Login
Login

Rectified 9-orthoplexes#Birectified 9-orthoplex

class=wikitable align=right width=500 style="margin-left:1em;"
align=center valign=top

|100px
9-orthoplex
{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}

|100px
Rectified 9-orthoplex
{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node_1|3|node}}

|100px
Birectified 9-orthoplex
{{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node_1|3|node|3|node}}

|100px
Trirectified 9-orthoplex
{{CDD|node|4|node|3|node|3|node|3|node|3|node_1|3|node|3|node|3|node}}

|100px
Quadrirectified 9-cube
{{CDD|node|4|node|3|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}

align=center valign=top

|100px
Trirectified 9-cube
{{CDD|node|4|node|3|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}

|100px
Birectified 9-cube
{{CDD|node|4|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node}}

|100px
Rectified 9-cube
{{CDD|node|4|node_1|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}

|100px
9-cube
{{CDD|node_1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|3|node}}

colspan=5|Orthogonal projections in A9 Coxeter plane

In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-orthoplex.

There are 9 rectifications of the 9-orthoplex. Vertices of the rectified 9-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 9-orthoplex are located in the triangular face centers of the 9-orthoplex. Vertices of the trirectified 9-orthoplex are located in the tetrahedral cell centers of the 9-orthoplex.

These polytopes are part of a family 511 uniform 9-polytopes with BC9 symmetry.

Rectified 9-orthoplex

class="wikitable" align="right" style="margin-left:10px" width="250"

!bgcolor=#e7dcc3 colspan=2|Rectified 9-orthoplex

bgcolor=#e7dcc3|Typeuniform 9-polytope
bgcolor=#e7dcc3|Schläfli symbolt1{37,4}
bgcolor=#e7dcc3|Coxeter-Dynkin diagrams{{CDD|node|3|node_1|3|node|3|node|3|node|3|node|3|node|3|node|4|node}}
{{CDD|node|3|node_1|3|node|3|node|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|Edges2016
bgcolor=#e7dcc3|Vertices144
bgcolor=#e7dcc3|Vertex figure7-orthoplex prism
bgcolor=#e7dcc3|Petrie polygonoctakaidecagon
bgcolor=#e7dcc3|Coxeter groupsC9, [4,37]
D9, [36,1,1]
bgcolor=#e7dcc3|Propertiesconvex

The rectified 9-orthoplex is the vertex figure for the demienneractic honeycomb.

: {{CDD|nodes_10ru|split2|node|3|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|3|node|4|node}}

= Alternate names =

  • rectified enneacross (Acronym riv) (Jonathan Bowers)Klitzing (o3x3o3o3o3o3o3o4o - riv)

= Construction =

There are two Coxeter groups associated with the rectified 9-orthoplex, one with the C9 or [4,37] Coxeter group, and a lower symmetry with two copies of 8-orthoplex facets, alternating, with the D9 or [36,1,1] Coxeter group.

= Cartesian coordinates =

Cartesian coordinates for the vertices of a rectified 9-orthoplex, centered at the origin, edge length \sqrt{2} are all permutations of:

: (±1,±1,0,0,0,0,0,0,0)

== Root vectors ==

Its 144 vertices represent the root vectors of the simple Lie group D9. The vertices can be seen in 3 hyperplanes, with the 36 vertices rectified 8-simplexs cells on opposite sides, and 72 vertices of an expanded 8-simplex passing through the center. When combined with the 18 vertices of the 9-orthoplex, these vertices represent the 162 root vectors of the B9 and C9 simple Lie groups.

= Images =

{{B9 Coxeter plane graphs|t7|200|NoA7=true|NoA5=true|NoA3=true}}

Birectified 9-orthoplex

= Alternate names =

  • Rectified 9-demicube
  • Birectified enneacross (Acronym brav) (Jonathan Bowers)Klitzing (o3o3x3o3o3o3o3o4o - brav)

= Images =

{{B9 Coxeter plane graphs|t6|200|NoA7=true|NoA5=true|NoA3=true}}

Trirectified 9-orthoplex

= Alternate names =

  • trirectified enneacross (Acronym tarv) (Jonathan Bowers)Klitzing (o3o3o3x3o3o3o3o4o - tarv)

= Images =

{{B9 Coxeter plane graphs|t5|200|NoA7=true|NoA5=true|NoA3=true}}

Notes

{{reflist}}

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}} [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 Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
  • {{KlitzingPolytopes|polyyotta.htm|9D|uniform polytopes (polyyotta)}} x3o3o3o3o3o3o3o4o - vee, o3x3o3o3o3o3o3o4o - riv, o3o3x3o3o3o3o3o4o - brav, o3o3o3x3o3o3o3o4o - tarv, o3o3o3o3x3o3o3o4o - nav, o3o3o3o3o3x3o3o4o - tarn, o3o3o3o3o3o3x3o4o - barn, o3o3o3o3o3o3o3x4o - ren, o3o3o3o3o3o3o3o4x - enne

External links

  • [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
  • [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

Category:9-polytopes