View source for Runcinated 8-simplexes

{| class=wikitable align=right width=400 style="margin-left:1em;"
|- align=center
|[[File:8-simplex t0.svg|100px]]<BR>[[8-simplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}
|[[File:8-simplex t03.svg|100px]]<BR>Runcinated 8-simplex<BR>{{CDD|node_1|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}
|[[File:8-simplex t14.svg|100px]]<BR>Biruncinated 8-simplex<BR>{{CDD|node|3|node_1|3|node|3|node|3|node_1|3|node|3|node|3|node}}
|[[File:8-simplex t25.svg|100px]]<BR>Triruncinated 8-simplex<BR>{{CDD|node|3|node|3|node_1|3|node|3|node|3|node_1|3|node|3|node}}
|- align=center
|[[File:8-simplex t013.svg|100px]]<BR>Runcitruncated 8-simplex<BR>{{CDD|node_1|3|node_1|3|node|3|node_1|3|node|3|node|3|node|3|node}}
|[[File:8-simplex t124.svg|100px]]<BR>Biruncitruncated 8-simplex<BR>{{CDD|node|3|node_1|3|node_1|3|node|3|node_1|3|node|3|node|3|node}}
|[[File:8-simplex t235.svg|100px]]<BR>Triruncitruncated 8-simplex<BR>{{CDD|node|3|node|3|node_1|3|node_1|3|node|3|node_1|3|node|3|node}}
|[[File:8-simplex t023.svg|100px]]<BR>Runcicantellated 8-simplex<BR>{{CDD|node_1|3|node|3|node_1|3|node_1|3|node|3|node|3|node|3|node}}
|- align=center
|[[File:8-simplex t134.svg|100px]]<BR>Biruncicantellated 8-simplex<BR>{{CDD|node|3|node_1|3|node|3|node_1|3|node_1|3|node|3|node|3|node}}
|[[File:8-simplex t0123.svg|100px]]<BR>Runcicantitruncated 8-simplex<BR>{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node|3|node}}
|[[File:8-simplex t1234.svg|100px]]<BR>Biruncicantitruncated 8-simplex<BR>{{CDD|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}
|[[File:8-simplex t2345.svg|100px]]<BR>Triruncicantitruncated 8-simplex<BR>{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node}}
|-
!colspan=4|[[Orthogonal projection]]s in A<sub>8</sub> [[Coxeter plane]]
|}

In eight-dimensional [[geometry]], a '''runcinated 8-simplex''' is a convex [[uniform 8-polytope]] with 3rd order [[Truncation (geometry)|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{{wbr}}truncated 8-simplex'' have a doubled symmetry, showing [18] order reflectional symmetry in the A<sub>8</sub> [[Coxeter plane]].

== Runcinated 8-simplex ==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
! style="background:#e7dcc3;" colspan="2"|Runcinated 8-simplex
|-
| style="background:#e7dcc3;"|Type||[[uniform 8-polytope]]
|-
| style="background:#e7dcc3;"|[[Schläfli symbol]]|| t<sub>0,3</sub>{3,3,3,3,3,3,3}
|-
| style="background:#e7dcc3;"|[[Coxeter-Dynkin diagram]]s||{{CDD|node_1|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}
|-
| style="background:#e7dcc3;"|6-faces||
|-
| style="background:#e7dcc3;"|5-faces||
|-
| style="background:#e7dcc3;"|4-faces||
|-
| style="background:#e7dcc3;"|Cells||
|-
| style="background:#e7dcc3;"|Faces||
|-
| style="background:#e7dcc3;"|Edges||4536 
|-
| style="background:#e7dcc3;"|Vertices||504 
|-
| style="background:#e7dcc3;"|[[Vertex figure]]||
|-
| style="background:#e7dcc3;"|[[Coxeter group]]||A<sub>8</sub>, [3<sup>7</sup>], order 362880
|-
| style="background:#e7dcc3;"|Properties||[[Convex polytope|convex]]
|}

=== Alternate names ===
* Runcinated enneazetton 
* Small prismated enneazetton (Acronym: spene) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/spene.htm (x3o3o3x3o3o3o3o - spene)]}}

=== Coordinates ===
The [[Cartesian coordinate]]s 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 [[Facet (geometry)|facets]] of the [[runcinated 9-orthoplex]].

