, , 430080
, - align=center BGCOLOR="#f0e0e0"
!118
, , , t_0,1,5,6{3⁶,4}, , Hexipentitruncated 8-orthoplex
Petiteritruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 1182720, , 215040
, - align=center BGCOLOR="#f0e0e0"
!119
, , , t_0,2,5,6{3⁶,4}, , Hexipenticantellated 8-orthoplex
Petiterirhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 2795520, , 430080
, - align=center BGCOLOR="#e0f0e0"
!120
, , , t_1,2,5,6{4,3⁶}, , Bipentisteritruncated 8-cube
Bitericellitrunki-octeractidiacosipentacontahexazetton, , , , , , , , , , , , , , 2150400, , 430080
, - align=center BGCOLOR="#f0e0e0"
!121
, , , t_0,3,5,6{3⁶,4}, , Hexipentiruncinated 8-orthoplex
Petiteriprismated diacosipentacontahexazetton, , , , , , , , , , , , , , 2795520, , 430080
, - align=center BGCOLOR="#e0e0f0"
!122
, , , t_1,2,4,6{4,3⁶}, , Bipentiruncitruncated 8-cube
Biteriprismatotruncated octeract, , , , , , , , , , , , , , 3870720, , 645120
, - align=center BGCOLOR="#e0e0f0"
!123
, , , t_1,2,4,5{4,3⁶}, , Bisteriruncitruncated 8-cube
Bicelliprismatotruncated octeract, , , , , , , , , , , , , , 1935360, , 430080
, - align=center BGCOLOR="#f0e0e0"
!124
, , , t_0,4,5,6{3⁶,4}, , Hexipentistericated 8-orthoplex
Petitericellated diacosipentacontahexazetton, , , , , , , , , , , , , , 1182720, , 215040
, - align=center BGCOLOR="#e0e0f0"
!125
, , , t_1,2,3,6{4,3⁶}, , Bipenticantitruncated 8-cube
Biterigreatorhombated octeract, , , , , , , , , , , , , , 2580480, , 430080
, - align=center BGCOLOR="#e0e0f0"
!126
, , , t_1,2,3,5{4,3⁶}, , Bistericantitruncated 8-cube
Bicelligreatorhombated octeract, , , , , , , , , , , , , , 2365440, , 430080
, - align=center BGCOLOR="#e0e0f0"
!127
, , , t_1,2,3,4{4,3⁶}, , Biruncicantitruncated 8-cube
Great biprismated octeract, , , , , , , , , , , , , , 860160, , 215040
, - align=center BGCOLOR="#f0e0e0"
!128
, , , t_0,1,2,7{3⁶,4}, , Hepticantitruncated 8-orthoplex
Exigreatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 516096, , 86016
, - align=center BGCOLOR="#f0e0e0"
!129
, , , t_0,1,3,7{3⁶,4}, , Heptiruncitruncated 8-orthoplex
Exiprismatotruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 1612800, , 215040
, - align=center BGCOLOR="#f0e0e0"
!130
, , , t_0,2,3,7{3⁶,4}, , Heptiruncicantellated 8-orthoplex
Exiprismatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 1290240, , 215040
, - align=center BGCOLOR="#e0e0f0"
!131
, , , t_0,4,5,6{4,3⁶}, , Hexipentistericated 8-cube
Petitericellated octeract, , , , , , , , , , , , , , 1182720, , 215040
, - align=center BGCOLOR="#f0e0e0"
!132
, , , t_0,1,4,7{3⁶,4}, , Heptisteritruncated 8-orthoplex
Exicellitruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 2293760, , 286720
, - align=center BGCOLOR="#f0e0e0"
!133
, , , t_0,2,4,7{3⁶,4}, , Heptistericantellated 8-orthoplex
Exicellirhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 3225600, , 430080
, - align=center BGCOLOR="#e0e0f0"
!134
, , , t_0,3,5,6{4,3⁶}, , Hexipentiruncinated 8-cube
Petiteriprismated octeract, , , , , , , , , , , , , , 2795520, , 430080
, - align=center BGCOLOR="#e0f0e0"
!135
, , , t_0,3,4,7{4,3⁶}, , Heptisteriruncinated 8-cube
Exicelliprismato-octeractidiacosipentacontahexazetton, , , , , , , , , , , , , , 1720320, , 286720
, - align=center BGCOLOR="#e0e0f0"
!136
, , , t_0,3,4,6{4,3⁶}, , Hexisteriruncinated 8-cube
Peticelliprismated octeract, , , , , , , , , , , , , , 2580480, , 430080
, - align=center BGCOLOR="#e0e0f0"
!137
, , , t_0,3,4,5{4,3⁶}, , Pentisteriruncinated 8-cube
Tericelliprismated octeract, , , , , , , , , , , , , , 1433600, , 286720
, - align=center BGCOLOR="#f0e0e0"
!138
, , , t_0,1,5,7{3⁶,4}, , Heptipentitruncated 8-orthoplex
Exiteritruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 1612800, , 215040
, - align=center BGCOLOR="#e0f0e0"
!139
, , , t_0,2,5,7{4,3⁶}, , Heptipenticantellated 8-cube
Exiterirhombi-octeractidiacosipentacontahexazetton, , , , , , , , , , , , , , 3440640, , 430080
, - align=center BGCOLOR="#e0e0f0"
!140
, , , t_0,2,5,6{4,3⁶}, , Hexipenticantellated 8-cube
Petiterirhombated octeract, , , , , , , , , , , , , , 2795520, , 430080
, - align=center BGCOLOR="#e0e0f0"
!141
, , , t_0,2,4,7{4,3⁶}, , Heptistericantellated 8-cube
Exicellirhombated octeract, , , , , , , , , , , , , , 3225600, , 430080
, - align=center BGCOLOR="#e0e0f0"
!142
, , , t_0,2,4,6{4,3⁶}, , Hexistericantellated 8-cube
Peticellirhombated octeract, , , , , , , , , , , , , , 4515840, , 645120
, - align=center BGCOLOR="#e0e0f0"
!143
, , , t_0,2,4,5{4,3⁶}, , Pentistericantellated 8-cube
Tericellirhombated octeract, , , , , , , , , , , , , , 2365440, , 430080
, - align=center BGCOLOR="#e0e0f0"
!144
, , , t_0,2,3,7{4,3⁶}, , Heptiruncicantellated 8-cube
Exiprismatorhombated octeract, , , , , , , , , , , , , , 1290240, , 215040
, - align=center BGCOLOR="#e0e0f0"
!145
, , , t_0,2,3,6{4,3⁶}, , Hexiruncicantellated 8-cube
Petiprismatorhombated octeract, , , , , , , , , , , , , , 2795520, , 430080
, - align=center BGCOLOR="#e0e0f0"
!146
, , , t_0,2,3,5{4,3⁶}, , Pentiruncicantellated 8-cube
Teriprismatorhombated octeract, , , , , , , , , , , , , , 2580480, , 430080
, - align=center BGCOLOR="#e0e0f0"
!147
, , , t_0,2,3,4{4,3⁶}, , Steriruncicantellated 8-cube
Celliprismatorhombated octeract, , , , , , , , , , , , , , 967680, , 215040
, - align=center BGCOLOR="#e0f0e0"
!148
, , , t_0,1,6,7{4,3⁶}, , Heptihexitruncated 8-cube
Exipetitrunki-octeractidiacosipentacontahexazetton, , , , , , , , , , , , , , 516096, , 86016
, - align=center BGCOLOR="#e0e0f0"
!149
, , , t_0,1,5,7{4,3⁶}, , Heptipentitruncated 8-cube
Exiteritruncated octeract, , , , , , , , , , , , , , 1612800, , 215040
, - align=center BGCOLOR="#e0e0f0"
!150
, , , t_0,1,5,6{4,3⁶}, , Hexipentitruncated 8-cube
Petiteritruncated octeract, , , , , , , , , , , , , , 1182720, , 215040
, - align=center BGCOLOR="#e0e0f0"
!151
, , , t_0,1,4,7{4,3⁶}, , Heptisteritruncated 8-cube
Exicellitruncated octeract, , , , , , , , , , , , , , 2293760, , 286720
, - align=center BGCOLOR="#e0e0f0"
!152
, , , t_0,1,4,6{4,3⁶}, , Hexisteritruncated 8-cube
Peticellitruncated octeract, , , , , , , , , , , , , , 3010560, , 430080
, - align=center BGCOLOR="#e0e0f0"
!153
, , , t_0,1,4,5{4,3⁶}, , Pentisteritruncated 8-cube
Tericellitruncated octeract, , , , , , , , , , , , , , 1433600, , 286720
, - align=center BGCOLOR="#e0e0f0"
!154
, , , t_0,1,3,7{4,3⁶}, , Heptiruncitruncated 8-cube
Exiprismatotruncated octeract, , , , , , , , , , , , , , 1612800, , 215040
, - align=center BGCOLOR="#e0e0f0"
!155
, , , t_0,1,3,6{4,3⁶}, , Hexiruncitruncated 8-cube
Petiprismatotruncated octeract, , , , , , , , , , , , , , 3225600, , 430080
, - align=center BGCOLOR="#e0e0f0"
!156
, , , t_0,1,3,5{4,3⁶}, , Pentiruncitruncated 8-cube
Teriprismatotruncated octeract, , , , , , , , , , , , , , 2795520, , 430080
, - align=center BGCOLOR="#e0e0f0"
!157
, , , t_0,1,3,4{4,3⁶}, , Steriruncitruncated 8-cube
Celliprismatotruncated octeract, , , , , , , , , , , , , , 967680, , 215040
, - align=center BGCOLOR="#e0e0f0"
!158
, , , t_0,1,2,7{4,3⁶}, , Hepticantitruncated 8-cube
Exigreatorhombated octeract, , , , , , , , , , , , , , 516096, , 86016
, - align=center BGCOLOR="#e0e0f0"
!159
, , , t_0,1,2,6{4,3⁶}, , Hexicantitruncated 8-cube
Petigreatorhombated octeract, , , , , , , , , , , , , , 1505280, , 215040
, - align=center BGCOLOR="#e0e0f0"
!160
, , , t_0,1,2,5{4,3⁶}, , Penticantitruncated 8-cube
Terigreatorhombated octeract, , , , , , , , , , , , , , 2007040, , 286720
, - align=center BGCOLOR="#e0e0f0"
!161
, , , t_0,1,2,4{4,3⁶}, , Stericantitruncated 8-cube
Celligreatorhombated octeract, , , , , , , , , , , , , , 1290240, , 215040
, - align=center BGCOLOR="#e0e0f0"
!162
, , , t_0,1,2,3{4,3⁶}, , Runcicantitruncated 8-cube
Great prismated octeract, , , , , , , , , , , , , , 344064, , 86016
, - align=center BGCOLOR="#f0e0e0"
!163
, , , t_0,1,2,3,4{3⁶,4}, , Steriruncicantitruncated 8-orthoplex
Great cellated diacosipentacontahexazetton, , , , , , , , , , , , , , 1075200, , 215040
, - align=center BGCOLOR="#f0e0e0"
!164
, , , t_0,1,2,3,5{3⁶,4}, , Pentiruncicantitruncated 8-orthoplex
Terigreatoprismated diacosipentacontahexazetton, , , , , , , , , , , , , , 4193280, , 645120
, - align=center BGCOLOR="#f0e0e0"
!165
, , , t_0,1,2,4,5{3⁶,4}, , Pentistericantitruncated 8-orthoplex
Tericelligreatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 3225600, , 645120
, - align=center BGCOLOR="#f0e0e0"
!166
, , , t_0,1,3,4,5{3⁶,4}, , Pentisteriruncitruncated 8-orthoplex
Tericelliprismatotruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 3225600, , 645120
, - align=center BGCOLOR="#f0e0e0"
!167
, , , t_0,2,3,4,5{3⁶,4}, , Pentisteriruncicantellated 8-orthoplex
Tericelliprismatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 3225600, , 645120
, - align=center BGCOLOR="#f0e0e0"
!168
, , , t_1,2,3,4,5{3⁶,4}, , Bisteriruncicantitruncated 8-orthoplex
Great bicellated diacosipentacontahexazetton, , , , , , , , , , , , , , 2903040, , 645120
, - align=center BGCOLOR="#f0e0e0"
!169
, , , t_0,1,2,3,6{3⁶,4}, , Hexiruncicantitruncated 8-orthoplex
Petigreatoprismated diacosipentacontahexazetton, , , , , , , , , , , , , , 5160960, , 860160
, - align=center BGCOLOR="#f0e0e0"
!170
, , , t_0,1,2,4,6{3⁶,4}, , Hexistericantitruncated 8-orthoplex
Peticelligreatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 7741440, , 1290240
, - align=center BGCOLOR="#f0e0e0"
!171
, , , t_0,1,3,4,6{3⁶,4}, , Hexisteriruncitruncated 8-orthoplex
Peticelliprismatotruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 7096320, , 1290240
, - align=center BGCOLOR="#f0e0e0"
!172
, , , t_0,2,3,4,6{3⁶,4}, , Hexisteriruncicantellated 8-orthoplex
Peticelliprismatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 7096320, , 1290240
, - align=center BGCOLOR="#f0e0e0"
!173
, , , t_1,2,3,4,6{3⁶,4}, , Bipentiruncicantitruncated 8-orthoplex
Biterigreatoprismated diacosipentacontahexazetton, , , , , , , , , , , , , , 6451200, , 1290240
, - align=center BGCOLOR="#f0e0e0"
!174
, , , t_0,1,2,5,6{3⁶,4}, , Hexipenticantitruncated 8-orthoplex
Petiterigreatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 4300800, , 860160
, - align=center BGCOLOR="#f0e0e0"
!175
, , , t_0,1,3,5,6{3⁶,4}, , Hexipentiruncitruncated 8-orthoplex
Petiteriprismatotruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 7096320, , 1290240
, - align=center BGCOLOR="#f0e0e0"
!176
, , , t_0,2,3,5,6{3⁶,4}, , Hexipentiruncicantellated 8-orthoplex
Petiteriprismatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 6451200, , 1290240
, - align=center BGCOLOR="#f0e0e0"
!177
, , , t_1,2,3,5,6{3⁶,4}, , Bipentistericantitruncated 8-orthoplex
Bitericelligreatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 5806080, , 1290240
, - align=center BGCOLOR="#f0e0e0"
!178
, , , t_0,1,4,5,6{3⁶,4}, , Hexipentisteritruncated 8-orthoplex
Petitericellitruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 4300800, , 860160
, - align=center BGCOLOR="#f0e0e0"
!179
, , , t_0,2,4,5,6{3⁶,4}, , Hexipentistericantellated 8-orthoplex
Petitericellirhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 7096320, , 1290240
, - align=center BGCOLOR="#e0e0f0"
!180
, , , t_1,2,3,5,6{4,3⁶}, , Bipentistericantitruncated 8-cube
Bitericelligreatorhombated octeract, , , , , , , , , , , , , , 5806080, , 1290240
, - align=center BGCOLOR="#f0e0e0"
!181
, , , t_0,3,4,5,6{3⁶,4}, , Hexipentisteriruncinated 8-orthoplex
Petitericelliprismated diacosipentacontahexazetton, , , , , , , , , , , , , , 4300800, , 860160
, - align=center BGCOLOR="#e0e0f0"
!182
, , , t_1,2,3,4,6{4,3⁶}, , Bipentiruncicantitruncated 8-cube
Biterigreatoprismated octeract, , , , , , , , , , , , , , 6451200, , 1290240
, - align=center BGCOLOR="#e0e0f0"
!183
, , , t_1,2,3,4,5{4,3⁶}, , Bisteriruncicantitruncated 8-cube
Great bicellated octeract, , , , , , , , , , , , , , 3440640, , 860160
, - align=center BGCOLOR="#f0e0e0"
!184
, , , t_0,1,2,3,7{3⁶,4}, , Heptiruncicantitruncated 8-orthoplex
Exigreatoprismated diacosipentacontahexazetton, , , , , , , , , , , , , , 2365440, , 430080
, - align=center BGCOLOR="#f0e0e0"
!185
, , , t_0,1,2,4,7{3⁶,4}, , Heptistericantitruncated 8-orthoplex
Exicelligreatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 5591040, , 860160
, - align=center BGCOLOR="#f0e0e0"
!186
, , , t_0,1,3,4,7{3⁶,4}, , Heptisteriruncitruncated 8-orthoplex
Exicelliprismatotruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 4730880, , 860160
, - align=center BGCOLOR="#f0e0e0"
!187
, , , t_0,2,3,4,7{3⁶,4}, , Heptisteriruncicantellated 8-orthoplex
Exicelliprismatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 4730880, , 860160
, - align=center BGCOLOR="#e0e0f0"
!188
, , , t_0,3,4,5,6{4,3⁶}, , Hexipentisteriruncinated 8-cube
Petitericelliprismated octeract, , , , , , , , , , , , , , 4300800, , 860160
, - align=center BGCOLOR="#f0e0e0"
!189
, , , t_0,1,2,5,7{3⁶,4}, , Heptipenticantitruncated 8-orthoplex
Exiterigreatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 5591040, , 860160
, - align=center BGCOLOR="#f0e0e0"
!190
, , , t_0,1,3,5,7{3⁶,4}, , Heptipentiruncitruncated 8-orthoplex
Exiteriprismatotruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 8386560, , 1290240
, - align=center BGCOLOR="#f0e0e0"
!191
, , , t_0,2,3,5,7{3⁶,4}, , Heptipentiruncicantellated 8-orthoplex
Exiteriprismatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 7741440, , 1290240
, - align=center BGCOLOR="#e0e0f0"
!192
, , , t_0,2,4,5,6{4,3⁶}, , Hexipentistericantellated 8-cube
Petitericellirhombated octeract, , , , , , , , , , , , , , 7096320, , 1290240
, - align=center BGCOLOR="#f0e0e0"
!193
, , , t_0,1,4,5,7{3⁶,4}, , Heptipentisteritruncated 8-orthoplex
Exitericellitruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 4730880, , 860160
, - align=center BGCOLOR="#e0e0f0"
!194
, , , t_0,2,3,5,7{4,3⁶}, , Heptipentiruncicantellated 8-cube
Exiteriprismatorhombated octeract, , , , , , , , , , , , , , 7741440, , 1290240
, - align=center BGCOLOR="#e0e0f0"
!195
, , , t_0,2,3,5,6{4,3⁶}, , Hexipentiruncicantellated 8-cube
Petiteriprismatorhombated octeract, , , , , , , , , , , , , , 6451200, , 1290240
, - align=center BGCOLOR="#e0e0f0"
!196
, , , t_0,2,3,4,7{4,3⁶}, , Heptisteriruncicantellated 8-cube
Exicelliprismatorhombated octeract, , , , , , , , , , , , , , 4730880, , 860160
, - align=center BGCOLOR="#e0e0f0"
!197
, , , t_0,2,3,4,6{4,3⁶}, , Hexisteriruncicantellated 8-cube
Peticelliprismatorhombated octeract, , , , , , , , , , , , , , 7096320, , 1290240
, - align=center BGCOLOR="#e0e0f0"
!198
, , , t_0,2,3,4,5{4,3⁶}, , Pentisteriruncicantellated 8-cube
Tericelliprismatorhombated octeract, , , , , , , , , , , , , , 3870720, , 860160
, - align=center BGCOLOR="#f0e0e0"
!199
, , , t_0,1,2,6,7{3⁶,4}, , Heptihexicantitruncated 8-orthoplex
Exipetigreatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 2365440, , 430080
, - align=center BGCOLOR="#f0e0e0"
!200
, , , t_0,1,3,6,7{3⁶,4}, , Heptihexiruncitruncated 8-orthoplex
Exipetiprismatotruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 5591040, , 860160
, - align=center BGCOLOR="#e0e0f0"
!201
, , , t_0,1,4,5,7{4,3⁶}, , Heptipentisteritruncated 8-cube
Exitericellitruncated octeract, , , , , , , , , , , , , , 4730880, , 860160
, - align=center BGCOLOR="#e0e0f0"
!202
, , , t_0,1,4,5,6{4,3⁶}, , Hexipentisteritruncated 8-cube
Petitericellitruncated octeract, , , , , , , , , , , , , , 4300800, , 860160
, - align=center BGCOLOR="#e0e0f0"
!203
, , , t_0,1,3,6,7{4,3⁶}, , Heptihexiruncitruncated 8-cube
Exipetiprismatotruncated octeract, , , , , , , , , , , , , , 5591040, , 860160
, - align=center BGCOLOR="#e0e0f0"
!204
, , , t_0,1,3,5,7{4,3⁶}, , Heptipentiruncitruncated 8-cube
Exiteriprismatotruncated octeract, , , , , , , , , , , , , , 8386560, , 1290240
, - align=center BGCOLOR="#e0e0f0"
!205
, , , t_0,1,3,5,6{4,3⁶}, , Hexipentiruncitruncated 8-cube
Petiteriprismatotruncated octeract, , , , , , , , , , , , , , 7096320, , 1290240
, - align=center BGCOLOR="#e0e0f0"
!206
, , , t_0,1,3,4,7{4,3⁶}, , Heptisteriruncitruncated 8-cube
Exicelliprismatotruncated octeract, , , , , , , , , , , , , , 4730880, , 860160
, - align=center BGCOLOR="#e0e0f0"
!207
, , , t_0,1,3,4,6{4,3⁶}, , Hexisteriruncitruncated 8-cube
Peticelliprismatotruncated octeract, , , , , , , , , , , , , , 7096320, , 1290240
, - align=center BGCOLOR="#e0e0f0"
!208
, , , t_0,1,3,4,5{4,3⁶}, , Pentisteriruncitruncated 8-cube
Tericelliprismatotruncated octeract, , , , , , , , , , , , , , 3870720, , 860160
, - align=center BGCOLOR="#e0e0f0"
!209
, , , t_0,1,2,6,7{4,3⁶}, , Heptihexicantitruncated 8-cube
Exipetigreatorhombated octeract, , , , , , , , , , , , , , 2365440, , 430080
, - align=center BGCOLOR="#e0e0f0"
!210
, , , t_0,1,2,5,7{4,3⁶}, , Heptipenticantitruncated 8-cube
Exiterigreatorhombated octeract, , , , , , , , , , , , , , 5591040, , 860160
, - align=center BGCOLOR="#e0e0f0"
!211
, , , t_0,1,2,5,6{4,3⁶}, , Hexipenticantitruncated 8-cube
Petiterigreatorhombated octeract, , , , , , , , , , , , , , 4300800, , 860160
, - align=center BGCOLOR="#e0e0f0"
!212
, , , t_0,1,2,4,7{4,3⁶}, , Heptistericantitruncated 8-cube
Exicelligreatorhombated octeract, , , , , , , , , , , , , , 5591040, , 860160
, - align=center BGCOLOR="#e0e0f0"
!213
, , , t_0,1,2,4,6{4,3⁶}, , Hexistericantitruncated 8-cube
Peticelligreatorhombated octeract, , , , , , , , , , , , , , 7741440, , 1290240
, - align=center BGCOLOR="#e0e0f0"
!214
, , , t_0,1,2,4,5{4,3⁶}, , Pentistericantitruncated 8-cube
Tericelligreatorhombated octeract, , , , , , , , , , , , , , 3870720, , 860160
, - align=center BGCOLOR="#e0e0f0"
!215
, , , t_0,1,2,3,7{4,3⁶}, , Heptiruncicantitruncated 8-cube
Exigreatoprismated octeract, , , , , , , , , , , , , , 2365440, , 430080
, - align=center BGCOLOR="#e0e0f0"
!216
, , , t_0,1,2,3,6{4,3⁶}, , Hexiruncicantitruncated 8-cube
Petigreatoprismated octeract, , , , , , , , , , , , , , 5160960, , 860160
, - align=center BGCOLOR="#e0e0f0"
!217
, , , t_0,1,2,3,5{4,3⁶}, , Pentiruncicantitruncated 8-cube
Terigreatoprismated octeract, , , , , , , , , , , , , , 4730880, , 860160
, - align=center BGCOLOR="#e0e0f0"
!218
, , , t_0,1,2,3,4{4,3⁶}, , Steriruncicantitruncated 8-cube
Great cellated octeract, , , , , , , , , , , , , , 1720320, , 430080
, - align=center BGCOLOR="#f0e0e0"
!219
, , , t_0,1,2,3,4,5{3⁶,4}, , Pentisteriruncicantitruncated 8-orthoplex
Great terated diacosipentacontahexazetton, , , , , , , , , , , , , , 5806080, , 1290240
, - align=center BGCOLOR="#f0e0e0"
!220
, , , t_0,1,2,3,4,6{3⁶,4}, , Hexisteriruncicantitruncated 8-orthoplex
Petigreatocellated diacosipentacontahexazetton, , , , , , , , , , , , , , 12902400, , 2580480
, - align=center BGCOLOR="#f0e0e0"
!221
, , , t_0,1,2,3,5,6{3⁶,4}, , Hexipentiruncicantitruncated 8-orthoplex
Petiterigreatoprismated diacosipentacontahexazetton, , , , , , , , , , , , , , 11612160, , 2580480
, - align=center BGCOLOR="#f0e0e0"
!222
, , , t_0,1,2,4,5,6{3⁶,4}, , Hexipentistericantitruncated 8-orthoplex
Petitericelligreatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 11612160, , 2580480
, - align=center BGCOLOR="#f0e0e0"
!223
, , , t_0,1,3,4,5,6{3⁶,4}, , Hexipentisteriruncitruncated 8-orthoplex
Petitericelliprismatotruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 11612160, , 2580480
, - align=center BGCOLOR="#f0e0e0"
!224
, , , t_0,2,3,4,5,6{3⁶,4}, , Hexipentisteriruncicantellated 8-orthoplex
Petitericelliprismatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 11612160, , 2580480
, - align=center BGCOLOR="#e0f0e0"
!225
, , , t_1,2,3,4,5,6{4,3⁶}, , Bipentisteriruncicantitruncated 8-cube
Great biteri-octeractidiacosipentacontahexazetton, , , , , , , , , , , , , , 10321920, , 2580480
, - align=center BGCOLOR="#f0e0e0"
!226
, , , t_0,1,2,3,4,7{3⁶,4}, , Heptisteriruncicantitruncated 8-orthoplex
Exigreatocellated diacosipentacontahexazetton, , , , , , , , , , , , , , 8601600, , 1720320
, - align=center BGCOLOR="#f0e0e0"
!227
, , , t_0,1,2,3,5,7{3⁶,4}, , Heptipentiruncicantitruncated 8-orthoplex
Exiterigreatoprismated diacosipentacontahexazetton, , , , , , , , , , , , , , 14192640, , 2580480
, - align=center BGCOLOR="#f0e0e0"
!228
, , , t_0,1,2,4,5,7{3⁶,4}, , Heptipentistericantitruncated 8-orthoplex
Exitericelligreatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 12902400, , 2580480
, - align=center BGCOLOR="#f0e0e0"
!229
, , , t_0,1,3,4,5,7{3⁶,4}, , Heptipentisteriruncitruncated 8-orthoplex
Exitericelliprismatotruncated diacosipentacontahexazetton, , , , , , , , , , , , , , 12902400, , 2580480
