8-cube · Paulipedia

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

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.^[1]^[2]

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

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	f_k	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

Projections

Orthographic projections
B₈		B₇

[16]		[14]
B₆		B₅

[12]		[10]
B₄	B₃		B₂

[8]	[6]		[4]
A₇	A₅		A₃

[8]	[6]		[4]

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:

Petrie polygon orthographic projections

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

References

Coxeter, Regular Polytopes, p. 12, Sec. 1.8 Configurations
Coxeter (1991), p. 117.
Klitzing, Richard. "o3o3o3o3o3o3o4x - octo".

H.S.M. Coxeter:
- Coxeter, Regular Polytopes, (3rd edition, 1973), Dover, ISBN 0-486-61480-8, pp. 294–295, Table I (iii): Regular Polytopes, three regular polytopes in n dimensions (n ≥ 5)
- Coxeter, H.S.M. (1991) [1974]. Regular Complex Polytopes. Cambridge University Press. ISBN 0-521-39490-2.
- 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, ISBN 978-0-471-01003-6
  - (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)
Klitzing, Richard. "8D uniform polytopes (polyzetta) o3o3o3o3o3o3o4x - octo".

External links

Weisstein, Eric W. "Hypercube". MathWorld.
Olshevsky, George. "Measure polytope". Glossary for Hyperspace. Archived from the original on 4 February 2007.
Multi-dimensional Glossary: hypercube Garrett Jones


Family	$A n$	$B n$	$I 2 (p)$ / $D n$	$E 6$ / $E 7$ / $E 8$ / $F 4$ / $G 2$	$H n$
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	1₂₂ • 2₂₁
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	1₃₂ • 2₃₁ • 3₂₁
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	1₄₂ • 2₄₁ • 4₂₁
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds • Polytope operations