Generalized vector space model: Difference between revisions

Revision as of 18:16, 14 January 2014

Orthogonal projections in B₇ Coxeter plane
7-orthoplex Template:CDD	Rectified 7-orthoplex Template:CDD	Birectified 7-orthoplex Template:CDD	Trirectified 7-orthoplex Template:CDD
Birectified 7-cube Template:CDD	Rectified 7-cube Template:CDD	7-cube Template:CDD

In seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex.

There are unique 7 degrees of rectifications, the zeroth being the 7-orthoplex, and the 6th and last being the 7-cube. Vertices of the rectified 7-orthoplex are located at the edge-centers of the 7-orthoplex. Vertices of the birectified 7-orthoplex are located in the triangular face centers of the 7-orthoplex. Vertices of the trirectified 7-orthoplex are located in the tetrahedral cell centers of the 7-orthoplex.

Rectified 7-orthoplex

Rectified 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	r{3,3,3,3,3,4}
Coxeter-Dynkin diagrams	Template:CDD Template:CDD
6-faces	142
5-faces	1344
4-faces	3360
Cells	3920
Faces	2520
Edges	840
Vertices	84
Vertex figure	5-orthoplex prism
Coxeter groups	C₇, [3,3,3,3,3,4] D₇, [3^4,1,1]
Properties	convex

The rectified 7-orthoplex is the vertex figure for the demihepteractic honeycomb. The rectified 7-orthoplex's 84 vertices represent the kissing number of a sphere-packing constructed from this honeycomb.

Template:CDD or Template:CDD

Alternate names

rectified heptacross
rectified hecatonicosoctaexon (Acronym rez) (Jonathan Bowers) - rectified 128-faceted polyexon^[1]

Images

Template:7-cube Coxeter plane graphs

Construction

There are two Coxeter groups associated with the rectified heptacross, one with the C₇ or [4,3,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D₇ or [3^4,1,1] Coxeter group.

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified heptacross, centered at the origin, edge length ${\sqrt {2}}\$ are all permutations of:

(±1,±1,0,0,0,0,0)

Root vectors

Its 84 vertices represent the root vectors of the simple Lie group D₇. The vertices can be seen in 3 hyperplanes, with the 21 vertices rectified 6-simplexs cells on opposite sides, and 42 vertices of an expanded 6-simplex passing through the center. When combined with the 14 vertices of the 7-orthoplex, these vertices represent the 98 root vectors of the B₇ and C₇ simple Lie groups.

Birectified 7-orthoplex

Birectified 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	2r{3,3,3,3,3,4}
Coxeter-Dynkin diagrams	Template:CDD Template:CDD
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure	{3}x{3,3,4}
Coxeter groups	C₇, [3,3,3,3,3,4] D₇, [3^4,1,1]
Properties	convex

Alternate names

Birectified heptacross
Birectified hecatonicosoctaexon (Acronym rez) (Jonathan Bowers) - birectified 128-faceted polyexon^[2]

Images

Template:7-cube Coxeter plane graphs

Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 7-orthoplex, centered at the origin, edge length ${\sqrt {2}}\$ are all permutations of:

(±1,±1,±1,0,0,0,0)

Trirectified 7-orthoplex

Trirectified 7-orthoplex
Type	uniform 7-polytope
Schläfli symbol	3r{3,3,3,3,3,4}
Coxeter-Dynkin diagrams	Template:CDD Template:CDD
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure	{3,3}x{3,4}
Coxeter groups	C₇, [3,3,3,3,3,4] D₇, [3^4,1,1]
Properties	convex

Alternate names

Trirectified heptacross
Trirectified hecatonicosoctaexon (trirectified 128-faceted polyexon)

Images

Template:7-cube Coxeter plane graphs

Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 7-orthoplex, centered at the origin, edge length ${\sqrt {2}}\$ are all permutations of:

(±1,±1,±1,±1,0,0,0)

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 o3o3x3o3o3o4o - rez, o3o3x3o3o3o4o - barz

External links

Template:Polytopes

↑ Klitzing, (o3o3x3o3o3o4o - rez)
↑ Klitzing, (o3o3x3o3o3o4o - barz)

[1] Klitzing, (o3o3x3o3o3o4o - rez)

[2] Klitzing, (o3o3x3o3o3o4o - barz)

[1]

[2]

