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 hexadeca-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 an 8-orthoplex and is a part of the infinite family of cross-polytopes.

Cartesian coordinates

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 < x_i < 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.

B₈	k-face	f_k	f₀	f₁	f₂	f₃	f₄	f₅	f₆	f₇	k-figure	Notes
A₇		f₀	256	8	28	56	70	56	28	8	[7-simplex\|]	B₈/A₇ = 2^8*8!/8! = 256
A₆A₁		f₁	2	1024	7	21	35	35	21	7	[6-simplex\|]	B₈/A₆A₁ = 2^8*8!/7!/2 = 1024
A₅B₂	[Square\|]	f₂	4	4	1792	6	15	20	15	6	[5-simplex\|]	B₈/A₅B₂ = 2^8*8!/6!/4/2 = 1792
A₄B₃	[Cube\|]	f₃	8	12	6	1792	5	10	10	5	[5-cell\|]	B₈/A₄B₃ = 2^8*8!/5!/8/3! = 1792
A₃B₄	[Tesseract\|]	f₄	16	32	24	8	1120	4	6	4	[tetrahedron\|]	B₈/A₃B₄ = 2^8*8!/4!/2^4/4! = 1120
A₂B₅	[5-cube\|]	f₅	32	80	80	40	10	448	3	3	[triangle\|]	B₈/A₂B₅ = 2^8*8!/3!/2^5/5! = 448
A₁B₆	[6-cube\|]	f₆	64	192	240	160	60	12	112	2		B₈/A₁B₆ = 2^8*8!/2/2^6/6! = 112
B₇	[7-cube\|]	f₇	128	448	672	560	280	84	14	16		B₈/B₇ = 2^8*8!/2^7/7! = 16

Derived polytopes

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

Related polytopes

The 8-cube is 8th in an infinite series of hypercube: