8-orthoplex

8-orthoplex Octacross
Orthogonal projection inside Petrie polygon
Type	Regular 8-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

In geometry, an 8-orthoplex or 8-cross polytope is a regular 8-polytope with 16 vertices, 112 edges, 448 triangle faces, 1120 tetrahedron cells, 1792 5-cells 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces.

It has two constructive forms, the first being regular with Schläfli symbol {3⁶,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3,3,3,3,3,3^1,1} or Coxeter symbol 5₁₁.

It is a part of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is an 8-hypercube, or octeract.

Alternate names

• Octacross, derived from combining the family name cross polytope with oct for eight (dimensions) in Greek
• Diacosipentacontahexazetton as a 256-facetted 8-polytope (polyzetton)

As a configuration

The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element. The configurations for dual polytopes can be seen by rotating the matrix elements by 180 degrees.^[1][2]

The diagonal f-vector numbers are derived through the Wythoff 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

Construction

There are two Coxeter groups associated with the 8-cube, one regular, dual of the octeract 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 an 8-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

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)

Every vertex pair is connected by an edge, except opposites.

Images

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 form 5₁₁ with the 8-simplex to form the 5₂₁ honeycomb.

References

• 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
    • (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.
• Klitzing, Richard. "8D uniform polytopes (polyzetta) x3o3o3o3o3o3o4o - ek".

External links

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