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{| class=wikitable align=right width=500
|- align=center
|[[File:8-cube_t7.svg|120px]]<BR>[[8-orthoplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}
|[[File:8-cube_t6.svg|120px]]<BR>Rectified 8-orthoplex<BR>{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node|4|node}}
|[[File:8-cube_t5.svg|120px]]<BR>Birectified 8-orthoplex<BR>{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node|4|node}}
|[[File:8-cube_t4.svg|120px]]<BR>Trirectified 8-orthoplex<BR>{{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node|4|node}}
|- align=center
|[[File:8-cube_t3.svg|120px]]<BR>[[Trirectified 8-cube]]<BR>{{CDD|node|3|node|3|node|3|node|3|node_1|3|node|3|node|4|node}}
|[[File:8-cube_t2.svg|120px]]<BR>[[Birectified 8-cube]]<BR>{{CDD|node|3|node|3|node|3|node|3|node|3|node_1|3|node|4|node}}
|[[File:8-cube_t1.svg|120px]]<BR>[[Rectified 8-cube]]<BR>{{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node_1|4|node}}
|[[File:8-cube_t0.svg|120px]]<BR>[[8-cube]]<BR>{{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node_1}}
|-
!colspan=4|[[Orthogonal projection]]s in A<sub>8</sub> [[Coxeter plane]]
|}
In eight-dimensional [[geometry]], a '''rectified 8-orthoplex''' is a convex [[uniform 8-polytope]], being a [[Rectification (geometry)|rectification]] of the regular [[8-orthoplex]].
 
There are unique 8 degrees of rectifications, the zeroth being the [[8-orthoplex]], and the 7th and last being the [[8-cube]]. Vertices of the rectified 8-orthoplex are located at the edge-centers of the 8-orthoplex. Vertices of the birectified 8-orthoplex are located in the triangular face centers of the 8-orthoplex. Vertices of the trirectified 8-orthoplex are located in the [[tetrahedron|tetrahedral]] cell centers of the 8-orthoplex.
 
== Rectified 8-orthoplex ==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Rectified 8-orthoplex
|-
|bgcolor=#e7dcc3|Type||[[uniform 8-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>1</sub>{3,3,3,3,3,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node|4|node}}<br>{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|split1|nodes}}
|-
|bgcolor=#e7dcc3|7-faces||272
|-
|bgcolor=#e7dcc3|6-faces||3072
|-
|bgcolor=#e7dcc3|5-faces||8960
|-
|bgcolor=#e7dcc3|4-faces||12544
|-
|bgcolor=#e7dcc3|Cells||10080
|-
|bgcolor=#e7dcc3|Faces||4928
|-
|bgcolor=#e7dcc3|Edges||1344
|-
|bgcolor=#e7dcc3|Vertices||112
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||6-orthoplex prism
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[hexakaidecagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||C<sub>8</sub>, [4,3<sup>6</sup>]<BR>D<sub>8</sub>, [3<sup>5,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
 
The rectified 8-orthoplex has 112 vertices. These represent the root vectors of the [[simple Lie group]] D<sub>8</sub>. The vertices can be seen in 3 [[hyperplane]]s, with the 28 vertices [[rectified 7-simplex]]s cells on opposite sides, and 56 vertices of an [[expanded 7-simplex]] passing through the center. When combined with the 16 vertices of the 8-orthoplex, these vertices represent the 128 root vectors of the B<sub>8</sub> and C<sub>8</sub> simple Lie groups.
=== Related polytopes ===
 
The ''rectified 8-orthoplex'' is the [[vertex figure]] for the [[demiocteractic honeycomb]].
: {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|split1|nodes}} or {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|3|node|4|node}}
 
=== Alternate names===
* rectified octacross
* rectified diacosipentacontahexazetton (Acronym: rek) (Jonathan Bowers)<ref>Klitzing, (o3x3o3o3o3o3o4o - rek)</ref>
 
=== Construction ===
 
There are two [[Coxeter group]]s associated with the ''rectified 8-orthoplex'', one with the C<sub>8</sub> or [4,3<sup>6</sup>] Coxeter group, and a lower symmetry with two copies of heptcross facets, alternating, with the D<sub>8</sub> or [3<sup>5,1,1</sup>] Coxeter group.
 
