8-cube


In geometry, an 8-cube is an eight-dimensional hypercube. It has 256 vertices, 1024 edges, 1792 square faces, 1792 cubic cells, 1120 tesseract 4-faces, 448 5-cube 5-faces, 112 6-cube 6-faces, and 16 7-cube 7-faces.
It is represented by Schläfli symbol, being composed of 3 7-cubes around each 6-face. It is called an octeract, a portmanteau of tesseract and oct for eight in Greek. It can also be called a regular hexdeca-8-tope or hexadecazetton, being an 8-dimensional polytope constructed from 16 regular facets.
It is a part of an infinite family of polytopes, called hypercubes. The dual of an 8-cube can be called a 8-orthoplex, and is a part of the infinite family of cross-polytopes.

Cartesian coordinates

for the vertices of an 8-cube centered at the origin and edge length 2 are
while the interior of the same consists of all points with -1 < xi < 1.

As a configuration

This configuration matrix represents the 8-cube. 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-cube. 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 one mirror at a time.
B8k-facefkf0f1f2f3f4f5f6f7k-figurenotes
A7f02568285670562887-simplex|B8/A7 = 2^8*8!/8! = 256
A6A1f12102472135352176-simplex|B8/A6A1 = 2^8*8!/7!/2 = 1024
A5B2Square|f2441792615201565-simplex|B8/A5B2 = 2^8*8!/6!/4/2 = 1792
A4B3Cube|f3812617925101055-cell|B8/A4B3 = 2^8*8!/5!/8/3! = 1792
A3B4Tesseract|f416322481120464tetrahedron|B8/A3B4 = 2^8*8!/4!/2^4/4! = 1120
A2B55-cube|f5328080401044833triangle|B8/A2B5 = 2^8*8!/3!/2^5/5! = 448
A1B66-cube|f66419224016060121122B8/A1B6 = 2^8*8!/2/2^6/6!= 112
B77-cube|f7128448672560280841416B8/B7 = 2^8*8!/2^7/7! = 16

Projections

Derived polytopes

Applying an alternation operation, deleting alternating vertices of the octeract, creates another uniform polytope, called a 8-demicube,, which has 16 demihepteractic and 128 8-simplex facets.