8-cube

8-cube Octeract
Orthogonal projection inside Petrie polygon
Type	Regular 8-polytope
Family	hypercube
Schläfli symbol	{4,36}
Coxeter-Dynkin diagrams	Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD Template:CDD
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

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.

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.Template:Sfnp

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

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

Projections

This 8-cube graph is an orthogonal projection. This orientation shows columns of vertices positioned a vertex-edge-vertex distance from one vertex on the left to one vertex on the right, and edges attaching adjacent columns of vertices. The numbers of vertices in each column are a row of Pascal's triangle: 1, 8, 28, 56, 70, 56, 28, 8, 1.

Derived polytopes

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.

Related polytopes

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

References

• H.S.M. Coxeter:
  • Coxeter, Regular Polytopes, (3rd edition, 1973), Dover, Template:Isbn, pp. 294–295, Table I (iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5)
  • Template:Cite book
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivić Weiss, Wiley-Interscience Publication, 1995, wiley.com, Template:Isbn
    • (Paper 22) H.S.M. Coxeter, Regular and Semi-Regular Polytopes I, [Math. Zeit. 46 (1940) 380–407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559–591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3–45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• Template:KlitzingPolytopes

External links

• Template:MathWorld
• Template:GlossaryForHyperspace
• Multi-dimensional Glossary: hypercube Garrett Jones