Rectified 7-cubes

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7-cube t0.svg
7-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.png
7-cube t1.svg
Rectified 7-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.png
7-cube t2.svg
Birectified 7-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.png
7-cube t3.svg
Trirectified 7-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.png
7-cube t4.svg
Birectified 7-orthoplex
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.png
7-cube t5.svg
Rectified 7-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.png
7-cube t6.svg
7-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.png
Orthogonal projections in BC7 Coxeter plane

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

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

Rectified 7-cube[edit]

Rectified 7-cube
Type uniform 7-polytope
Schläfli symbol r{4,3,3,3,3,3}
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.png
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure 5-simplex prism
Coxeter groups BC7, [3,3,3,3,3,4]
Properties convex

Alternate names[edit]

  • rectified hepteract (Acronym resa) (Jonathan Bowers)[1]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph 7-cube t1.svg 7-cube t1 B6.svg 7-cube t1 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t1 B4.svg 7-cube t1 B3.svg 7-cube t1 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t1 A5.svg 7-cube t1 A3.svg
Dihedral symmetry [6] [4]

Cartesian coordinates[edit]

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

(±1,±1,±1,±1,±1,±1,0)

Birectified 7-cube[edit]

Birectified 7-cube
Type uniform 7-polytope
Schläfli symbol 2r{4,3,3,3,3,3}
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.png
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure {3}x{3,3,3}
Coxeter groups BC7, [3,3,3,3,3,4]
Properties convex

Alternate names[edit]

  • Birectified hepteract (Acronym bersa) (Jonathan Bowers)[2]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph 7-cube t2.svg 7-cube t2 B6.svg 7-cube t2 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t2 B4.svg 7-cube t2 B3.svg 7-cube t2 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t2 A5.svg 7-cube t2 A3.svg
Dihedral symmetry [6] [4]

Cartesian coordinates[edit]

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

(±1,±1,±1,±1,±1,0,0)

Trirectified 7-cube[edit]

Trirectified 7-cube
Type uniform 7-polytope
Schläfli symbol 3r{4,3,3,3,3,3}
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.png
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure {3,3}x{3,3}
Coxeter groups BC7, [3,3,3,3,3,4]
Properties convex

Alternate names[edit]

  • Trirectified hepteract
  • Trirectified 7-orthoplex
  • Trirectified heptacross (Acronym sez) (Jonathan Bowers)[3]

Images[edit]

orthographic projections
Coxeter plane B7 / A6 B6 / D7 B5 / D6 / A4
Graph 7-cube t3.svg 7-cube t3 B6.svg 7-cube t3 B5.svg
Dihedral symmetry [14] [12] [10]
Coxeter plane B4 / D5 B3 / D4 / A2 B2 / D3
Graph 7-cube t3 B4.svg 7-cube t3 B3.svg 7-cube t3 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane A5 A3
Graph 7-cube t3 A5.svg 7-cube t3 A3.svg
Dihedral symmetry [6] [4]

Cartesian coordinates[edit]

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

(±1,±1,±1,±1,0,0,0)

= Related polytopes[edit]

2-isotopic hypercubes
Dim. 2 3 4 5 6 7 8
Name t{4} r{4,3} 2t{4,3,3} 2r{4,3,3,3} 3t{4,3,3,3,3} 3r{4,3,3,3,3,3} 4t{4,3,3,3,3,3,3}
Coxeter
diagram
CDel label4.pngCDel branch 11.png CDel node 1.pngCDel split1-43.pngCDel nodes.png CDel branch 11.pngCDel 4a3b.pngCDel nodes.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 4a3b.pngCDel nodes.png CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 4a3b.pngCDel nodes.png CDel node 1.pngCDel split1.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 4a3b.pngCDel nodes.png CDel branch 11.pngCDel 3ab.pngCDel nodes.pngCDel 3ab.pngCDel nodes.pngCDel 4a3b.pngCDel nodes.png
Images Truncated square.png 3-cube t1.svgCuboctahedron.png 4-cube t12.svgSchlegel half-solid bitruncated 8-cell.png 5-cube t2.svg5-cube t2 A3.svg 6-cube t23.svg6-cube t23 A5.svg 7-cube t3.svg7-cube t3 A5.svg 8-cube t34.svg8-cube t34 A7.svg ...
Facets {3} Regular polygon 3 annotated.svg
{4} Regular polygon 4 annotated.svg
t{3,3} Uniform polyhedron-33-t01.png
t{3,4} Uniform polyhedron-43-t12.png
r{3,3,3} Schlegel half-solid rectified 5-cell.png
r{3,3,4} Schlegel wireframe 24-cell.png
2t{3,3,3,3} 5-simplex t12.svg
2t{3,3,3,4} 5-cube t23.svg
2r{3,3,3,3,3} 6-simplex t2.svg
2r{3,3,3,3,4} 6-cube t4.svg
3t{3,3,3,3,3,3} 7-simplex t23.svg
3t{3,3,3,3,3,4} 7-cube t45.svg
Vertex
figure
Cuboctahedron vertfig.png
Rectangle
Bitruncated 8-cell verf.png
Disphenoid
Birectified penteract verf.png
{3}×{4} duoprism
{3,3}×{3,4} duoprism

Notes[edit]

  1. ^ Klitzing, (o3o3o3o3o3x4o - rasa)
  2. ^ Klitzing, (o3o3o3o3x3o4o - bersa)
  3. ^ Klitzing, (o3o3o3x3o3o4o - sez)

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.
  • Richard Klitzing, 7D, uniform polytopes (polyexa) o3o3o3x3o3o4o - sez, o3o3o3o3x3o4o - bersa, o3o3o3o3o3x4o - rasa

External links[edit]