8-cube

8-cube Octeract
Orthogonal projection inside Petrie polygon
Type	Regular 8-polytope
Family	hypercube
Schläfli symbol	{4,36}
Coxeter-Dynkin diagrams
7-faces	16 {4,35}
6-faces	112 {4,34}
5-faces	448 {4,33}
4-faces	1120 {4,32}
Cells	1792 {4,3}
Faces	1792 {4}
Edges	1024
Vertices	256
Vertex figure	7-simplex
Petrie polygon	hexadecagon
Coxeter group	C8, [36,4]
Dual	8-orthoplex
Properties	convex, Hanner polytope

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 {4,3⁶}, being composed of 3 7-cubes around each 6-face. It is called an octeract, a portmanteau of tesseract (the 4-cube) and oct for eight (dimensions) 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 for the vertices of an 8-cube centered at the origin and edge length 2 are

(±1,±1,±1,±1,±1,±1,±1,±1)

while the interior of the same consists of all points (x₀, x₁, x₂, x₃, x₄, x₅, x₆, x₇) with −1 < x_i < 1.

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.¹²

${\displaystyle {\begin{bmatrix}{\begin{matrix}256&8&28&56&70&56&28&8\\2&1024&7&21&35&35&21&7\\4&4&1792&6&15&20&15&6\\8&12&6&1792&5&10&10&5\\16&32&24&8&1120&4&6&4\\32&80&80&40&10&448&3&3\\64&192&240&160&60&12&112&2\\128&448&672&560&280&84&14&16\end{matrix}}\end{bmatrix}}}$

The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.³

B8		k-face	fk	f0	f1	f2	f3	f4	f5	f6	f7	k-figure	Notes
A7		( )	f0	256	8	28	56	70	56	28	8	{3,3,3,3,3,3}	B8/A7 = 2^8·8!/8! = 256
A6A1		{ }	f1	2	1024	7	21	35	35	21	7	{3,3,3,3,3}	B8/A6A1 = 2^8·8!/7!/2 = 1024
A5B2		{4}	f2	4	4	1792	6	15	20	15	6	{3,3,3,3}	B8/A5B2 = 2^8·8!/6!/4/2 = 1792
A4B3		{4,3}	f3	8	12	6	1792	5	10	10	5	{3,3,3}	B8/A4B3 = 2^8·8!/5!/8/3! = 1792
A3B4		{4,3,3}	f4	16	32	24	8	1120	4	6	4	{3,3}	B8/A3B4 = 2^8·8!/4!/2^4/4! = 1120
A2B5		{4,3,3,3}	f5	32	80	80	40	10	448	3	3	{3}	B8/A2B5 = 2^8·8!/3!/2^5/5! = 448
A1B6		{4,3,3,3,3}	f6	64	192	240	160	60	12	112	2	{ }	B8/A1B6 = 2^8·8!/2/2^6/6! = 112
B7		{4,3,3,3,3,3}	f7	128	448	672	560	280	84	14	16	( )	B8/B7 = 2^8·8!/2^7/7! = 16

Orthographic projections

B8			B7
[16]			[14]
B6			B5
[12]			[10]
B4		B3		B2
[8]		[6]		[4]
A7		A5		A3
[8]		[6]		[4]

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

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

Petrie polygon orthographic projections

Line segment	Square	Cube	4-cube	5-cube	6-cube	7-cube	8-cube	9-cube	10-cube