Lifting theory: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
Newer edit →
en>Yobot
m WP:CHECKWIKI error fixes + general fixes using AWB (7896)
 
No edit summary
Line 1: Line 1:
Hi [http://www.encyclopedia.com/searchresults.aspx?q=close+friend close friend]. Let me introduce myself. I am Brunilda Hayman. It really is not a widespread detail but what I like carrying out is to engage in croquet but I haven't designed a dime with it. I've constantly cherished living in Oregon. For a long time I've been working as an information officer but I prepare on changing it. You can discover my web site in this article: http://www.eeg-portal.de/cgi-bin/suchen.pl?suchworte=Yasmin+Gallstones+Aspect+Results<br><br>my web blog Oferta Salomon ([http://www.eeg-portal.de/cgi-bin/suchen.pl?suchworte=Yasmin+Gallstones+Aspect+Results http://www.eeg-portal.de])
{| class=wikitable align=right width=450
|- align=center
|[[File:6-cube_t5.svg|150px]]<BR>[[6-orthoplex]]<BR>{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}
|[[File:6-cube_t4.svg|150px]]<BR>Rectified 6-orthoplex<BR>{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}
|[[File:6-cube_t3.svg|150px]]<BR>Birectified 6-orthoplex<BR>{{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node}}
|- align=center
|[[File:6-cube_t2.svg|150px]]<BR>[[Birectified 6-cube]]<BR>{{CDD|node|3|node|3|node|3|node|3|node_1|3|node|4|node}}
|[[File:6-cube_t1.svg|150px]]<BR>[[Rectified 6-cube]]<BR>{{CDD|node|3|node|3|node|3|node|3|node|3|node_1|4|node}}
|[[File:6-cube_t0.svg|150px]]<BR>[[6-cube]]<BR>{{CDD|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}
|-
!colspan=4|[[Orthogonal projection]]s in B<sub>6</sub> [[Coxeter plane]]
|}
In six-dimensional [[geometry]], a '''rectified 6-orthoplex''' is a convex [[uniform 6-polytope]], being a [[Rectification (geometry)|rectification]] of the regular [[6-orthoplex]].
 
There are unique 6 degrees of rectifications, the zeroth being the [[6-orthoplex]], and the 6th and last being the [[6-cube]]. Vertices of the rectified 6-orthoplex are located at the edge-centers of the 6-orthoplex. Vertices of the birectified 6-orthoplex are located in the triangular face centers of the 6-orthoplex.
 
== Rectified 6-orthoplex==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Rectified hexacross
|-
|bgcolor=#e7dcc3|Type||[[uniform polypeton]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| r{3,3,3,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node|3|node_1|3|node|3|node|3|node|4|node}}<br>{{CDD|node|3|node_1|3|node|3|node|split1|nodes}}
|-
|bgcolor=#e7dcc3|5-faces||76 total:<BR>64 [[rectified 5-simplex]]<BR>12 [[5-orthoplex]]
|-
|bgcolor=#e7dcc3|4-faces||576 total:<BR>192 [[rectified 5-cell]]<BR>384 [[5-cell]]
|-
|bgcolor=#e7dcc3|Cells||1200 total:<BR>240 [[octahedron]]<BR>960 [[tetrahedron]]
|-
|bgcolor=#e7dcc3|Faces||1120 total:<BR>160 and 960 triangles
|-
|bgcolor=#e7dcc3|Edges||480
|-
|bgcolor=#e7dcc3|Vertices||60
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||16-cell prism
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[Dodecagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||B<sub>6</sub>, [3,3,3,3,4]<BR>D<sub>6</sub>, [3<sup>3,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
 
The ''rectified 6-orthoplex'' is the [[vertex figure]] for the [[demihexeractic honeycomb]].
:{{CDD|nodes_10ru|split2|node|3|node|3|node|split1|nodes}} or {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|4|node}}
 
=== Alternate names===
* rectified hexacross
* rectified hexacontitetrapeton (acronym: rag) (Jonathan Bowers)
 
=== Construction ===
There are two [[Coxeter group]]s associated with the ''rectified hexacross'', one with the C<sub>6</sub> or [4,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D<sub>6</sub> or [3<sup>3,1,1</sup>] Coxeter group.
 
=== Cartesian coordinates ===
[[Cartesian coordinates]] for the vertices of a rectified hexacross, centered at the origin, edge length <math> \sqrt{2}\ </math> are all permutations of:
: (±1,±1,0,0,0,0)
 
==== Root vectors ====
 
The 60 vertices represent the root vectors of the [[simple Lie group]] D<sub>6</sub>. The vertices can be seen in 3 [[hyperplane]]s, with the 15 vertices [[rectified 5-simplex]]s cells on opposite sides, and 30 vertices of an [[expanded 5-simplex]] passing through the center. When combined with the 12 vertices of the 6-orthoplex, these vertices represent the 72 root vectors of the B<sub>6</sub> and C<sub>6</sub> simple Lie groups.
 