=== Cartesian coordinates ===
[[Cartesian coordinates]] for the vertices of a rectified 8-orthoplex, centered at the origin, edge length <math>\sqrt{2}</math> are all permutations of:
: (±1,±1,0,0,0,0,0,0)
 
=== Images ===
{{8-cube Coxeter plane graphs|t6|150}}
 
== Birectified 8-orthoplex==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Birectified 8-orthoplex
|-
|bgcolor=#e7dcc3|Type||[[uniform 8-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>2</sub>{3,3,3,3,3,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node|4|node}}<br>{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|split1|nodes}}
|-
|bgcolor=#e7dcc3|7-faces||
|-
|bgcolor=#e7dcc3|6-faces||
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||
|-
|bgcolor=#e7dcc3|Vertices||
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||{3,3,3,4}x{3}
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||C<sub>8</sub>, [3,3,3,3,3,3,4]<BR>D<sub>8</sub>, [3<sup>5,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
 
=== Alternate names===
* birectified octacross
* birectified diacosipentacontahexazetton (Acronym: bark) (Jonathan Bowers)<ref>Klitzing, (o3o3x3o3o3o3o4o - bark)</ref>
=== Cartesian coordinates ===
[[Cartesian coordinates]] for the vertices of a birectified 8-orthoplex, centered at the origin, edge length <math>\sqrt{2}</math> are all permutations of:
: (±1,±1,±1,0,0,0,0,0)
 
=== Images ===
{{8-cube Coxeter plane graphs|t5|150}}
 
== Trirectified 8-orthoplex==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Trirectified 8-orthoplex
|-
|bgcolor=#e7dcc3|Type||[[uniform 8-polytope]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>3</sub>{3,3,3,3,3,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node|4|node}}<br>{{CDD|node|3|node|3|node|3|node_1|3|node|3|node|split1|nodes}}
|-
|bgcolor=#e7dcc3|7-faces||
|-
|bgcolor=#e7dcc3|6-faces||
|-
|bgcolor=#e7dcc3|5-faces||
|-
|bgcolor=#e7dcc3|4-faces||
|-
|bgcolor=#e7dcc3|Cells||
|-
|bgcolor=#e7dcc3|Faces||
|-
|bgcolor=#e7dcc3|Edges||
|-
|bgcolor=#e7dcc3|Vertices||
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||{3,3,4}x{3,3}
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||C<sub>8</sub>, [3,3,3,3,3,3,4]<BR>D<sub>8</sub>, [3<sup>5,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The '''trirectified 8-orthoplex''' can [[tessellation|tessellate]] space in the [[quadrirectified 8-cubic honeycomb]].
 
=== Alternate names===
* trirectified octacross
* trirectified diacosipentacontahexazetton (acronym: tark) (Jonathan Bowers)<ref>Klitzing, (o3o3o3x3o3o3o4o - tark)</ref>
 
=== Cartesian coordinates ===
[[Cartesian coordinates]] for the vertices of a trirectified 8-orthoplex, centered at the origin, edge length <math>\sqrt{2}</math> are all permutations of:
: (±1,±1,±1,±1,0,0,0,0)
 
=== Images ===
{{8-cube Coxeter plane graphs|t4|150}}
 
== 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''', editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
*** (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)}} o3x3o3o3o3o3o4o - rek, o3o3x3o3o3o3o4o - bark, o3o3o3x3o3o3o4o - tark
 
== External links ==
*{{GlossaryForHyperspace | anchor=Cross | title=Cross polytope }}
* [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
 
{{Polytopes}}
 
[[Category:8-polytopes]]

Revision as of 01:55, 25 July 2013


8-orthoplex
Template:CDD
Rectified 8-orthoplex
Template:CDD
Birectified 8-orthoplex
Template:CDD
Trirectified 8-orthoplex
Template:CDD

Trirectified 8-cube
Template:CDD
Birectified 8-cube
Template:CDD
Rectified 8-cube
Template:CDD
8-cube
Template:CDD
Orthogonal projections in A8 Coxeter plane

In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex.