, - align=center BGCOLOR="#e0f0e0"
!230
, , , t_0,2,3,4,5,7{4,3⁶}, , Heptipentisteriruncicantellated 8-cube
Exitericelliprismatorhombi-octeractidiacosipentacontahexazetton, , , , , , , , , , , , , , 12902400, , 2580480
, - align=center BGCOLOR="#e0e0f0"
!231
, , , t_0,2,3,4,5,6{4,3⁶}, , Hexipentisteriruncicantellated 8-cube
Petitericelliprismatorhombated octeract, , , , , , , , , , , , , , 11612160, , 2580480
, - align=center BGCOLOR="#f0e0e0"
!232
, , , t_0,1,2,3,6,7{3⁶,4}, , Heptihexiruncicantitruncated 8-orthoplex
Exipetigreatoprismated diacosipentacontahexazetton, , , , , , , , , , , , , , 8601600, , 1720320
, - align=center BGCOLOR="#f0e0e0"
!233
, , , t_0,1,2,4,6,7{3⁶,4}, , Heptihexistericantitruncated 8-orthoplex
Exipeticelligreatorhombated diacosipentacontahexazetton, , , , , , , , , , , , , , 14192640, , 2580480
, - align=center BGCOLOR="#e0f0e0"
!234
, , , t_0,1,3,4,6,7{4,3⁶}, , Heptihexisteriruncitruncated 8-cube
Exipeticelliprismatotrunki-octeractidiacosipentacontahexazetton, , , , , , , , , , , , , , 12902400, , 2580480
, - align=center BGCOLOR="#e0e0f0"
!235
, , , t_0,1,3,4,5,7{4,3⁶}, , Heptipentisteriruncitruncated 8-cube
Exitericelliprismatotruncated octeract, , , , , , , , , , , , , , 12902400, , 2580480
, - align=center BGCOLOR="#e0e0f0"
!236
, , , t_0,1,3,4,5,6{4,3⁶}, , Hexipentisteriruncitruncated 8-cube
Petitericelliprismatotruncated octeract, , , , , , , , , , , , , , 11612160, , 2580480
, - align=center BGCOLOR="#e0f0e0"
!237
, , , t_0,1,2,5,6,7{4,3⁶}, , Heptihexipenticantitruncated 8-cube
Exipetiterigreatorhombi-octeractidiacosipentacontahexazetton, , , , , , , , , , , , , , 8601600, , 1720320
, - align=center BGCOLOR="#e0e0f0"
!238
, , , t_0,1,2,4,6,7{4,3⁶}, , Heptihexistericantitruncated 8-cube
Exipeticelligreatorhombated octeract, , , , , , , , , , , , , , 14192640, , 2580480
, - align=center BGCOLOR="#e0e0f0"
!239
, , , t_0,1,2,4,5,7{4,3⁶}, , Heptipentistericantitruncated 8-cube
Exitericelligreatorhombated octeract, , , , , , , , , , , , , , 12902400, , 2580480
, - align=center BGCOLOR="#e0e0f0"
!240
, , , t_0,1,2,4,5,6{4,3⁶}, , Hexipentistericantitruncated 8-cube
Petitericelligreatorhombated octeract, , , , , , , , , , , , , , 11612160, , 2580480
, - align=center BGCOLOR="#e0e0f0"
!241
, , , t_0,1,2,3,6,7{4,3⁶}, , Heptihexiruncicantitruncated 8-cube
Exipetigreatoprismated octeract, , , , , , , , , , , , , , 8601600, , 1720320
, - align=center BGCOLOR="#e0e0f0"
!242
, , , t_0,1,2,3,5,7{4,3⁶}, , Heptipentiruncicantitruncated 8-cube
Exiterigreatoprismated octeract, , , , , , , , , , , , , , 14192640, , 2580480
, - align=center BGCOLOR="#e0e0f0"
!243
, , , t_0,1,2,3,5,6{4,3⁶}, , Hexipentiruncicantitruncated 8-cube
Petiterigreatoprismated octeract, , , , , , , , , , , , , , 11612160, , 2580480
, - align=center BGCOLOR="#e0e0f0"
!244
, , , t_0,1,2,3,4,7{4,3⁶}, , Heptisteriruncicantitruncated 8-cube
Exigreatocellated octeract, , , , , , , , , , , , , , 8601600, , 1720320
, - align=center BGCOLOR="#e0e0f0"
!245
, , , t_0,1,2,3,4,6{4,3⁶}, , Hexisteriruncicantitruncated 8-cube
Petigreatocellated octeract, , , , , , , , , , , , , , 12902400, , 2580480
, - align=center BGCOLOR="#e0e0f0"
!246
, , , t_0,1,2,3,4,5{4,3⁶}, , Pentisteriruncicantitruncated 8-cube
Great terated octeract, , , , , , , , , , , , , , 6881280, , 1720320
, - align=center BGCOLOR="#f0e0e0"
!247
, , , t_{0,1,2,3,4,5,6}{3⁶,4}, , Hexipentisteriruncicantitruncated 8-orthoplex
Great petated diacosipentacontahexazetton, , , , , , , , , , , , , , 20643840, , 5160960
, - align=center BGCOLOR="#f0e0e0"
!248
, , , t_{0,1,2,3,4,5,7}{3⁶,4}, , Heptipentisteriruncicantitruncated 8-orthoplex
Exigreatoterated diacosipentacontahexazetton, , , , , , , , , , , , , , 23224320, , 5160960
, - align=center BGCOLOR="#f0e0e0"
!249
, , , t_{0,1,2,3,4,6,7}{3⁶,4}, , Heptihexisteriruncicantitruncated 8-orthoplex
Exipetigreatocellated diacosipentacontahexazetton, , , , , , , , , , , , , , 23224320, , 5160960
, - align=center BGCOLOR="#f0e0e0"
!250
, , , t_{0,1,2,3,5,6,7}{3⁶,4}, , Heptihexipentiruncicantitruncated 8-orthoplex
Exipetiterigreatoprismated diacosipentacontahexazetton, , , , , , , , , , , , , , 23224320, , 5160960
, - align=center BGCOLOR="#e0e0f0"
!251
, , , t_{0,1,2,3,5,6,7}{4,3⁶}, , Heptihexipentiruncicantitruncated 8-cube
Exipetiterigreatoprismated octeract, , , , , , , , , , , , , , 23224320, , 5160960
, - align=center BGCOLOR="#e0e0f0"
!252
, , , t_{0,1,2,3,4,6,7}{4,3⁶}, , Heptihexisteriruncicantitruncated 8-cube
Exipetigreatocellated octeract, , , , , , , , , , , , , , 23224320, , 5160960
, - align=center BGCOLOR="#e0e0f0"
!253
, , , t_{0,1,2,3,4,5,7}{4,3⁶}, , Heptipentisteriruncicantitruncated 8-cube
Exigreatoterated octeract, , , , , , , , , , , , , , 23224320, , 5160960
, - align=center BGCOLOR="#e0e0f0"
!254
, , , t_{0,1,2,3,4,5,6}{4,3⁶}, , Hexipentisteriruncicantitruncated 8-cube
Great petated octeract, , , , , , , , , , , , , , 20643840, , 5160960
, - align=center BGCOLOR="#e0f0e0"
!255
, , , t_{0,1,2,3,4,5,6,7}{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 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.

{, class="wikitable collapsible collapsed"
!colspan=15, D₈ uniform polytopes
, -
!rowspan=2, #
!rowspan=2, Coxeter-Dynkin diagram
!rowspan=2, Name
!rowspan=2, Base point
(Alternately signed)
!colspan=8, Element counts
!rowspan=2, Circumrad
, -
!7, , 6, , 5, , 4, , 3, , 2, , 1, , 0
, - align=center
!1
, ,
= , , 8-demicube
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
, - align=center
!2
, ,
= , , cantic 8-cube
h₂{4,3,3,3,3,3,3}, , (1,1,3,3,3,3,3,3), , , , , , , , , , , , , , 23296, , 3584, , 2.6457512
, - align=center
!3
, ,
= , , runcic 8-cube
h₃{4,3,3,3,3,3,3}, , (1,1,1,3,3,3,3,3), , , , , , , , , , , , , , 64512, , 7168, , 2.4494896
, - align=center
!4
, ,
= , , steric 8-cube
h₄{4,3,3,3,3,3,3}, , (1,1,1,1,3,3,3,3), , , , , , , , , , , , , , 98560, , 8960, , 2.2360678
, - align=center
!5
, ,
= , , pentic 8-cube
h₅{4,3,3,3,3,3,3}, , (1,1,1,1,1,3,3,3), , , , , , , , , , , , , , 89600, , 7168, , 1.9999999
, - align=center
!6
, ,
= , , hexic 8-cube
h₆{4,3,3,3,3,3,3}, , (1,1,1,1,1,1,3,3), , , , , , , , , , , , , , 48384, , 3584, , 1.7320508
, - align=center
!7
, ,
= , , heptic 8-cube
h₇{4,3,3,3,3,3,3}, , (1,1,1,1,1,1,1,3), , , , , , , , , , , , , , 14336, , 1024, , 1.4142135
, - align=center
!8
, ,
= , , runcicantic 8-cube
h_2,3{4,3,3,3,3,3,3}, , (1,1,3,5,5,5,5,5), , , , , , , , , , , , , , 86016, , 21504, , 4.1231055
, - align=center
!9
, ,
= , , stericantic 8-cube
h_2,4{4,3,3,3,3,3,3}, , (1,1,3,3,5,5,5,5), , , , , , , , , , , , , , 349440, , 53760, , 3.8729835
, - align=center
!10
, ,
= , , steriruncic 8-cube
h_3,4{4,3,3,3,3,3,3}, , (1,1,1,3,5,5,5,5), , , , , , , , , , , , , , 179200, , 35840, , 3.7416575
, - align=center
!11
, ,
= , , penticantic 8-cube
h_2,5{4,3,3,3,3,3,3}, , (1,1,3,3,3,5,5,5), , , , , , , , , , , , , , 573440, , 71680, , 3.6055512
, - align=center
!12
, ,
= , , pentiruncic 8-cube
h_3,5{4,3,3,3,3,3,3}, , (1,1,1,3,3,5,5,5), , , , , , , , , , , , , , 537600, , 71680, , 3.4641016
, - align=center
!13
, ,
= , , pentisteric 8-cube
h_4,5{4,3,3,3,3,3,3}, , (1,1,1,1,3,5,5,5), , , , , , , , , , , , , , 232960, , 35840, , 3.3166249
, - align=center
!14
, ,
= , , hexicantic 8-cube
h_2,6{4,3,3,3,3,3,3}, , (1,1,3,3,3,3,5,5), , , , , , , , , , , , , , 456960, , 53760, , 3.3166249
, - align=center
!15
, ,
= , , hexicruncic 8-cube
h_3,6{4,3,3,3,3,3,3}, , (1,1,1,3,3,3,5,5), , , , , , , , , , , , , , 645120, , 71680, , 3.1622777
, - align=center
!16
, ,
= , , hexisteric 8-cube
h_4,6{4,3,3,3,3,3,3}, , (1,1,1,1,3,3,5,5), , , , , , , , , , , , , , 483840, , 53760, , 3
, - align=center
!17
, ,
= , , hexipentic 8-cube
h_5,6{4,3,3,3,3,3,3}, , (1,1,1,1,1,3,5,5), , , , , , , , , , , , , , 182784, , 21504, , 2.8284271
, - align=center
!18
, ,
= , , hepticantic 8-cube
h_2,7{4,3,3,3,3,3,3}, , (1,1,3,3,3,3,3,5), , , , , , , , , , , , , , 172032, , 21504, , 3
, - align=center
!19
, ,
= , , heptiruncic 8-cube
h_3,7{4,3,3,3,3,3,3}, , (1,1,1,3,3,3,3,5), , , , , , , , , , , , , , 340480, , 35840, , 2.8284271
, - align=center
!20
, ,
= , , heptsteric 8-cube
h_4,7{4,3,3,3,3,3,3}, , (1,1,1,1,3,3,3,5), , , , , , , , , , , , , , 376320, , 35840, , 2.6457512
, - align=center
!21
, ,
= , , heptipentic 8-cube
h_5,7{4,3,3,3,3,3,3}, , (1,1,1,1,1,3,3,5), , , , , , , , , , , , , , 236544, , 21504, , 2.4494898
, - align=center
!22
, ,
= , , heptihexic 8-cube
h_6,7{4,3,3,3,3,3,3}, , (1,1,1,1,1,1,3,5), , , , , , , , , , , , , , 78848, , 7168, , 2.236068
, - align=center
!23
, ,
= , , steriruncicantic 8-cube
h_2,3,4{4,3⁶}, , (1,1,3,5,7,7,7,7), , , , , , , , , , , , , , 430080, , 107520, , 5.3851647
, - align=center
!24
, ,
= , , pentiruncicantic 8-cube
h_2,3,5{4,3⁶}, , (1,1,3,5,5,7,7,7), , , , , , , , , , , , , , 1182720, , 215040, , 5.0990195
, - align=center
!25
, ,
= , , pentistericantic 8-cube
h_2,4,5{4,3⁶}, , (1,1,3,3,5,7,7,7), , , , , , , , , , , , , , 1075200, , 215040, , 4.8989797
, - align=center
!26
, ,
= , , pentisterirunic 8-cube
h_3,4,5{4,3⁶}, , (1,1,1,3,5,7,7,7), , , , , , , , , , , , , , 716800, , 143360, , 4.7958317
, - align=center