=== Images ===
{{8-simplex Coxeter plane graphs|t03|120}}

== Biruncinated 8-simplex ==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
! style="background:#e7dcc3;" colspan="2"|Biruncinated 8-simplex
|-
| style="background:#e7dcc3;"|Type||[[uniform 8-polytope]]
|-
| style="background:#e7dcc3;"|[[Schläfli symbol]]|| t<sub>1,4</sub>{3,3,3,3,3,3,3}
|-
| style="background:#e7dcc3;"|[[Coxeter-Dynkin diagram]]||{{CDD|node|3|node_1|3|node|3|node|3|node_1|3|node|3|node|3|node}}
|-
| style="background:#e7dcc3;"|7-faces||
|-
| style="background:#e7dcc3;"|6-faces||
|-
| style="background:#e7dcc3;"|5-faces||
|-
| style="background:#e7dcc3;"|4-faces||
|-
| style="background:#e7dcc3;"|Cells||
|-
| style="background:#e7dcc3;"|Faces||
|-
| style="background:#e7dcc3;"|Edges||11340
|-
| style="background:#e7dcc3;"|Vertices||1260
|-
| style="background:#e7dcc3;"|[[Vertex figure]]||
|-
| style="background:#e7dcc3;"|[[Coxeter group]]||A<sub>8</sub>, [3<sup>7</sup>], order 362880
|-
| style="background:#e7dcc3;"|Properties||[[Convex polytope|convex]]
|}

=== Alternate names ===
* Biruncinated enneazetton 
* Small biprismated enneazetton (Acronym: sabpene) (Jonathan Bowers)<ref>Klitzing (o3x3o3o3x3o3o3o - sabpene)</ref>

=== Coordinates ===
The [[Cartesian coordinate]]s 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 [[Facet (geometry)|facets]] of the [[biruncinated 9-orthoplex]].

=== Images ===
{{8-simplex Coxeter plane graphs|t14|120}}

== Triruncinated 8-simplex ==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
! style="background:#e7dcc3;" colspan="2"|Triruncinated 8-simplex
|-
| style="background:#e7dcc3;"|Type||[[uniform 8-polytope]]
|-
| style="background:#e7dcc3;"|[[Schläfli symbol]]|| t<sub>2,5</sub>{3,3,3,3,3,3,3}
|-
| style="background:#e7dcc3;"|[[Coxeter-Dynkin diagram]]s||{{CDD|node|3|node|3|node_1|3|node|3|node|3|node_1|3|node|3|node}}
|-
| style="background:#e7dcc3;"|7-faces||
|-
| style="background:#e7dcc3;"|6-faces||
|-
| style="background:#e7dcc3;"|5-faces||
|-
| style="background:#e7dcc3;"|4-faces||
|-
| style="background:#e7dcc3;"|Cells||
|-
| style="background:#e7dcc3;"|Faces||
|-
| style="background:#e7dcc3;"|Edges||15120 
|-
| style="background:#e7dcc3;"|Vertices||1680 
|-
| style="background:#e7dcc3;"|[[Vertex figure]]||
|-
| style="background:#e7dcc3;"|[[Coxeter group]]||A<sub>8</sub>×2, <nowiki>[[</nowiki>3<sup>7</sup>]], order 725760 
|-
| style="background:#e7dcc3;"|Properties||[[Convex polytope|convex]]
|}

=== Alternate names ===
* Triruncinated enneazetton 
* Small triprismated enneazetton (Acronym: satpeb) (Jonathan Bowers){{sfn|Klitzing|at=[https://bendwavy.org/klitzing/incmats/satpeb.htm (o3o3x3o3o3x3o3o - satpeb)]}}

=== Coordinates ===
The [[Cartesian coordinate]]s 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 [[Facet (geometry)|facets]] of the [[triruncinated 9-orthoplex]].

