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-cell 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

Octacross, derived from combining the family name cross polytope with oct for eight in GreekDiacosipentacontahexazetton as a 256-facetted 8-polytope, acronym: ek

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.
The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing individual mirrors.

B₈	k-face	f_k	f₀	f₁	f₂	f₃	f₄	f₅	f₆	f₇	k-figure	notes
B₇		f₀	16	14	84	280	560	672	448	128	[7-orthoplex\|]	B₈/B₇ = 2^8*8!/2^7/7! = 16
A₁B₆		f₁	2	112	12	60	160	240	192	64	[6-orthoplex\|]	B₈/A₁B₆ = 2^8*8!/2/2^6/6! = 112
A₂B₅	[triangle\|]	f₂	3	3	448	10	40	80	80	32	[5-orthoplex\|]	B₈/A₂B₅ = 2^8*8!/3!/2^5/5! = 448
A₃B₄	[Tetrahedron\|]	f₃	4	6	4	1120	8	24	32	16	[16-cell\|]	B₈/A₃B₄ = 2^8*8!/4!/2^4/4! = 1120
A₄B₃	[5-cell\|]	f₄	5	10	10	5	1792	6	12	8	[octahedron\|]	B₈/A₄B₃ = 2^8*8!/5!/8/3! = 1792
A₅B₂	[5-simplex\|]	f₅	6	15	20	15	6	1792	4	4	[square\|]	B₈/A₅B₂ = 2^8*8!/6!/4/2 = 1792
A₆A₁	[6-simplex\|]	f₆	7	21	35	35	21	7	1024	2		B₈/A₆A₁ = 2^8*8!/7!/2 = 1024
A₇	[7-simplex\|]	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 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.

Name	Coxeter diagram	Schläfli symbol	Symmetry	Order	Vertex figure
regular 8-orthoplex				10321920
Quasiregular 8-orthoplex				5160960
8-fusil		8		256

Cartesian coordinates

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.