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.
It has two constructive forms, the first being regular with Schläfli symbol, and the second with alternately labeled facets, with Schläfli symbol 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

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.
The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing individual mirrors.
B8k-facefkf0f1f2f3f4f5f6f7k-figurenotes
B7f01614842805606724481287-orthoplex|B8/B7 = 2^8*8!/2^7/7! = 16
A1B6f121121260160240192646-orthoplex|B8/A1B6 = 2^8*8!/2/2^6/6! = 112
A2B5triangle|f23344810408080325-orthoplex|B8/A2B5 = 2^8*8!/3!/2^5/5! = 448
A3B4Tetrahedron|f34641120824321616-cell|B8/A3B4 = 2^8*8!/4!/2^4/4! = 1120
A4B35-cell|f451010517926128octahedron|B8/A4B3 = 2^8*8!/5!/8/3! = 1792
A5B25-simplex|f561520156179244square|B8/A5B2 = 2^8*8!/6!/4/2 = 1792
A6A16-simplex|f6721353521710242B8/A6A1 = 2^8*8!/7!/2 = 1024
A77-simplex|f7828567056288256B8/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 symmetry group, and a half symmetry with two copies of 7-simplex facets, alternating, with the D8 or symmetry group. A lowest symmetry construction is based on a dual of an 8-orthotope, called an 8-fusil.
NameCoxeter diagramSchläfli symbolSymmetryOrderVertex figure
regular 8-orthoplex10321920
Quasiregular 8-orthoplex5160960
8-fusil8256

Cartesian coordinates

for the vertices of an 8-cube, centered at the origin are
Every vertex pair is connected by an edge, except opposites.

Images

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