Truncated 8-orthoplexes

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8-cube t7.svg
8-orthoplex
CDel 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
8-cube t67.svg
Truncated 8-orthoplex
CDel node 1.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
8-cube t56.svg
Bitruncated 8-orthoplex
CDel node.pngCDel 3.pngCDel node 1.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
8-cube t45.svg
Tritruncated 8-orthoplex
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
8-cube t34.svg
Quadritruncated 8-cube
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
8-cube t23.svg
Tritruncated 8-cube
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
8-cube t12.svg
Bitruncated 8-cube
CDel 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 1.pngCDel 4.pngCDel node.png
8-cube t01.svg
Truncated 8-cube
CDel 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 4.pngCDel node 1.png
8-cube t0.svg
8-cube
CDel 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 1.png
Orthogonal projections in B8 Coxeter plane

In eight-dimensional geometry, a truncated 8-orthoplex is a convex uniform 8-polytope, being a truncation of the regular 8-orthoplex.

There are 7 truncation for the 8-orthoplex. Vertices of the truncation 8-orthoplex are located as pairs on the edge of the 8-orthoplex. Vertices of the bitruncated 8-orthoplex are located on the triangular faces of the 8-orthoplex. Vertices of the tritruncated 7-orthoplex are located inside the tetrahedral cells of the 8-orthoplex. The final truncations are best expressed relative to the 8-cube.

Truncated 8-orthoplex[edit]

Truncated 8-orthoplex
Type uniform 8-polytope
Schläfli symbol t0,1{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams CDel node 1.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 node 1.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

6-faces
5-faces
4-faces
Cells
Faces
Edges 1456
Vertices 224
Vertex figure Elongated 6-orthoplex pyramid
Coxeter groups B8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Properties convex

Alternate names[edit]

  • Truncated octacross (acronym tek) (Jonthan Bowers)[1]

Construction[edit]

There are two Coxeter groups associated with the truncated 8-orthoplex, one with the C8 or [4,3,3,3,3,3,3] Coxeter group, and a lower symmetry with the D8 or [35,1,1] Coxeter group.

Coordinates[edit]

Cartesian coordinates for the vertices of a truncated 8-orthoplex, centered at the origin, are all 224 vertices are sign (4) and coordinate (56) permutations of

(±2,±1,0,0,0,0,0,0)

Images[edit]

orthographic projections
B8 B7
8-cube t67.svg 8-cube t67 B7.svg
[16] [14]
B6 B5
8-cube t67 B6.svg 8-cube t67 B5.svg
[12] [10]
B4 B3 B2
8-cube t67 B4.svg 8-cube t67 B3.svg 8-cube t67 B2.svg
[8] [6] [4]
A7 A5 A3
8-cube t67 A7.svg 8-cube t67 A5.svg 8-cube t67 A3.svg
[8] [6] [4]

Bitruncated 8-orthoplex[edit]

Bitruncated 8-orthoplex
Type uniform 8-polytope
Schläfli symbol t1,2{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node 1.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 node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png

6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Properties convex

Alternate names[edit]

  • Bitruncated octacross (acronym batek) (Jonthan Bowers)[2]

Coordinates[edit]

Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutations of

(±2,±2,±1,0,0,0,0,0)

Images[edit]

orthographic projections
B8 B7
8-cube t56.svg 8-cube t56 B7.svg
[16] [14]
B6 B5
8-cube t56 B6.svg 8-cube t56 B5.svg
[12] [10]
B4 B3 B2
8-cube t56 B4.svg 8-cube t56 B3.svg 8-cube t56 B2.svg
[8] [6] [4]
A7 A5 A3
8-cube t56 A7.svg 8-cube t56 A5.svg 8-cube t56 A3.svg
[8] [6] [4]

Tritruncated 8-orthoplex[edit]

Tritruncated 8-orthoplex
Type uniform 8-polytope
Schläfli symbol t2,3{3,3,3,3,3,3,4}
Coxeter-Dynkin diagrams CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.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 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png

6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups B8, [3,3,3,3,3,3,4]
D8, [35,1,1]
Properties convex

Alternate names[edit]

  • Tritruncated octacross (acronym tatek) (Jonthan Bowers)[3]

Coordinates[edit]

Cartesian coordinates for the vertices of a bitruncated 8-orthoplex, centered at the origin, are all sign and coordinate permutations of

(±2,±2,±2,±1,0,0,0,0)

Images[edit]

orthographic projections
B8 B7
8-cube t45.svg 8-cube t45 B7.svg
[16] [14]
B6 B5
8-cube t45 B6.svg 8-cube t45 B5.svg
[12] [10]
B4 B3 B2
8-cube t45 B4.svg 8-cube t45 B3.svg 8-cube t45 B2.svg
[8] [6] [4]
A7 A5 A3
8-cube t45 A7.svg 8-cube t45 A5.svg 8-cube t45 A3.svg
[8] [6] [4]

Notes[edit]

  1. ^ Klitizing, (x3x3o3o3o3o3o4o - tek)
  2. ^ Klitizing, (o3x3x3o3o3o3o4o - batek)
  3. ^ Klitizing, (o3o3x3x3o3o3o4o - tatek)

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)
  • Klitzing, Richard. "8D uniform polytopes (polyzetta)".  x3x3o3o3o3o3o4o - tek, o3x3x3o3o3o3o4o - batek, o3o3x3x3o3o3o4o - tatek

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds