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)

As a configuration[edit]

The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.[1][2]

The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing individual mirrors. [3]

B8 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 k-face fk f0 f1 f2 f3 f4 f5 f6 f7 k-figure notes
B7 CDel node x.pngCDel 2.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 ( ) f0 16 14 84 280 560 672 448 128 {3,3,3,3,3,4} B8/B7 = 2^8*8!/2^7/7! = 16
A1B6 CDel node 1.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png { } f1 2 112 12 60 160 240 192 64 {3,3,3,3,4} B8/A1B6 = 2^8*8!/2/2^6/6! = 112
A2B5 CDel node 1.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png {3} f2 3 3 448 10 40 80 80 32 {3,3,3,4} B8/A2B5 = 2^8*8!/3!/2^5/5! = 448
A3B4 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png {3,3} f3 4 6 4 1120 8 24 32 16 {3,3,4} B8/A3B4 = 2^8*8!/4!/2^4/4! = 1120
A4B3 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png {3,3,3} f4 5 10 10 5 1792 6 12 8 {3,4} B8/A4B3 = 2^8*8!/5!/8/3! = 1792
A5B2 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.pngCDel 4.pngCDel node.png {3,3,3,3} f5 6 15 20 15 6 1792 4 4 {4} B8/A5B2 = 2^8*8!/6!/4/2 = 1792
A6A1 CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node x.pngCDel 2.pngCDel node.png {3,3,3,3,3} f6 7 21 35 35 21 7 1024 2 { } B8/A6A1 = 2^8*8!/7!/2 = 1024
A7 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 2.pngCDel node x.png {3,3,3,3,3,3} f7 8 28 56 70 56 28 8 256 ( ) B8/A7 = 2^8*8!/8! = 256

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
Quasiregular 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]

  1. ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
  2. ^ Coxeter, Complex Regular Polytopes, p.117
  3. ^ Klitzing, Richard. "x3o3o3o3o3o3o4o - ek".
  • 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.
  • Klitzing, Richard. "8D uniform polytopes (polyzetta) x3o3o3o3o3o3o4o - ek".

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / 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