!27
, ,
= , , hexiruncicantic 8-cube
h_2,3,6{4,3⁶}, , (1,1,3,5,5,5,7,7), , , , , , , , , , , , , , 1290240, , 215040, , 4.7958317
, - align=center
!28
, ,
= , , hexistericantic 8-cube
h_2,4,6{4,3⁶}, , (1,1,3,3,5,5,7,7), , , , , , , , , , , , , , 2096640, , 322560, , 4.5825758
, - align=center
!29
, ,
= , , hexisterirunic 8-cube
h_3,4,6{4,3⁶}, , (1,1,1,3,5,5,7,7), , , , , , , , , , , , , , 1290240, , 215040, , 4.472136
, - align=center
!30
, ,
= , , hexipenticantic 8-cube
h_2,5,6{4,3⁶}, , (1,1,3,3,3,5,7,7), , , , , , , , , , , , , , 1290240, , 215040, , 4.3588991
, - align=center
!31
, ,
= , , hexipentirunic 8-cube
h_3,5,6{4,3⁶}, , (1,1,1,3,3,5,7,7), , , , , , , , , , , , , , 1397760, , 215040, , 4.2426405
, - align=center
!32
, ,
= , , hexipentisteric 8-cube
h_4,5,6{4,3⁶}, , (1,1,1,1,3,5,7,7), , , , , , , , , , , , , , 698880, , 107520, , 4.1231055
, - align=center
!33
, ,
= , , heptiruncicantic 8-cube
h_2,3,7{4,3⁶}, , (1,1,3,5,5,5,5,7), , , , , , , , , , , , , , 591360, , 107520, , 4.472136
, - align=center
!34
, ,
= , , heptistericantic 8-cube
h_2,4,7{4,3⁶}, , (1,1,3,3,5,5,5,7), , , , , , , , , , , , , , 1505280, , 215040, , 4.2426405
, - align=center
!35
, ,
= , , heptisterruncic 8-cube
h_3,4,7{4,3⁶}, , (1,1,1,3,5,5,5,7), , , , , , , , , , , , , , 860160, , 143360, , 4.1231055
, - align=center
!36
, ,
= , , heptipenticantic 8-cube
h_2,5,7{4,3⁶}, , (1,1,3,3,3,5,5,7), , , , , , , , , , , , , , 1612800, , 215040, , 4
, - align=center
!37
, ,
= , , heptipentiruncic 8-cube
h_3,5,7{4,3⁶}, , (1,1,1,3,3,5,5,7), , , , , , , , , , , , , , 1612800, , 215040, , 3.8729835
, - align=center
!38
, ,
= , , heptipentisteric 8-cube
h_4,5,7{4,3⁶}, , (1,1,1,1,3,5,5,7), , , , , , , , , , , , , , 752640, , 107520, , 3.7416575
, - align=center
!39
, ,
= , , heptihexicantic 8-cube
h_2,6,7{4,3⁶}, , (1,1,3,3,3,3,5,7), , , , , , , , , , , , , , 752640, , 107520, , 3.7416575
, - align=center
!40
, ,
= , , heptihexiruncic 8-cube
h_3,6,7{4,3⁶}, , (1,1,1,3,3,3,5,7), , , , , , , , , , , , , , 1146880, , 143360, , 3.6055512
, - align=center
!41
, ,
= , , heptihexisteric 8-cube
h_4,6,7{4,3⁶}, , (1,1,1,1,3,3,5,7), , , , , , , , , , , , , , 913920, , 107520, , 3.4641016
, - align=center
!42
, ,
= , , heptihexipentic 8-cube
h_5,6,7{4,3⁶}, , (1,1,1,1,1,3,5,7), , , , , , , , , , , , , , 365568, , 43008, , 3.3166249
, - align=center
!43
, ,
= , , pentisteriruncicantic 8-cube
h_2,3,4,5{4,3⁶}, , (1,1,3,5,7,9,9,9), , , , , , , , , , , , , , 1720320, , 430080, , 6.4031243
, - align=center
!44
, ,
= , , hexisteriruncicantic 8-cube
h_2,3,4,6{4,3⁶}, , (1,1,3,5,7,7,9,9), , , , , , , , , , , , , , 3225600, , 645120, , 6.0827627
, - align=center
!45
, ,
= , , hexipentiruncicantic 8-cube
h_2,3,5,6{4,3⁶}, , (1,1,3,5,5,7,9,9), , , , , , , , , , , , , , 2903040, , 645120, , 5.8309517
, - align=center
!46
, ,
= , , hexipentistericantic 8-cube
h_2,4,5,6{4,3⁶}, , (1,1,3,3,5,7,9,9), , , , , , , , , , , , , , 3225600, , 645120, , 5.6568542
, - align=center
!47
, ,
= , , hexipentisteriruncic 8-cube
h_3,4,5,6{4,3⁶}, , (1,1,1,3,5,7,9,9), , , , , , , , , , , , , , 2150400, , 430080, , 5.5677648
, - align=center
!48
, ,
= , , heptsteriruncicantic 8-cube
h_2,3,4,7{4,3⁶}, , (1,1,3,5,7,7,7,9), , , , , , , , , , , , , , 2150400, , 430080, , 5.7445626
, - align=center
!49
, ,
= , , heptipentiruncicantic 8-cube
h_2,3,5,7{4,3⁶}, , (1,1,3,5,5,7,7,9), , , , , , , , , , , , , , 3548160, , 645120, , 5.4772258
, - align=center
!50
, ,
= , , heptipentistericantic 8-cube
h_2,4,5,7{4,3⁶}, , (1,1,3,3,5,7,7,9), , , , , , , , , , , , , , 3548160, , 645120, , 5.291503
, - align=center
!51
, ,
= , , heptipentisteriruncic 8-cube
h_3,4,5,7{4,3⁶}, , (1,1,1,3,5,7,7,9), , , , , , , , , , , , , , 2365440, , 430080, , 5.1961527
, - align=center
!52
, ,
= , , heptihexiruncicantic 8-cube
h_2,3,6,7{4,3⁶}, , (1,1,3,5,5,5,7,9), , , , , , , , , , , , , , 2150400, , 430080, , 5.1961527
, - align=center
!53
, ,
= , , heptihexistericantic 8-cube
h_2,4,6,7{4,3⁶}, , (1,1,3,3,5,5,7,9), , , , , , , , , , , , , , 3870720, , 645120, , 5