===Images===
{{6-cube Coxeter plane graphs|t5|150}}
 
== Birectified 6-orthoplex==
{| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Birectified 6-orthoplex
|-
|bgcolor=#e7dcc3|Type||[[uniform polypeton]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]|| 2r{3,3,3,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||{{CDD|node|3|node|3|node_1|3|node|3|node|4|node}}<br>{{CDD|node|3|node_1|3|node|3|node|split1|nodes}}
|-
|bgcolor=#e7dcc3|5-faces||76
|-
|bgcolor=#e7dcc3|4-faces||636
|-
|bgcolor=#e7dcc3|Cells||2160
|-
|bgcolor=#e7dcc3|Faces||2880
|-
|bgcolor=#e7dcc3|Edges||1440
|-
|bgcolor=#e7dcc3|Vertices||160
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[equilateral triangle|{3}]]×[[octahedron|{3,4}]] duoprism
|-
|bgcolor=#e7dcc3|[[Petrie polygon]]||[[Dodecagon]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||B<sub>6</sub>, [3,3,3,3,4]<BR>D<sub>6</sub>, [3<sup>3,1,1</sup>]
|-
|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
|}
The '''birectified 6-orthoplex''' can tessellation space in the [[trirectified 6-cubic honeycomb]].
 
=== Alternate names===
* birectified hexacross
* birectified hexacontitetrapeton (acronym: brag) (Jonathan Bowers)
 
=== Cartesian coordinates ===
[[Cartesian coordinates]] for the vertices of a rectified hexacross, centered at the origin, edge length <math> \sqrt{2}\ </math> are all permutations of:
: (±1,±1,±1,0,0,0)
 
===Images===
{{6-cube Coxeter plane graphs|t4|150}}
 
== Related polytopes ==
 
These polytopes are a part a family of 63 [[Uniform_polypeton|uniform polypeta]] generated from the B<sub>6</sub> [[Coxeter plane]], including the regular [[6-cube]] or [[6-orthoplex]].
 
{{Hexeract 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''', 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|polypeta.htm|6D uniform polytopes (polypeta)| o3x3o3o3o4o - rag}}
 
== 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:6-polytopes]]

Revision as of 01:10, 26 August 2013


6-orthoplex
Template:CDD
Rectified 6-orthoplex
Template:CDD
Birectified 6-orthoplex
Template:CDD

Birectified 6-cube
Template:CDD
Rectified 6-cube
Template:CDD
6-cube
Template:CDD
Orthogonal projections in B6 Coxeter plane

In six-dimensional geometry, a rectified 6-orthoplex is a convex uniform 6-polytope, being a rectification of the regular 6-orthoplex.

There are unique 6 degrees of rectifications, the zeroth being the 6-orthoplex, and the 6th and last being the 6-cube. Vertices of the rectified 6-orthoplex are located at the edge-centers of the 6-orthoplex. Vertices of the birectified 6-orthoplex are located in the triangular face centers of the 6-orthoplex.

Rectified 6-orthoplex

Rectified hexacross
Type uniform polypeton
Schläfli symbol r{3,3,3,3,4}
Coxeter-Dynkin diagrams Template:CDD
Template:CDD
5-faces 76 total:
64 rectified 5-simplex
12 5-orthoplex
4-faces 576 total:
192 rectified 5-cell
384 5-cell
Cells 1200 total:
240 octahedron
960 tetrahedron
Faces 1120 total:
160 and 960 triangles
Edges 480
Vertices 60
Vertex figure 16-cell prism
Petrie polygon Dodecagon
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

The rectified 6-orthoplex is the vertex figure for the demihexeractic honeycomb.

Template:CDD or Template:CDD

Alternate names

  • rectified hexacross
  • rectified hexacontitetrapeton (acronym: rag) (Jonathan Bowers)

Construction

There are two Coxeter groups associated with the rectified hexacross, one with the C6 or [4,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D6 or [33,1,1] Coxeter group.

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified hexacross, centered at the origin, edge length are all permutations of:

(±1,±1,0,0,0,0)

Root vectors

The 60 vertices represent the root vectors of the simple Lie group D6. The vertices can be seen in 3 hyperplanes, with the 15 vertices rectified 5-simplexs cells on opposite sides, and 30 vertices of an expanded 5-simplex passing through the center. When combined with the 12 vertices of the 6-orthoplex, these vertices represent the 72 root vectors of the B6 and C6 simple Lie groups.

Images

Template:6-cube Coxeter plane graphs

Birectified 6-orthoplex

Birectified 6-orthoplex
Type uniform polypeton
Schläfli symbol 2r{3,3,3,3,4}
Coxeter-Dynkin diagrams Template:CDD
Template:CDD
5-faces 76
4-faces 636
Cells 2160
Faces 2880
Edges 1440
Vertices 160
Vertex figure {3}×{3,4} duoprism
Petrie polygon Dodecagon
Coxeter groups B6, [3,3,3,3,4]
D6, [33,1,1]
Properties convex

The birectified 6-orthoplex can tessellation space in the trirectified 6-cubic honeycomb.

Alternate names

  • birectified hexacross
  • birectified hexacontitetrapeton (acronym: brag) (Jonathan Bowers)

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified hexacross, centered at the origin, edge length are all permutations of:

(±1,±1,±1,0,0,0)

Images

Template:6-cube Coxeter plane graphs

Related polytopes

These polytopes are a part a family of 63 uniform polypeta generated from the B6 Coxeter plane, including the regular 6-cube or 6-orthoplex.

Template:Hexeract family

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

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

External links

Template:Polytopes

Retrieved from "https://en.formulasearchengine.com/index.php?title=Lifting_theory&oldid=25179"