There are unique 8 degrees of rectifications, the zeroth being the 8-orthoplex, and the 7th and last being the 8-cube. Vertices of the rectified 8-orthoplex are located at the edge-centers of the 8-orthoplex. Vertices of the birectified 8-orthoplex are located in the triangular face centers of the 8-orthoplex. Vertices of the trirectified 8-orthoplex are located in the tetrahedral cell centers of the 8-orthoplex.

Rectified 8-orthoplex

Rectified 8-orthoplex
Type uniform 8-polytope
Schläfli symbol t1{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams Template:CDD
Template:CDD
7-faces 272
6-faces 3072
5-faces 8960
4-faces 12544
Cells 10080
Faces 4928
Edges 1344
Vertices 112
Vertex figure 6-orthoplex prism
Petrie polygon hexakaidecagon
Coxeter groups C8, [4,36]
D8, [35,1,1]
Properties convex

The rectified 8-orthoplex has 112 vertices. These represent the root vectors of the simple Lie group D8. The vertices can be seen in 3 hyperplanes, with the 28 vertices rectified 7-simplexs cells on opposite sides, and 56 vertices of an expanded 7-simplex passing through the center. When combined with the 16 vertices of the 8-orthoplex, these vertices represent the 128 root vectors of the B8 and C8 simple Lie groups.

The rectified 8-orthoplex is the vertex figure for the demiocteractic honeycomb.

Template:CDD or Template:CDD

Alternate names

  • rectified octacross
  • rectified diacosipentacontahexazetton (Acronym: rek) (Jonathan Bowers)[1]

Construction

There are two Coxeter groups associated with the rectified 8-orthoplex, one with the C8 or [4,36] Coxeter group, and a lower symmetry with two copies of heptcross facets, alternating, with the D8 or [35,1,1] Coxeter group.

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified 8-orthoplex, centered at the origin, edge length 2 are all permutations of:

(±1,±1,0,0,0,0,0,0)

Images

Template:8-cube Coxeter plane graphs

Birectified 8-orthoplex

Birectified 8-orthoplex
Type uniform 8-polytope
Schläfli symbol t2{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams Template:CDD
Template:CDD
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure {3,3,3,4}x{3}
Coxeter groups C8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Properties convex

Alternate names

  • birectified octacross
  • birectified diacosipentacontahexazetton (Acronym: bark) (Jonathan Bowers)[2]

Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 8-orthoplex, centered at the origin, edge length 2 are all permutations of:

(±1,±1,±1,0,0,0,0,0)

Images

Template:8-cube Coxeter plane graphs

Trirectified 8-orthoplex

Trirectified 8-orthoplex
Type uniform 8-polytope
Schläfli symbol t3{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams Template:CDD
Template:CDD
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure {3,3,4}x{3,3}
Coxeter groups C8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Properties convex

The trirectified 8-orthoplex can tessellate space in the quadrirectified 8-cubic honeycomb.

Alternate names

  • trirectified octacross
  • trirectified diacosipentacontahexazetton (acronym: tark) (Jonathan Bowers)[3]

Cartesian coordinates

Cartesian coordinates for the vertices of a trirectified 8-orthoplex, centered at the origin, edge length 2 are all permutations of:

(±1,±1,±1,±1,0,0,0,0)

Images

Template:8-cube Coxeter plane graphs

Notes

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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, editied 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.
  • Template:KlitzingPolytopes o3x3o3o3o3o3o4o - rek, o3o3x3o3o3o3o4o - bark, o3o3o3x3o3o3o4o - tark

Template:Polytopes

  1. ↑ Klitzing, (o3x3o3o3o3o3o4o - rek)
  2. ↑ Klitzing, (o3o3x3o3o3o3o4o - bark)
  3. ↑ Klitzing, (o3o3o3x3o3o3o4o - tark)
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