Rectified 10-cubes

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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 BC10 Coxeter plane

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

There are 10 rectifications of the 10-cube, with the zeroth being the 10-cube itself. Vertices of the rectified 10-cube are located at the edge-centers of the 10-cube. Vertices of the birectified 10-cube are located in the square face centers of the 10-cube. Vertices of the trirectified 10-cube are located in the cubic cell centers of the 10-cube. The others are more simply constructed relative to the 10-cube dual polytpoe, the 10-orthoplex.

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

Rectified 10-cube[edit]

Rectified 10-orthoplex
Type uniform 10-polytope
Schläfli symbol t1{38,4}
Coxeter-Dynkin diagrams 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
CDel nodes 11.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.png
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 46080
Vertices 5120
Vertex figure 8-simplex prism
Coxeter groups C10, [4,38]
D10, [37,1,1]
Properties convex

Alternate names[edit]

  • Rectified dekeract (Acronym rade) (Jonathan Bowers)[1]

Cartesian coordinates[edit]

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

(±1,±1,±1,±1,±1,±1,±1,±1,±1,0)

Images[edit]

Orthographic projections
B10 B9 B8
10-cube t1.svg 10-cube t1 B9.svg 10-cube t1 B8.svg
[20] [18] [16]
B7 B6 B5
10-cube t1 B7.svg 10-cube t1 B6.svg 10-cube t1 B5.svg
[14] [12] [10]
B4 B3 B2
10-cube t1 B4.svg 10-cube t1 B3.svg 10-cube t1 B2.svg
[8] [6] [4]

Birectified 10-cube[edit]

Birectified 10-orthoplex
Type uniform 10-polytope
Coxeter symbol 0711
Schläfli symbol t2{38,4}
Coxeter-Dynkin diagrams 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
CDel nodes.pngCDel split2.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
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 184320
Vertices 11520
Vertex figure {4}x{36}
Coxeter groups C10, [4,38]
D10, [37,1,1]
Properties convex

Alternate names[edit]

  • Birectified dekeract (Acronym brade) (Jonathan Bowers)[2]

Cartesian coordinates[edit]

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

(±1,±1,±1,±1,±1,±1,±1,±1,0,0)

Images[edit]

Orthographic projections
B10 B9 B8
10-cube t2.svg 10-cube t2 B9.svg 10-cube t2 B8.svg
[20] [18] [16]
B7 B6 B5
10-cube t2 B7.svg 10-cube t2 B6.svg 10-cube t2 B5.svg
[14] [12] [10]
B4 B3 B2
10-cube t2 B4.svg 10-cube t2 B3.svg 10-cube t2 B2.svg
[8] [6] [4]

Trirectified 10-cube[edit]

Trirectified 10-orthoplex
Type uniform 10-polytope
Schläfli symbol t3{38,4}
Coxeter-Dynkin diagrams 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
CDel nodes.pngCDel split2.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
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 322560
Vertices 15360
Vertex figure {4,3}x{35}
Coxeter groups C10, [4,38]
D10, [37,1,1]
Properties convex

Alternate names[edit]

  • Tririrectified dekeract (Acronym trade) (Jonathan Bowers)[3]

Cartesian coordinates[edit]

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

(±1,±1,±1,±1,±1,±1,±1,0,0,0)

Images[edit]

Orthographic projections
B10 B9 B8
10-cube t3.svg 10-cube t3 B9.svg 10-cube t3 B8.svg
[20] [18] [16]
B7 B6 B5
10-cube t3 B7.svg 10-cube t3 B6.svg 10-cube t3 B5.svg
[14] [12] [10]
B4 B3 B2
10-cube t3 B4.svg 10-cube t3 B3.svg 10-cube t3 B2.svg
[8] [6] [4]

Quadrirectified 10-cube[edit]

Quadrirectified 10-orthoplex
Type uniform 10-polytope
Schläfli symbol t4{38,4}
Coxeter-Dynkin diagrams 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
CDel nodes.pngCDel split2.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
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges 322560
Vertices 13440
Vertex figure {4,3,3}x{34}
Coxeter groups C10, [4,38]
D10, [37,1,1]
Properties convex

Alternate names[edit]

  • Quadrirectified dekeract
  • Quadrirectified decacross (Acronym trade) (Jonathan Bowers)[4]

Cartesian coordinates[edit]

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

(±1,±1,±1,±1,±1,±1,0,0,0,0)

Images[edit]

Orthographic projections
B10 B9 B8
10-cube t4.svg 10-cube t4 B9.svg 10-cube t4 B8.svg
[20] [18] [16]
B7 B6 B5
10-cube t4 B7.svg 10-cube t4 B6.svg 10-cube t4 B5.svg
[14] [12] [10]
B4 B3 B2
10-cube t4 B4.svg 10-cube t4 B3.svg 10-cube t4 B2.svg
[8] [6] [4]

Notes[edit]

  1. ^ Klitzing, (o3o3o3o3o3o3o3o3x4o - rade)
  2. ^ Klitzing, (o3o3o3o3o3o3o3x3o4o - brade)
  3. ^ Klitzing, (o3o3o3o3o3o3x3o3o4o - trade)
  4. ^ Klitzing, (o3o3o3o3o3x3o3o3o4o - terade)

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]