=== Images ===
{{8-simplex2 Coxeter plane graphs|t25|120}}

== Runcitruncated 8-simplex ==
{{CDD|node_1|3|node_1|3|node|3|node_1|3|node|3|node|3|node|3|node}}
Acronym: potane (Jonathan Bowers)<ref>Klitzing (x3x3o3x3o3o3o3o - potane)</ref>

=== Images ===
{{8-simplex2 Coxeter plane graphs|t013|120}}

== Biruncitruncated 8-simplex ==
{{CDD|node|3|node_1|3|node_1|3|node|3|node_1|3|node|3|node|3|node}}
Acronym: biptene (Jonathan Bowers)<ref>Klitzing (o3x3x3o3x3o3o3o - biptene)</ref>

=== Images ===
{{8-simplex2 Coxeter plane graphs|t124|120}}

== Triruncitruncated 8-simplex ==
{{CDD|node|3|node|3|node_1|3|node_1|3|node|3|node_1|3|node|3|node}}
Acronym: toprane (Jonathan Bowers)<ref>Klitzing (o3o3x3x3o3x3o3o - toprane)</ref>

=== Images ===
{{8-simplex2 Coxeter plane graphs|t235|120}}

== Runcicantellated 8-simplex ==
{{CDD|node_1|3|node|3|node_1|3|node_1|3|node|3|node|3|node|3|node}}
Acronym: prene (Jonathan Bowers)<ref>Klitzing (x3o3x3x3o3o3o3o - prene)</ref>

=== Images ===
{{8-simplex2 Coxeter plane graphs|t023|120}}

== Biruncicantellated 8-simplex ==
{{CDD|node|3|node_1|3|node|3|node_1|3|node_1|3|node|3|node|3|node}}
Acronym: biprene (Jonathan Bowers)<ref>Klitzing (o3x3o3x3x3o3o3o - biprene)</ref>

=== Images ===
{{8-simplex2 Coxeter plane graphs|t134|120}}

== Runcicantitruncated 8-simplex ==
{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node|3|node}}
Acronym: gapene (Jonathan Bowers)<ref>Klitzing (x3x3x3x3o3o3o3o - gapene)</ref>

=== Images ===
{{8-simplex2 Coxeter plane graphs|t0123|120}}

== Biruncicantitruncated 8-simplex ==
{{CDD|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}
Acronym: gabpene (Jonathan Bowers)<ref>Klitzing (o3x3x3x3x3o3o3o - gabpene)</ref>

=== Images ===
{{8-simplex2 Coxeter plane graphs|t1234|120}}

== Triruncicantitruncated 8-simplex ==
{{CDD|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node}}
Acronym: gatpeb (Jonathan Bowers)<ref>Klitzing (o3o3x3x3x3x3o3o - gatpeb)</ref>

=== Images ===
{{8-simplex2 Coxeter plane graphs|t2345|120}}

== Related polytopes ==
The 11 presented polytopes are in the family of 135 [[8-polytope#The A8 family|uniform 8-polytopes]] with A<sub>8</sub> symmetry.
{{Enneazetton family}}

== Notes ==
{{reflist}}

== References ==
* [[Harold Scott MacDonald Coxeter|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, [https://www.wiley.com/en-us/Kaleidoscopes-p-9780471010036 wiley.com], {{isbn|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]
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. 
* {{KlitzingPolytopes|polyzetta.htm|8D uniform polytopes (polyzetta) with acronyms}} x3o3o3x3o3o3o3o - spene, o3x3o3o3x3o3o3o - sabpene, o3o3x3o3o3x3o3o - satpeb, x3x3o3x3o3o3o3o - potane, o3x3x3o3x3o3o3o3 - biptene, o3o3x3x3o3x3o3o - toprane, x3o3x3x3o3o3o3o - prene, o3x3o3x3x3o3o3o - biprene, x3x3x3x3o3o3o3o3 - gapene, o3x3x3x3x3o3o3o - gabpene, o3o3x3x3x3x3o3o - gatpeb {{sfn whitelist| CITEREFKlitzing}}

== External links ==
* [https://web.archive.org/web/20070310205351/http://members.cox.net/hedrondude/topes.htm Polytopes of Various Dimensions]
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

[[Category:8-polytopes]]