8-orthoplex

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.

8-orthoplex
Octacross

Orthogonal projection
inside Petrie polygon
TypeRegular 8-polytope
Familyorthoplex
Schläfli symbol{36,4}
{3,3,3,3,3,31,1}
Coxeter-Dynkin diagrams
7-faces256 {36}
6-faces1024 {35}
5-faces1792 {34}
4-faces1792 {33}
Cells1120 {3,3}
Faces448 {3}
Edges112
Vertices16
Vertex figure7-orthoplex
Petrie polygonhexadecagon
Coxeter groupsC8, [36,4]
D8, [35,1,1]
Dual8-cube
Propertiesconvex

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
  • 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

This configuration matrix represents the 8-orthoplex. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces, 6-faces and 7-faces. The diagonal numbers say how many of each element occur in the whole 8-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.[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]

B8k-facefkf0f1f2f3f4f5f6f7k-figurenotes
B7( ) f0 161484280560672448128{3,3,3,3,3,4} B8/B7 = 2^8*8!/2^7/7! = 16
A1B6{ } f1 2112126016024019264{3,3,3,3,4} B8/A1B6 = 2^8*8!/2/2^6/6! = 112
A2B5{3} f2 334481040808032{3,3,3,4} B8/A2B5 = 2^8*8!/3!/2^5/5! = 448
A3B4{3,3} f3 46411208243216{3,3,4} B8/A3B4 = 2^8*8!/4!/2^4/4! = 1120
A4B3{3,3,3} f4 51010517926128{3,4} B8/A4B3 = 2^8*8!/5!/8/3! = 1792
A5B2{3,3,3,3} f5 61520156179244{4} B8/A5B2 = 2^8*8!/6!/4/2 = 1792
A6A1{3,3,3,3,3} f6 721353521710242{ } B8/A6A1 = 2^8*8!/7!/2 = 1024
A7{3,3,3,3,3,3} f7 828567056288256( ) B8/A7 = 2^8*8!/8! = 256

Construction

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 coordinates

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
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.

gollark: Repetitive content can have really good compression ratios.
gollark: Wayland appears technically superior in some ways and worse in others (no consistent standards for asking compositors to do screenshots or whatever, and all window managers reinventing compositor bits), but also support is quite bad.
gollark: Oh yes, some distros ship it with that.
gollark: nginx mostly just uses one.
gollark: In what way?

References
  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
      • (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".
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 Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.