8-orthoplex

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8-orthoplex
Octacross
8-orthoplex.svg
Orthogonal projection
inside Petrie polygon
Type Regular 8-polytope
Family orthoplex
Schläfli symbol {36,4}
{3,3,3,3,3,31,1}
Coxeter-Dynkin diagrams 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
CDel 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 256 {36}7-simplex t0.svg
6-faces 1024 {35}6-simplex t0.svg
5-faces 1792 {34}5-simplex t0.svg
4-faces 1792 {33}4-simplex t0.svg
Cells 1120 {3,3}3-simplex t0.svg
Faces 448 {3}2-simplex t0.svg
Edges 112
Vertices 16
Vertex figure 7-orthoplex
Petrie polygon hexadecagon
Coxeter groups C8, [36,4]
D8, [35,1,1]
Dual 8-cube
Properties convex

In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces.

It has two constructive forms, the first being regular with Schläfli symbol {36,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,3,31,1} or Coxeter symbol 511.

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is an 8-hypercube, or octeract.

Alternate names[edit]

  • Octacross, derived from combining the family name cross polytope with oct for eight (dimensions) in Greek
  • Diacosipentacontahexazetton as a 256-facetted 8-polytope (polyzetton)

Construction[edit]

There are two Coxeter groups associated with the 8-cube, one regular, dual of the octeract with the C8 or [4,3,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 7-simplex facets, alternating, with the D8 or [35,1,1] symmetry group.A lowest symmetry construction is based on a dual of an 8-orthotope, called an 8-fusil.

Name Coxeter diagram Schläfli symbol Symmetry Order Vertex figure
regular 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 {3,3,3,3,3,3,4} [3,3,3,3,3,3,4] 10321920 CDel 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
Alternate 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 split1.pngCDel nodes.png {3,3,3,3,3,31,1} [3,3,3,3,3,31,1] 5160960 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.png
8-fusil CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png 8{} [27] 256 CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.pngCDel 2.pngCDel node f1.png

Cartesian coordinates[edit]

Cartesian coordinates for the vertices of an 8-cube, centered at the origin are

(±1,0,0,0,0,0,0,0), (0,±1,0,0,0,0,0,0), (0,0,±1,0,0,0,0,0), (0,0,0,±1,0,0,0,0),
(0,0,0,0,±1,0,0,0), (0,0,0,0,0,±1,0,0), (0,0,0,0,0,0,0,±1), (0,0,0,0,0,0,0,±1)

Every vertex pair is connected by an edge, except opposites.

Images[edit]

orthographic projections
B8 B7
8-cube t7.svg 8-cube t7 B7.svg
[16] [14]
B6 B5
8-cube t7 B6.svg 8-cube t7 B5.svg
[12] [10]
B4 B3 B2
8-cube t7 B4.svg 8-cube t7 B3.svg 8-cube t7 B2.svg
[8] [6] [4]
A7 A5 A3
8-cube t7 A7.svg 8-cube t7 A5.svg 8-cube t7 A3.svg
[8] [6] [4]

It is used in its alternated form 511 with the 8-simplex to form the 521 honeycomb.

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, 8D uniform polytopes (polyzetta), x3o3o3o3o3o3o4o - ek

External links[edit]