Lorentz space: Difference between revisions
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{| class=wikitable align=right width=500 | |||
|- align=center valign=top | |||
|[[File:5-cube t4.svg|100px]]<BR>[[5-orthoplex]]<BR>{{CDD|node|3|node_1|3|node|3|node|3|node|4|node}} | |||
|[[File:5-cube t3.svg|100px]]<BR>Rectified 5-orthoplex<BR>{{CDD|node|3|node_1|3|node|3|node|3|node|4|node}} | |||
|[[File:5-cube t2.svg|100px]]<BR>[[Birectified 5-cube]]<BR>{{CDD|node|3|node|3|node|3|node_1|3|node|4|node}} | |||
|[[File:5-cube t1.svg|100px]]<BR>[[Rectified 5-cube]]<BR>{{CDD|node|3|node|3|node|3|node|3|node|3|node_1|4|node}} | |||
|[[File:5-cube t0.svg|100px]]<BR>[[5-cube]]<BR>{{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node_1}} | |||
|- | |||
!colspan=5|[[Orthogonal projection]]s in B<sub>5</sub> [[Coxeter plane]] | |||
|} | |||
In five-dimensional [[geometry]], a '''rectified 5-orthoplex''' is a convex [[uniform 5-polytope]], being a [[Rectification (geometry)|rectification]] of the regular [[5-orthoplex]]. | |||
There are 5 degrees of rectifications for any 5-polytope, the zeroth here being the [[5-orthoplex]] itself, and the 4th and last being the [[5-cube]]. Vertices of the rectified 5-orthoplex are located at the edge-centers of the 5-orthoplex. Vertices of the birectified 5-orthoplex are located in the triangular face centers of the 5-orthoplex. | |||
== Rectified 5-orthoplex== | |||
{| class="wikitable" align="right" style="margin-left:10px" width="250" | |||
!bgcolor=#e7dcc3 colspan=2|Rectified pentacross | |||
|- | |||
|bgcolor=#e7dcc3|Type||[[uniform polyteron]] | |||
|- | |||
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| t<sub>1</sub>{3,3,3,4} | |||
|- | |||
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node|3|node_1|3|node|3|node|4|node}}<br>{{CDD|node|3|node_1|3|node|split1|nodes}} | |||
|- | |||
|bgcolor=#e7dcc3|Hypercells||42 total:<BR>10 [[16-cell|{3,3,4}]]<BR>32 [[Rectified 5-cell|t<sub>1</sub>{3,3,3}]] | |||
|- | |||
|bgcolor=#e7dcc3|Cells||240 total:<BR>80 [[octahedron|{3,4}]]<BR>160 [[tetrahedron|{3,3}]] | |||
|- | |||
|bgcolor=#e7dcc3|Faces||400 total:<BR>80+320 [[triangle|{3}]] | |||
|- | |||
|bgcolor=#e7dcc3|Edges||240 | |||
|- | |||
|bgcolor=#e7dcc3|Vertices||40 | |||
|- | |||
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Rectified pentacross verf.png|40px]]<BR>[[Octahedral prism]] | |||
|- | |||
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[Decagon]] | |||
|- | |||
|bgcolor=#e7dcc3|[[Coxeter group]]s||BC<sub>5</sub>, [3,3,3,4]<BR>D<sub>5</sub>, [3<sup>2,1,1</sup>] | |||
|- | |||
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]] | |||
|} | |||
Its 40 vertices represent the root vectors of the [[simple Lie group]] D<sub>5</sub>. The vertices can be seen in 3 [[hyperplane]]s, with the 10 vertices [[rectified 5-cell]]s cells on opposite sides, and 20 vertices of a [[runcinated 5-cell]] passing through the center. When combined with the 10 vertices of the [[5-orthoplex]], these vertices represent the 50 root vectors of the B<sub>5</sub> and C<sub>5</sub> simple Lie groups. | |||
=== Alternate names=== | |||
* rectified pentacross | |||
* rectified triacontiditeron (32-faceted polyteron) | |||
=== Construction === | |||
There are two [[Coxeter group]]s associated with the ''rectified pentacross'', one with the C<sub>5</sub> or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of ''16-cell'' facets, alternating, with the D<sub>5</sub> or [3<sup>2,1,1</sup>] Coxeter group. | |||