@@ Line 1: / Line 1: @@
-Hello from France. I'm glad to came across you. My first name is Augustina. <br>I live in a town called Choisy-Le-Roi in western France.<br>I was also born in Choisy-Le-Roi 21 years ago. Married in July year 2010. I'm working at the backery.<br><br>Have a look at my web blog [http://www.bataviarr.com/profile/sahoule Women mountain bike sizing.]
+{| class=wikitable align=right width=500
+|- align=center
+|[[File:7-cube_t6.svg|120px]]<BR>[[7-orthoplex]]<BR>{{CDD|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}
+|[[File:7-cube_t5.svg|120px]]<BR>Rectified 7-orthoplex<BR>{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}
+|[[File:7-cube_t4.svg|120px]]<BR>Birectified 7-orthoplex<BR>{{CDD|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}
+|[[File:7-cube_t3.svg|120px]]<BR>Trirectified 7-orthoplex<BR>{{CDD|node|3|node|3|node|3|node_1|3|node|3|node|3|node}}
+|- align=center
+|[[File:7-cube_t2.svg|120px]]<BR>[[Birectified 7-cube]]<BR>{{CDD|node|3|node|3|node|3|node|3|node_1|3|node|4|node}}
+|[[File:7-cube_t1.svg|120px]]<BR>[[Rectified 7-cube]]<BR>{{CDD|node|3|node|3|node|3|node|3|node|3|node_1|4|node}}
+|[[File:7-cube_t0.svg|120px]]<BR>[[7-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>7</sub> [[Coxeter plane]]
+|}
+In seven-dimensional [[geometry]], a '''rectified 7-orthoplex''' is a convex [[uniform 7-polytope]], being a [[Rectification (geometry)|rectification]] of the regular [[7-orthoplex]].
+There are unique 7 degrees of rectifications, the zeroth being the [[7-orthoplex]], and the 6th and last being the [[7-cube]]. Vertices of the rectified 7-orthoplex are located at the edge-centers of the 7-orthoplex. Vertices of the birectified 7-orthoplex are located in the triangular face centers of the 7-orthoplex. Vertices of the trirectified 7-orthoplex are located in the [[tetrahedron|tetrahedral]] cell centers of the 7-orthoplex.
+== Rectified 7-orthoplex ==
+{| class="wikitable" align="right" style="margin-left:10px" width="250"
+!bgcolor=#e7dcc3 colspan=2|Rectified 7-orthoplex
+|-
+|bgcolor=#e7dcc3|Type||[[uniform 7-polytope]]
+|-
+|bgcolor=#e7dcc3|[[Schläfli symbol]]|| r{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|4|node}}<br>{{CDD|node|3|node_1|3|node|3|node|3|node|split1|nodes}}
+|-
+|bgcolor=#e7dcc3|6-faces||142
+|-
+|bgcolor=#e7dcc3|5-faces||1344
+|-
+|bgcolor=#e7dcc3|4-faces||3360
+|-
+|bgcolor=#e7dcc3|Cells||3920
+|-
+|bgcolor=#e7dcc3|Faces||2520
+|-
+|bgcolor=#e7dcc3|Edges||840
+|-
+|bgcolor=#e7dcc3|Vertices||84
+|-
+|bgcolor=#e7dcc3|[[Vertex figure]]||5-orthoplex prism
+|-
+|bgcolor=#e7dcc3|[[Coxeter group]]s||C<sub>7</sub>, [3,3,3,3,3,4]<BR>D<sub>7</sub>, [3<sup>4,1,1</sup>]
+|-
+|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
+|}
+The ''rectified 7-orthoplex'' is the [[vertex figure]] for the [[demihepteractic honeycomb]]. The rectified 7-orthoplex's 84 vertices represent the [[kissing number]] of a sphere-packing constructed from this honeycomb.
+:{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|split1|nodes}} or {{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|4|node}}
+=== Alternate names===
+* rectified heptacross
+* rectified hecatonicosoctaexon (Acronym rez) (Jonathan Bowers) - rectified 128-faceted polyexon<ref>Klitzing, (o3o3x3o3o3o4o - rez)</ref>
+=== Images ===
+{{7-cube Coxeter plane graphs|t5|150}}
+=== Construction ===
+There are two [[Coxeter group]]s associated with the ''rectified heptacross'', one with the C<sub>7</sub> or [4,3,3,3,3,3] Coxeter group, and a lower symmetry with two copies of pentacross facets, alternating, with the D<sub>7</sub> or [3<sup>4,1,1</sup>] Coxeter group.
+=== Cartesian coordinates ===
