9-orthoplex

In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells 4-faces, 5376 5-simplex 5-faces, 4608 6-simplex 6-faces, 2304 7-simplex 7-faces, and 512 8-simplex 8-faces.

Regular 9-orthoplex Ennecross
Orthogonal projection inside Petrie polygon
Type	Regular 9-polytope
Family	orthoplex
Schläfli symbol	{3^7,4} {3^6,3^1,1}
Coxeter-Dynkin diagrams
8-faces	512 {3^7}
7-faces	2304 {3^6}
6-faces	4608 {3^5}
5-faces	5376 {3^4}
4-faces	4032 {3^3}
Cells	2016 {3,3}
Faces	672 {3}
Edges	144
Vertices	18
Vertex figure	Octacross
Petrie polygon	Octadecagon
Coxeter groups	C9, [3^7,4] D9, [3^6,1,1]
Dual	9-cube
Properties	convex

It has two constructed forms, the first being regular with Schläfli symbol {3⁷,4}, and the second with alternately labeled (checkerboarded) facets, with Schläfli symbol {3⁶,3^1,1} or Coxeter symbol 6₁₁.

It is one of an infinite family of polytopes, called cross-polytopes or orthoplexes. The dual polytope is the 9-hypercube or enneract.

Alternate names

• Enneacross, derived from combining the family name cross polytope with ennea for nine (dimensions) in Greek
• Pentacosidodecayotton as a 512-facetted 9-polytope (polyyotton)

Construction

There are two Coxeter groups associated with the 9-orthoplex, one regular, dual of the enneract with the C₉ or [4,3⁷] symmetry group, and a lower symmetry with two copies of 8-simplex facets, alternating, with the D₉ or [3^6,1,1] symmetry group.

Cartesian coordinates

Cartesian coordinates for the vertices of a 9-orthoplex, centered at the origin, are

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

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

Images

orthographic projections

B9		B8		B7
[18]		[16]		[14]
B6			B5
[12]			[10]
B4		B3		B2
[8]		[6]		[4]
A7		A5		A3
—		—		—
[8]		[6]		[4]

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. "9D uniform polytopes (polyyotta) x3o3o3o3o3o3o3o4o - vee".

External links

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

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 4-polytope	5-cell	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: Polytope families • Regular polytope • List of regular polytopes and compounds