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8-orthoplex

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.

Contents

Alternate namesEdit

  • 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 configurationEdit

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                 k-face fk f0 f1 f2 f3 f4 f5 f6 f7 k-figure notes
B7                 ( ) 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                 { } 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                 {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                 {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                 {3,3,3} f4 5 10 10 5 1792 6 12 8 {3,4} B8/A4B3 = 2^8*8!/5!/8/3! = 1792
A5B2                 {3,3,3,3} f5 6 15 20 15 6 1792 4 4 {4} B8/A5B2 = 2^8*8!/6!/4/2 = 1792
A6A1                 {3,3,3,3,3} f6 7 21 35 35 21 7 1024 2 { } B8/A6A1 = 2^8*8!/7!/2 = 1024
A7                 {3,3,3,3,3,3} f7 8 28 56 70 56 28 8 256 ( ) B8/A7 = 2^8*8!/8! = 256

ConstructionEdit

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                 {3,3,3,3,3,3,4} [3,3,3,3,3,3,4] 10321920              
Quasiregular 8-orthoplex               {3,3,3,3,3,31,1} [3,3,3,3,3,31,1] 5160960            
8-fusil                 8{} [27] 256              

Cartesian coordinatesEdit

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.

ImagesEdit

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

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

ReferencesEdit

  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 linksEdit