![]() 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Runcinated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Biruncinated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Triruncinated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Runcitruncated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Biruncitruncated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Triruncitruncated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Runcicantellated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Biruncicantellated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Runcicantitruncated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Biruncicantitruncated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() Triruncicantitruncated 8-simplex ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Orthogonal projections in A8 Coxeter plane |
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In eight-dimensional geometry, a runcinated 8-simplex is a convex uniform 8-polytope with 3rd order truncations (runcination) of the regular 8-simplex.
There are eleven unique runcinations of the 8-simplex, including permutations of truncation and cantellation. The triruncinated 8-simplex and triruncicanti
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Transcription
Runcinated 8-simplex
Runcinated 8-simplex | |
---|---|
Type | uniform 8-polytope |
Schläfli symbol | t0,3{3,3,3,3,3,3,3} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 4536 |
Vertices | 504 |
Vertex figure | |
Coxeter group | A8, [37], order 362880 |
Properties | convex |
Alternate names
- Runcinated enneazetton
- Small prismated enneazetton (Acronym: spene) (Jonathan Bowers)[1]
Coordinates
The Cartesian coordinates of the vertices of the runcinated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,0,1,1,1,2). This construction is based on facets of the runcinated 9-orthoplex.
Images
Ak Coxeter plane | A8 | A7 | A6 | A5 |
---|---|---|---|---|
Graph | ![]() |
![]() |
![]() |
![]() |
Dihedral symmetry | [9] | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
![]() |
![]() | |
Dihedral symmetry | [5] | [4] | [3] |
Biruncinated 8-simplex
Biruncinated 8-simplex | |
---|---|
Type | uniform 8-polytope |
Schläfli symbol | t1,4{3,3,3,3,3,3,3} |
Coxeter-Dynkin diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7-faces | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 11340 |
Vertices | 1260 |
Vertex figure | |
Coxeter group | A8, [37], order 362880 |
Properties | convex |
Alternate names
- Biruncinated enneazetton
- Small biprismated enneazetton (Acronym: sabpene) (Jonathan Bowers)[2]
Coordinates
The Cartesian coordinates of the vertices of the biruncinated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,0,1,1,1,2,2). This construction is based on facets of the biruncinated 9-orthoplex.
Images
Ak Coxeter plane | A8 | A7 | A6 | A5 |
---|---|---|---|---|
Graph | ![]() |
![]() |
![]() |
![]() |
Dihedral symmetry | [9] | [8] | [7] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
![]() |
![]() | |
Dihedral symmetry | [5] | [4] | [3] |
Triruncinated 8-simplex
Triruncinated 8-simplex | |
---|---|
Type | uniform 8-polytope |
Schläfli symbol | t2,5{3,3,3,3,3,3,3} |
Coxeter-Dynkin diagrams | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7-faces | |
6-faces | |
5-faces | |
4-faces | |
Cells | |
Faces | |
Edges | 15120 |
Vertices | 1680 |
Vertex figure | |
Coxeter group | A8×2, [[37]], order 725760 |
Properties | convex |
Alternate names
- Triruncinated enneazetton
- Small triprismated enneazetton (Acronym: satpeb) (Jonathan Bowers)[3]
Coordinates
The Cartesian coordinates of the vertices of the triruncinated 8-simplex can be most simply positioned in 9-space as permutations of (0,0,0,1,1,1,2,2,2). This construction is based on facets of the triruncinated 9-orthoplex.
Images
Ak Coxeter plane | A8 | A7 | A6 | A5 |
---|---|---|---|---|
Graph | ![]() |
![]() |
![]() |
![]() |
Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
![]() |
![]() | |
Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Runcitruncated 8-simplex
Images
Ak Coxeter plane | A8 | A7 | A6 | A5 |
---|---|---|---|---|
Graph | ![]() |
![]() |
![]() |
![]() |
Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
![]() |
![]() | |
Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Biruncitruncated 8-simplex
Images
Ak Coxeter plane | A8 | A7 | A6 | A5 |
---|---|---|---|---|
Graph | ![]() |
![]() |
![]() |
![]() |
Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
![]() |
![]() | |
Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Triruncitruncated 8-simplex
Images
Ak Coxeter plane | A8 | A7 | A6 | A5 |
---|---|---|---|---|
Graph | ![]() |
![]() |
![]() |
![]() |
Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
![]() |
![]() | |
Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Runcicantellated 8-simplex
Images
Ak Coxeter plane | A8 | A7 | A6 | A5 |
---|---|---|---|---|
Graph | ![]() |
![]() |
![]() |
![]() |
Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
![]() |
![]() | |
Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Biruncicantellated 8-simplex
Images
Ak Coxeter plane | A8 | A7 | A6 | A5 |
---|---|---|---|---|
Graph | ![]() |
![]() |
![]() |
![]() |
Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
![]() |
![]() | |
Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Runcicantitruncated 8-simplex
Images
Ak Coxeter plane | A8 | A7 | A6 | A5 |
---|---|---|---|---|
Graph | ![]() |
![]() |
![]() |
![]() |
Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
![]() |
![]() | |
Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Biruncicantitruncated 8-simplex
Images
Ak Coxeter plane | A8 | A7 | A6 | A5 |
---|---|---|---|---|
Graph | ![]() |
![]() |
![]() |
![]() |
Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
![]() |
![]() | |
Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Triruncicantitruncated 8-simplex
Images
Ak Coxeter plane | A8 | A7 | A6 | A5 |
---|---|---|---|---|
Graph | ![]() |
![]() |
![]() |
![]() |
Dihedral symmetry | [[9]] = [18] | [8] | [[7]] = [14] | [6] |
Ak Coxeter plane | A4 | A3 | A2 | |
Graph | ![]() |
![]() |
![]() | |
Dihedral symmetry | [[5]] = [10] | [4] | [[3]] = [6] |
Related polytopes
This polytope is one of 135 uniform 8-polytopes with A8 symmetry.
A8 polytopes | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
![]() t0 |
![]() t1 |
![]() t2 |
![]() t3 |
![]() t01 |
![]() t02 |
![]() t12 |
![]() t03 |
![]() t13 |
![]() t23 |
![]() t04 |
![]() t14 |
![]() t24 |
![]() t34 |
![]() t05 |
![]() t15 |
![]() t25 |
![]() t06 |
![]() t16 |
![]() t07 |
![]() t012 |
![]() t013 |
![]() t023 |
![]() t123 |
![]() t014 |
![]() t024 |
![]() t124 |
![]() t034 |
![]() t134 |
![]() t234 |
![]() t<sub>015</sub> |
![]() t<sub>025</sub> |
![]() t125 |
![]() t<sub>035</sub> |
![]() t135 |
![]() t<sub>235</sub> |
![]() t<sub>045</sub> |
![]() t145 |
![]() t<sub>016</sub> |
![]() t<sub>026</sub> |
![]() t<sub>126</sub> |
![]() t<sub>036</sub> |
![]() t<sub>136</sub> |
![]() t<sub>046</sub> |
![]() t<sub>056</sub> |
![]() t<sub>017</sub> |
![]() t<sub>027</sub> |
![]() t<sub>037</sub> |
![]() t0123 |
![]() t0124 |
![]() t0134 |
![]() t0234 |
![]() t1234 |
![]() t<sub>0125</sub> |
![]() t<sub>0135</sub> |
![]() t<sub>0235</sub> |
![]() t1235 |
![]() t<sub>0145</sub> |
![]() t<sub>0245</sub> |
![]() t1245 |
![]() t<sub>0345</sub> |
![]() t1345 |
![]() t2345 |
![]() t<sub>0126</sub> |
![]() t<sub>0136</sub> |
![]() t<sub>0236</sub> |
![]() t<sub>1236</sub> |
![]() t<sub>0146</sub> |
![]() t<sub>0246</sub> |
![]() t<sub>1246</sub> |
![]() t<sub>0346</sub> |
![]() t<sub>1346</sub> |
![]() t<sub>0156</sub> |
![]() t<sub>0256</sub> |
![]() t<sub>1256</sub> |
![]() t<sub>0356</sub> |
![]() t<sub>0456</sub> |
![]() t<sub>0127</sub> |
![]() t<sub>0137</sub> |
![]() t<sub>0237</sub> |
![]() t<sub>0147</sub> |
![]() t<sub>0247</sub> |
![]() t<sub>0347</sub> |
![]() t<sub>0157</sub> |
![]() t<sub>0257</sub> |
![]() t<sub>0167</sub> |
![]() t01234 |
![]() t<sub>01235</sub> |
![]() t<sub>01245</sub> |
![]() t<sub>01345</sub> |
![]() t<sub>02345</sub> |
![]() t12345 |
![]() t<sub>01236</sub> |
![]() t<sub>01246</sub> |
![]() t<sub>01346</sub> |
![]() t<sub>02346</sub> |
![]() t<sub>12346</sub> |
![]() t<sub>01256</sub> |
![]() t<sub>01356</sub> |
![]() t<sub>02356</sub> |
![]() t<sub>12356</sub> |
![]() t<sub>01456</sub> |
![]() t<sub>02456</sub> |
![]() t<sub>03456</sub> |
![]() t<sub>01237</sub> |
![]() t<sub>01247</sub> |
![]() t<sub>01347</sub> |
![]() t<sub>02347</sub> |
![]() t<sub>01257</sub> |
![]() t<sub>01357</sub> |
![]() t<sub>02357</sub> |
![]() t<sub>01457</sub> |
![]() t<sub>01267</sub> |
![]() t<sub>01367</sub> |
![]() t<sub>012345</sub> |
![]() t<sub>012346</sub> |
![]() t<sub>012356</sub> |
![]() t<sub>012456</sub> |
![]() t<sub>013456</sub> |
![]() t<sub>023456</sub> |
![]() t<sub>123456</sub> |
![]() t<sub>012347</sub> |
![]() t<sub>012357</sub> |
![]() t<sub>012457</sub> |
![]() t<sub>013457</sub> |
![]() t<sub>023457</sub> |
![]() t<sub>012367</sub> |
![]() t<sub>012467</sub> |
![]() t<sub>013467</sub> |
![]() t<sub>012567</sub> |
![]() t<sub>0123456</sub> |
![]() t<sub>0123457</sub> |
![]() t<sub>0123467</sub> |
![]() t<sub>0123567</sub> |
![]() t01234567 |
Notes
References
- H.S.M. Coxeter:
- H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
- Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
- (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 Uniform Polytopes, Manuscript (1991)
- N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
- Klitzing, Richard. "8D uniform polytopes (polyzetta)". x3o3o3x3o3o3o3o - spene, o3x3o3o3x3o3o3o - sabpene, o3o3x3o3o3x3o3o - satpeb
External links
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