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

It is represented by

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]

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

The diagonal

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

Projections

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 number of vertices in each column represents rows in Pascal's triangle , being 1:8:28:56:70:56:28:8:1.

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

demihepteractic

and 128 8-simplex facets.

Related polytopes

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

Petrie polygon orthographic projections

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

References

1. ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
2. ^ Coxeter, Complex Regular Polytopes, p.117
3. ^ Klitzing, Richard. "o3o3o3o3o3o3o4x - octo".

• H.S.M. Coxeter:
  • Coxeter,
    ISBN 0-486-61480-8
    , p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
    ISBN 978-0-471-01003-6 [1]
    • (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

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
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	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: List of regular polytopes and compounds