8-orthoplex
8-orthoplex Octacross | |
---|---|
![]() 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 | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
7-faces | 256 {36}![]() |
6-faces | 1024 {35}![]() |
5-faces | 1792 {34}![]() |
4-faces | 1792 {33}![]() |
Cells | 1120 {3,3}![]() |
Faces | 448 {3}![]() |
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[]
- 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]
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 |
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[]
B8 | B7 | ||||
---|---|---|---|---|---|
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[16] | [14] | ||||
B6 | B5 | ||||
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[12] | [10] | ||||
B4 | B3 | B2 | |||
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[8] | [6] | [4] | |||
A7 | A5 | A3 | |||
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[8] | [6] | [4] |
It is used in its alternated form 511 with the 8-simplex to form the 521 honeycomb.
References[]
- ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
- ^ Coxeter, Complex Regular Polytopes, p.117
- ^ 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[]
- Olshevsky, George. "Cross polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
- Polytopes of Various Dimensions
- Multi-dimensional Glossary
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 polychoron | Pentachoron | 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 |
- 8-polytopes