8-orthoplex Octacross
Orthogonal projection insidePetrie polygon
Type	Regular8-polytope
Family	orthoplex
Schläfli symbol	{36,4} {3,3,3,3,3,31,1}
Coxeter-Dynkin diagrams
7-faces	256{36}
6-faces	1024{35}
5-faces	1792{34}
4-faces	1792{33}
Cells	1120{3,3}
Faces	448{3}
Edges	112
Vertices	16
Vertex figure	7-orthoplex
Petrie polygon	hexadecagon
Coxeter groups	C8, [36,4] D8, [35,1,1]
Dual	8-cube
Properties	convex,Hanner polytope

Ingeometry, an8-orthoplex or 8-cross polytope is a regular8-polytope with 16vertices, 112edges, 448 trianglefaces, 1120 tetrahedroncells, 17925-cells4-faces, 17925-faces, 10246-faces, and 2567-faces.

It has two constructive forms, the first being regular withSchläfli symbol {3⁶,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,3,3^1,1} orCoxeter symbol5₁₁.

It is a part of an infinite family of polytopes, calledcross-polytopes ororthoplexes. Thedual polytope is an 8-hypercube, orocteract.

• Octacross, derived from combining the family namecross polytope withoct for eight (dimensions) inGreek
• Diacosipentacontahexazetton as a 256-facetted8-polytope (polyzetton)

Thisconfiguration matrix represents the 8-orthoplex. 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-orthoplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.^[1][2]

The diagonalf-vector numbers are derived through theWythoff construction, dividing the full group order of a subgroup order by removing individual mirrors.^[3]

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

There are twoCoxeter groups associated with the 8-cube, oneregular,dual of theocteract with the C₈ or [4,3,3,3,3,3,3] symmetry group, and a half symmetry with two copies of 7-simplex facets, alternating, with the D₈ or [3^5,1,1] symmetry group. A lowest symmetry construction is based on a dual of an 8-orthotope, called an8-fusil.

Name	Coxeter diagram	Schläfli symbol	Symmetry	Order	Vertex figure
regular 8-orthoplex		{3,3,3,3,3,3,4}	[3,3,3,3,3,3,4]	10321920
Quasiregular 8-orthoplex		{3,3,3,3,3,31,1}	[3,3,3,3,3,31,1]	5160960
8-fusil		8{}	[27]	256

Cartesian coordinates for the vertices of an 8-cube, centered at the origin are

(±1,0,0,0,0,0,0,0), (0,±1,0,0,0,0,0,0), (0,0,±1,0,0,0,0,0), (0,0,0,±1,0,0,0,0),

(0,0,0,0,±1,0,0,0), (0,0,0,0,0,±1,0,0), (0,0,0,0,0,0,0,±1), (0,0,0,0,0,0,0,±1)

Everyvertex pair is connected by anedge, except opposites.

orthographic projections

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

It is used in its alternated form5₁₁ with the8-simplex to form the5₂₁ honeycomb.

• H.S.M. Coxeter:
  • H.S.M. Coxeter,Regular Polytopes, 3rd Edition, Dover New York, 1973
  • 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 JohnsonUniform Polytopes, Manuscript (1991)
  • N.W. Johnson:The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard."8D uniform polytopes (polyzetta) x3o3o3o3o3o3o4o - ek".

• Olshevsky, George."Cross polytope".Glossary for Hyperspace. Archived fromthe original on 4 February 2007.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary

• 8-polytopes

• Articles with short description
• Short description matches Wikidata