Surf Wiki
Save to docs
general/10-polytopes

From Surf Wiki (app.surf) — the open knowledge base

Rectified 10-simplexes

Orthogonal projections in A9 Coxeter plane

In ten-dimensional geometry, a rectified 10-simplex is a convex uniform 10-polytope, being a rectification of the regular 10-simplex.

These polytopes are part of a family of 527 uniform 10-polytopes with A10 symmetry.

There are unique 5 degrees of rectifications including the zeroth, the 10-simplex itself. Vertices of the rectified 10-simplex are located at the edge-centers of the 10-simplex. Vertices of the birectified 10-simplex are located in the triangular face centers of the 10-simplex. Vertices of the trirectified 10-simplex are located in the tetrahedral cell centers of the 10-simplex. Vertices of the quadrirectified 10-simplex are located in the 5-cell centers of the 10-simplex.

Rectified 10-simplex

Rectified 10-simplex
Type
Schläfli symbol
Coxeter-Dynkin diagrams
9-faces
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Petrie polygon
Coxeter groups
Properties

The rectified 10-simplex is the vertex figure of the 11-demicube.

Alternate names

  • Rectified hendecaxennon (Acronym ru) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of the rectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,0,1,1). This construction is based on facets of the rectified 11-orthoplex.

Images

Birectified 10-simplex

Birectified 10-simplex
Type
Schläfli symbol
Coxeter-Dynkin diagrams
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties

Alternate names

  • Birectified hendecaxennon (Acronym bru) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of the birectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,0,1,1,1). This construction is based on facets of the birectified 11-orthoplex.

Images

Trirectified 10-simplex

Trirectified 10-simplex
Type
Schläfli symbol
Coxeter-Dynkin diagrams
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties

Alternate names

  • Trirectified hendecaxennon (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of the trirectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,0,1,1,1,1). This construction is based on facets of the trirectified 11-orthoplex.

Images

Quadrirectified 10-simplex

Quadrirectified 10-simplex
Type
Schläfli symbol
Coxeter-Dynkin diagrams
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups
Properties

Alternate names

  • Quadrirectified hendecaxennon (Acronym teru) (Jonathan Bowers)

Coordinates

The Cartesian coordinates of the vertices of the quadrirectified 10-simplex can be most simply positioned in 11-space as permutations of (0,0,0,0,0,0,1,1,1,1,1). This construction is based on facets of the quadrirectified 11-orthoplex.

Images

Notes

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, 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)
  • x3o3o3o3o3o3o3o3o3o - ux, o3x3o3o3o3o3o3o3o3o - ru, o3o3x3o3o3o3o3o3o3o - bru, o3o3o3x3o3o3o3o3o3o - tru, o3o3o3o3x3o3o3o3o3o - teru

References

  1. Klitzing, (o3x3o3o3o3o3o3o3o3o - ru)
  2. Klitzing, (o3o3x3o3o3o3o3o3o3o - bru)
  3. Klitzing, (o3o3o3x3o3o3o3o3o3o - tru)
  4. Klitzing, (o3o3o3o3x3o3o3o3o3o - teru)
Info: Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

10-polytopes
Want to explore this topic further?

Ask Mako anything about Rectified 10-simplexes — get instant answers, deeper analysis, and related topics.

Research with Mako

Free with your Surf account

Content sourced from Wikipedia, available under CC BY-SA 4.0.

This content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.

Report