8-orthoplex Octacross | |
---|---|
Orthogonal projection inside Petrie polygon | |
Type | Regular 8-polytope |
Family | orthoplex |
Schläfli symbol | {3^{6},4} {3,3,3,3,3,3^{1,1}} |
Coxeter-Dynkin diagrams | |
7-faces | 256 {3^{6}} |
6-faces | 1024 {3^{5}} |
5-faces | 1792 {3^{4}} |
4-faces | 1792 {3^{3}} |
Cells | 1120 {3,3} |
Faces | 448 {3} |
Edges | 112 |
Vertices | 16 |
Vertex figure | 7-orthoplex |
Petrie polygon | hexadecagon |
Coxeter groups | C_{8}, [3^{6},4] D_{8}, [3^{5,1,1}] |
Dual | 8-cube |
Properties | convex, Hanner polytope |
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 {3^{6},4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,3,3^{1,1}} or Coxeter symbol 5_{11}.
It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is an 8-hypercube, or octeract.
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]}
B_{8} | k-face | f_{k} | f_{0} | f_{1} | f_{2} | f_{3} | f_{4} | f_{5} | f_{6} | f_{7} | k-figure | notes | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
B_{7} | ( ) | f_{0} | 16 | 14 | 84 | 280 | 560 | 672 | 448 | 128 | {3,3,3,3,3,4} | B_{8}/B_{7} = 2^8*8!/2^7/7! = 16 | |
A_{1}B_{6} | { } | f_{1} | 2 | 112 | 12 | 60 | 160 | 240 | 192 | 64 | {3,3,3,3,4} | B_{8}/A_{1}B_{6} = 2^8*8!/2/2^6/6! = 112 | |
A_{2}B_{5} | {3} | f_{2} | 3 | 3 | 448 | 10 | 40 | 80 | 80 | 32 | {3,3,3,4} | B_{8}/A_{2}B_{5} = 2^8*8!/3!/2^5/5! = 448 | |
A_{3}B_{4} | {3,3} | f_{3} | 4 | 6 | 4 | 1120 | 8 | 24 | 32 | 16 | {3,3,4} | B_{8}/A_{3}B_{4} = 2^8*8!/4!/2^4/4! = 1120 | |
A_{4}B_{3} | {3,3,3} | f_{4} | 5 | 10 | 10 | 5 | 1792 | 6 | 12 | 8 | {3,4} | B_{8}/A_{4}B_{3} = 2^8*8!/5!/8/3! = 1792 | |
A_{5}B_{2} | {3,3,3,3} | f_{5} | 6 | 15 | 20 | 15 | 6 | 1792 | 4 | 4 | {4} | B_{8}/A_{5}B_{2} = 2^8*8!/6!/4/2 = 1792 | |
A_{6}A_{1} | {3,3,3,3,3} | f_{6} | 7 | 21 | 35 | 35 | 21 | 7 | 1024 | 2 | { } | B_{8}/A_{6}A_{1} = 2^8*8!/7!/2 = 1024 | |
A_{7} | {3,3,3,3,3,3} | f_{7} | 8 | 28 | 56 | 70 | 56 | 28 | 8 | 256 | ( ) | B_{8}/A_{7} = 2^8*8!/8! = 256 |
There are two Coxeter groups associated with the 8-cube, one regular, dual of the octeract with the C_{8} or [4,3,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 7-simplex facets, alternating, with the D_{8} or [3^{5,1,1}] 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 | {3,3,3,3,3,3^{1,1}} | [3,3,3,3,3,3^{1,1}] | 5160960 | ||
8-fusil | 8{} | [2^{7}] | 256 |
Cartesian coordinates for the vertices of an 8-cube, centered at the origin are
Every vertex pair is connected by an edge, except opposites.
B_{8} | B_{7} | ||||
---|---|---|---|---|---|
[16] | [14] | ||||
B_{6} | B_{5} | ||||
[12] | [10] | ||||
B_{4} | B_{3} | B_{2} | |||
[8] | [6] | [4] | |||
A_{7} | A_{5} | A_{3} | |||
[8] | [6] | [4] |
It is used in its alternated form 5_{11} with the 8-simplex to form the 5_{21} honeycomb.