== Cartesian coordinates == | |||
[[Cartesian coordinates]] for the vertices of a rectified pentacross, centered at the origin, edge length <math> \sqrt{2}\ </math> are all permutations of: | |||
: (±1,±1,0,0,0) | |||
===Images=== | |||
{{5-cube Coxeter plane graphs|t3|150}} | |||
== Related polytopes == | |||
The ''rectified 5-orthoplex'' is the vertex figure for the [[5-demicube honeycomb]]: | |||
:{{CDD|nodes_10ru|split2|node|3|node|split1|nodes}} or {{CDD|nodes_10ru|split2|node|3|node|3|node|4|node}} | |||
This polytope is one of 31 [[Uniform_polyteron#Uniform_polyteron|uniform polytera]] generated from the regular [[5-cube]] or [[5-orthoplex]]. | |||
{{Penteract 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, 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|polytera.htm|5D|uniform polytopes (polytera)}} o3x3o3o4o - rat | |||
== 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:5-polytopes]] |
Revision as of 17:24, 21 June 2013
5-orthoplex Template:CDD |
Rectified 5-orthoplex Template:CDD |
Birectified 5-cube Template:CDD |
Rectified 5-cube Template:CDD |
5-cube Template:CDD |
Orthogonal projections in B5 Coxeter plane |
---|
In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.
There are 5 degrees of rectifications for any 5-polytope, the zeroth here being the 5-orthoplex itself, and the 4th and last being the 5-cube. Vertices of the rectified 5-orthoplex are located at the edge-centers of the 5-orthoplex. Vertices of the birectified 5-orthoplex are located in the triangular face centers of the 5-orthoplex.
Rectified 5-orthoplex
Rectified pentacross | |
---|---|
Type | uniform polyteron |
Schläfli symbol | t1{3,3,3,4} |
Coxeter-Dynkin diagrams | Template:CDD Template:CDD |
Hypercells | 42 total: 10 {3,3,4} 32 t1{3,3,3} |
Cells | 240 total: 80 {3,4} 160 {3,3} |
Faces | 400 total: 80+320 {3} |
Edges | 240 |
Vertices | 40 |
Vertex figure | Octahedral prism |
Petrie polygon | Decagon |
Coxeter groups | BC5, [3,3,3,4] D5, [32,1,1] |
Properties | convex |
Its 40 vertices represent the root vectors of the simple Lie group D5. The vertices can be seen in 3 hyperplanes, with the 10 vertices rectified 5-cells cells on opposite sides, and 20 vertices of a runcinated 5-cell passing through the center. When combined with the 10 vertices of the 5-orthoplex, these vertices represent the 50 root vectors of the B5 and C5 simple Lie groups.
Alternate names
- rectified pentacross
- rectified triacontiditeron (32-faceted polyteron)
Construction
There are two Coxeter groups associated with the rectified pentacross, one with the C5 or [4,3,3,3] Coxeter group, and a lower symmetry with two copies of 16-cell facets, alternating, with the D5 or [32,1,1] Coxeter group.
Cartesian coordinates
Cartesian coordinates for the vertices of a rectified pentacross, centered at the origin, edge length are all permutations of:
- (±1,±1,0,0,0)
Images
Template:5-cube Coxeter plane graphs
Related polytopes
The rectified 5-orthoplex is the vertex figure for the 5-demicube honeycomb:
This polytope is one of 31 uniform polytera generated from the regular 5-cube or 5-orthoplex.
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, 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.
- Template:KlitzingPolytopes o3x3o3o4o - rat