Rectified 10-orthoplexes

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10-cube t8.svg
10-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
10-cube t7.svg
Rectified 10-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
10-cube t6.svg
Birectified 10-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
10-cube t5.svg
Trirectified 10-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
10-cube t5.svg
Quadirectified 10-orthoplex
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
10-cube t4.svg
Quadrirectified 10-cube
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
10-cube t3.svg
Trirectified 10-cube
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
10-cube t2.svg
Birectified 10-cube
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
10-cube t1.svg
Rectified 10-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
10-cube t0.svg
10-cube
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Orthogonal projections in A10 Coxeter plane

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

There are 10 rectifications of the 10-orthoplex. Vertices of the rectified 10-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 10-orthoplex are located in the triangular face centers of the 10-orthoplex. Vertices of the trirectified 10-orthoplex are located in the tetrahedral cell centers of the 10-orthoplex.

These polytopes are part of a family 1023 uniform 10-polytopes with BC10 symmetry.

Rectified 10-orthoplex[edit]

Rectified 10-orthoplex
Type uniform 10-polytope
Schläfli symbol t1{38,4}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 2880
Vertices 180
Vertex figure 8-orthoplex prism
Petrie polygon icosagon
Coxeter groups C10, [4,38]
D10, [37,1,1]
Properties convex

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

Rectified 10-orthoplex[edit]

The rectified 10-orthoplex is the vertex figure for the demidekeractic honeycomb.

CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png or CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png

Alternate names[edit]

  • rectified decacross (Acronym rake) (Jonathan Bowers)[1]

Construction[edit]

There are two Coxeter groups associated with the rectified 10-orthoplex, one with the C10 or [4,38] Coxeter group, and a lower symmetry with two copies of 9-orthoplex facets, alternating, with the D10 or [37,1,1] Coxeter group.

Cartesian coordinates[edit]

Cartesian coordinates for the vertices of a rectified 10-orthoplex, centered at the origin, edge length \sqrt{2} are all permutations of:

(±1,±1,0,0,0,0,0,0,0,0)

Root vectors[edit]

Its 180 vertices represent the root vectors of the simple Lie group D10. The vertices can be seen in 3 hyperplanes, with the 45 vertices rectified 9-simplices facets on opposite sides, and 90 vertices of an expanded 9-simplex passing through the center. When combined with the 20 vertices of the 9-orthoplex, these vertices represent the 200 root vectors of the simple Lie group B10.

Images[edit]

Orthographic projections
B10 B9 B8
10-cube t8.svg 10-cube t8 B9.svg 10-cube t8 B8.svg
[20] [18] [16]
B7 B6 B5
10-cube t8 B7.svg 10-cube t8 B6.svg 10-cube t8 B5.svg
[14] [12] [10]
B4 B3 B2
10-cube t8 B4.svg 10-cube t8 B3.svg 10-cube t8 B2.svg
[8] [6] [4]

Birectified 10-orthoplex[edit]

Birectified 10-orthoplex
Type uniform 10-polytope
Schläfli symbol t2{38,4}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups C10, [4,38]
D10, [37,1,1]
Properties convex

Alternate names[edit]

  • Birectified decacross

Cartesian coordinates[edit]

Cartesian coordinates for the vertices of a birectified 10-orthoplex, centered at the origin, edge length \sqrt{2} are all permutations of:

(±1,±1,±1,0,0,0,0,0,0,0)

Images[edit]

Orthographic projections
B10 B9 B8
10-cube t7.svg 10-cube t7 B9.svg 10-cube t7 B8.svg
[20] [18] [16]
B7 B6 B5
10-cube t7 B7.svg 10-cube t7 B6.svg 10-cube t7 B5.svg
[14] [12] [10]
B4 B3 B2
10-cube t7 B4.svg 10-cube t7 B3.svg 10-cube t7 B2.svg
[8] [6] [4]

Trirectified 10-orthoplex[edit]

Trirectified 10-orthoplex
Type uniform 10-polytope
Schläfli symbol t3{38,4}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups C10, [4,38]
D10, [37,1,1]
Properties convex

Alternate names[edit]

  • Trirectified decacross (Acronym trake) (Jonathan Bowers)[2]

Cartesian coordinates[edit]

Cartesian coordinates for the vertices of a trirectified 10-orthoplex, centered at the origin, edge length \sqrt{2} are all permutations of:

(±1,±1,±1,±1,0,0,0,0,0,0)

Images[edit]

Orthographic projections
B10 B9 B8
10-cube t6.svg 10-cube t6 B9.svg 10-cube t6 B8.svg
[20] [18] [16]
B7 B6 B5
10-cube t6 B7.svg 10-cube t6 B6.svg 10-cube t6 B5.svg
[14] [12] [10]
B4 B3 B2
10-cube t6 B4.svg 10-cube t6 B3.svg 10-cube t6 B2.svg
[8] [6] [4]

Quadrirectified 10-orthoplex[edit]

Quadrirectified 10-orthoplex
Type uniform 10-polytope
Schläfli symbol t4{38,4}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups C10, [4,38]
D10, [37,1,1]
Properties convex

Alternate names[edit]

  • Quadrirectified decacross (Acronym brake) (Jonthan Bowers)[3]

Cartesian coordinates[edit]

Cartesian coordinates for the vertices of a quadrirectified 10-orthoplex, centered at the origin, edge length \sqrt{2} are all permutations of:

(±1,±1,±1,±1,±1,0,0,0,0,0)

Images[edit]

Orthographic projections
B10 B9 B8
10-cube t6.svg 10-cube t6 B9.svg 10-cube t6 B8.svg
[20] [18] [16]
B7 B6 B5
10-cube t6 B7.svg 10-cube t6 B6.svg 10-cube t6 B5.svg
[14] [12] [10]
B4 B3 B2
10-cube t6 B4.svg 10-cube t6 B3.svg 10-cube t6 B2.svg
[8] [6] [4]

Notes[edit]

  1. ^ Klitzing, (o3x3o3o3o3o3o3o3o4o - rake)
  2. ^ Klitzing, (o3o3o3x3o3o3o3o3o4o - trake)
  3. ^ Klitzing, (o3o3x3o3o3o3o3o3o4o - brake)

References[edit]

  • 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. (1966)
  • Richard Klitzing, 10D, uniform polytopes (polyxenna) x3o3o3o3o3o3o3o3o4o - ka, o3x3o3o3o3o3o3o3o4o - rake, o3o3x3o3o3o3o3o3o4o - brake, o3o3o3x3o3o3o3o3o4o - trake, o3o3o3o3x3o3o3o3o4o - terake, o3o3o3o3o3x3o3o3o4o - terade, o3o3o3o3o3o3x3o3o4o - trade, o3o3o3o3o3o3o3x3o4o - brade, o3o3o3o3o3o3o3o3x4o - rade, o3o3o3o3o3o3o3o3o4x - deker

External links[edit]