8-orthoplex

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-cell 4-faces, 1792 5-faces, 1024 6-faces, and 256 7-faces.

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, Hanner polytope

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), acronym: ek^[1]

As a configuration

This configuration 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.^[2][3]

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

The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing individual mirrors.^[1]

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.