+[[Cartesian coordinates]] for the vertices of a rectified heptacross, centered at the origin, edge length <math> \sqrt{2}\ </math> are all permutations of:
+: (±1,±1,0,0,0,0,0)
+==== Root vectors ====
+Its 84 vertices represent the root vectors of the [[simple Lie group]] D<sub>7</sub>. The vertices can be seen in 3 [[hyperplane]]s, with the 21 vertices [[rectified 6-simplex]]s cells on opposite sides, and 42 vertices of an [[expanded 6-simplex]] passing through the center.  When combined with the 14 vertices of the 7-orthoplex, these vertices represent the 98 root vectors of the B<sub>7</sub> and C<sub>7</sub> simple Lie groups.
+== Birectified 7-orthoplex ==
+{| class="wikitable" align="right" style="margin-left:10px" width="250"
+!bgcolor=#e7dcc3 colspan=2|Birectified 7-orthoplex
+|-
+|bgcolor=#e7dcc3|Type||[[uniform 7-polytope]]
+|-
+|bgcolor=#e7dcc3|[[Schläfli symbol]]|| 2r{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|4|node}}<br>{{CDD|node|3|node|3|node_1|3|node|3|node|split1|nodes}}
+|-
+|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}x{3,3,4}
+|-
+|bgcolor=#e7dcc3|[[Coxeter group]]s||C<sub>7</sub>, [3,3,3,3,3,4]<BR>D<sub>7</sub>, [3<sup>4,1,1</sup>]
+|-
+|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
+|}
+=== Alternate names===
+* Birectified heptacross
+* Birectified hecatonicosoctaexon (Acronym rez) (Jonathan Bowers) - birectified 128-faceted polyexon<ref>Klitzing, (o3o3x3o3o3o4o - barz)</ref>
+=== Images ===
+{{7-cube Coxeter plane graphs|t4|150}}
+=== Cartesian coordinates ===
+[[Cartesian coordinates]] for the vertices of a birectified 7-orthoplex, centered at the origin, edge length <math> \sqrt{2}\ </math> are all permutations of:
+: (±1,±1,±1,0,0,0,0)
+== Trirectified 7-orthoplex ==
+{| class="wikitable" align="right" style="margin-left:10px" width="250"
+!bgcolor=#e7dcc3 colspan=2|Trirectified 7-orthoplex
+|-
+|bgcolor=#e7dcc3|Type||[[uniform 7-polytope]]
+|-
+|bgcolor=#e7dcc3|[[Schläfli symbol]]|| 3r{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|4|node}}<br>{{CDD|node|3|node|3|node|3|node_1|3|node|split1|nodes}}
+|-
+|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}x{3,4}
+|-
+|bgcolor=#e7dcc3|[[Coxeter group]]s||C<sub>7</sub>, [3,3,3,3,3,4]<BR>D<sub>7</sub>, [3<sup>4,1,1</sup>]
+|-
+|bgcolor=#e7dcc3|Properties||[[Convex polytope|convex]]
+|}
+=== Alternate names===
+* Trirectified heptacross
+* Trirectified hecatonicosoctaexon (trirectified 128-faceted polyexon)
+=== Images ===
+{{7-cube Coxeter plane graphs|t3|150}}
+=== Cartesian coordinates ===
+[[Cartesian coordinates]] for the vertices of a birectified 7-orthoplex, centered at the origin, edge length <math> \sqrt{2}\ </math> are all permutations of:
+: (±1,±1,±1,±1,0,0,0)
+== 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|polyexa.htm|7D|uniform polytopes (polyexa)}} o3o3x3o3o3o4o - rez, o3o3x3o3o3o4o - barz
+== 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:7-polytopes]]

Generalized vector space model: Difference between revisions

Revision as of 18:16, 14 January 2014

Contents

Rectified 7-orthoplex

Alternate names

Images

Construction

Cartesian coordinates

Root vectors

Birectified 7-orthoplex

Alternate names

Images

Cartesian coordinates

Trirectified 7-orthoplex

Alternate names

Images

Cartesian coordinates

Notes

References

External links

Navigation menu

Generalized vector space model: Difference between revisions

Revision as of 18:16, 14 January 2014

Rectified 7-orthoplex

Alternate names

Images

Construction

Cartesian coordinates

Root vectors

Birectified 7-orthoplex

Alternate names

Images

Cartesian coordinates

Trirectified 7-orthoplex

Alternate names

Images

Cartesian coordinates

Notes

References

External links

Navigation menu

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