, - align=center
!54
, ,
= , , heptihexisteriruncic 8-cube
h_3,4,6,7{4,3⁶}, , (1,1,1,3,5,5,7,9), , , , , , , , , , , , , , 2365440, , 430080, , 4.8989797
, - align=center
!55
, ,
= , , heptihexipenticantic 8-cube
h_2,5,6,7{4,3⁶}, , (1,1,3,3,3,5,7,9), , , , , , , , , , , , , , 2580480, , 430080, , 4.7958317
, - align=center
!56
, ,
= , , heptihexipentiruncic 8-cube
h_3,5,6,7{4,3⁶}, , (1,1,1,3,3,5,7,9), , , , , , , , , , , , , , 2795520, , 430080, , 4.6904159
, - align=center
!57
, ,
= , , heptihexipentisteric 8-cube
h_4,5,6,7{4,3⁶}, , (1,1,1,1,3,5,7,9), , , , , , , , , , , , , , 1397760, , 215040, , 4.5825758
, - align=center
!58
, ,
= , , hexipentisteriruncicantic 8-cube
h_2,3,4,5,6{4,3⁶}, , (1,1,3,5,7,9,11,11), , , , , , , , , , , , , , 5160960, , 1290240, , 7.1414285
, - align=center
!59
, ,
= , , heptipentisteriruncicantic 8-cube
h_2,3,4,5,7{4,3⁶}, , (1,1,3,5,7,9,9,11), , , , , , , , , , , , , , 5806080, , 1290240, , 6.78233
, - align=center
!60
, ,
= , , heptihexisteriruncicantic 8-cube
h_2,3,4,6,7{4,3⁶}, , (1,1,3,5,7,7,9,11), , , , , , , , , , , , , , 5806080, , 1290240, , 6.480741
, - align=center
!61
, ,
= , , heptihexipentiruncicantic 8-cube
h_2,3,5,6,7{4,3⁶}, , (1,1,3,5,5,7,9,11), , , , , , , , , , , , , , 5806080, , 1290240, , 6.244998
, - align=center
!62
, ,
= , , heptihexipentistericantic 8-cube
h_2,4,5,6,7{4,3⁶}, , (1,1,3,3,5,7,9,11), , , , , , , , , , , , , , 6451200, , 1290240, , 6.0827627
, - align=center
!63
, ,
= , , heptihexipentisteriruncic 8-cube
h_3,4,5,6,7{4,3⁶}, , (1,1,1,3,5,7,9,11), , , , , , , , , , , , , , 4300800, , 860160, , 6.0000000
, - align=center
!64
, ,
= , , heptihexipentisteriruncicantic 8-cube
h_2,3,4,5,6,7{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 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.

{, class="wikitable collapsible collapsed"
!colspan=15, E₈ uniform polytopes
, -
!rowspan=2, #
!rowspan=2, Coxeter-Dynkin diagram

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

Regular and uniform honeycombs

There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 7-space:
{, class="wikitable"
, -
!#
!colspan=2, Coxeter group
! Coxeter diagram
!Forms
, - align=center
, 1, , ${\tilde{A_7$, , ^[8/sup>">.html" ;"title="[8">[8/sup>">">, 29
, - align=center
, 2, , ${\tilde{C_7$, , , , 135
, - align=center
, 3, , ${\tilde{B_7$, , [4,3⁴,3^1,1, , , 191 (64 new)
, - align=center
, 4, , ${\tilde{D_7$, , , , 77 (10 new)
, - align=center
, 5, , ${\tilde{E_7$, , , , 143

Regular and uniform tessellations include:
* ${\tilde{A_7$ 29 uniquely ringed forms, including:
** 7-simplex honeycomb: {3<sup>[8/sup>}
* ${\tilde{C_7$ 135 uniquely ringed forms, including:
** Regular 7-cube honeycomb">.html" ;"title="7-simplex honeycomb: {3[8">7-simplex honeycomb: {3[8/sup>}
* ${\tilde{C_7$ 135 uniquely ringed forms, including:
** Regular 7-cube honeycomb: {4,34,4} = {4,34,31,1}, =
*${\tilde{B_7$ 191 uniquely ringed forms, 127 shared with ${\tilde{C_7$, and 64 new, including:
** 7-demicube honeycomb: h{4,34,4} = {31,1,34,4}, =
* ${\tilde{D_7$, [31,1,33,31,1]: 77 unique ring permutations, and 10 are new, the first Coxeter called a quarter 7-cubic honeycomb.
** , , , , , , , , ,
* ${\tilde{E_7$ 143 uniquely ringed forms, including:
** 1₃₃ honeycomb: {3,33,3},
** 3₃₁ honeycomb: {3,3,3,33,1},</sup>

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.

{, class=wikitable
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">align=right">${\bar{Q_7$ = align=${\bar{S_7$ = align=${\bar{T_7$ = T. Gosset: ''On the Regular and Semi-Regular Figures in Space of n Dimensions'', 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
* Harold Scott MacDonald Coxeter">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,
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]
* Norman Johnson (mathematician), N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. Dissertation, University of Toronto, 1966
*

External links

Polytope names

{{DEFAULTSORT:8-Polytope
8-polytopes