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.
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.
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16
{\displaystyle {\begin{bmatrix}{\begin{matrix}16&14&84&280&560&672&448&128\\2&112&12&60&160&240&192&64\\3&3&448&10&40&80&80&32\\4&6&4&1120&8&24&32&16\\5&10&10&5&1792&6&12&8\\6&15&20&15&6&1792&4&4\\7&21&35&35&21&7&1024&2\\8&28&56&70&56&28&8&256\end{matrix}}\end{bmatrix}}}
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] 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.
Cartesian coordinates for the vertices of an 8-cube, centered at the origin are Every vertex pair is connected by an edge, except opposites.
It is used in its alternated form 511 with the 8-simplex to form the 521 honeycomb.