Rectified 9-orthoplexes

9-orthoplex	Rectified 9-orthoplex	Birectified 9-orthoplex	Trirectified 9-orthoplex	Quadrirectified 9-cube
Trirectified 9-cube	Birectified 9-cube	Rectified 9-cube	9-cube
Orthogonal projections in A9 Coxeter plane

In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-orthoplex.

There are 9 rectifications of the 9-orthoplex. Vertices of the rectified 9-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 9-orthoplex are located in the triangular face centers of the 9-orthoplex. Vertices of the trirectified 9-orthoplex are located in the tetrahedral cell centers of the 9-orthoplex.

These polytopes are part of a family 511 uniform 9-polytopes with BC₉ symmetry.

Rectified 9-orthoplex

Rectified 9-orthoplex
Type	uniform 9-polytope
Schläfli symbol	t1{3^7,4}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	2016
Vertices	144
Vertex figure	7-orthoplex prism
Petrie polygon	octakaidecagon
Coxeter groups	C9, [4,3^7] D9, [3^6,1,1]
Properties	convex

The rectified 9-orthoplex is the vertex figure for the demienneractic honeycomb.

or

Rectified 9-orthoplex

Alternate names

• rectified enneacross (Acronym riv) (Jonathan Bowers)^[1]

Construction

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

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified 9-orthoplex, centered at the origin, edge length ${\sqrt {2}}$ are all permutations of:

(±1,±1,0,0,0,0,0,0,0)

Root vectors

Its 144 vertices represent the root vectors of the simple Lie group D₉. The vertices can be seen in 3 hyperplanes, with the 36 vertices rectified 8-simplexs cells on opposite sides, and 72 vertices of an expanded 8-simplex passing through the center. When combined with the 18 vertices of the 9-orthoplex, these vertices represent the 162 root vectors of the B₉ and C₉ simple Lie groups.

Images

orthographic projections

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

Birectified 9-orthoplex

Alternate names

• Rectified 9-demicube
• Birectified enneacross (Acronym brav) (Jonathan Bowers)^[2]

Images

orthographic projections

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

Trirectified 9-orthoplex

Alternate names

• trirectified enneacross (Acronym tarv) (Jonathan Bowers)^[3]

Images

orthographic projections

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

Notes

1. Klitzing (o3x3o3o3o3o3o3o4o - riv)
2. Klitzing (o3o3x3o3o3o3o3o4o - brav)
3. Klitzing (o3o3o3x3o3o3o3o4o - tarv)

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. (1966)
• Klitzing, Richard. "9D uniform polytopes (polyyotta)". x3o3o3o3o3o3o3o4o - vee, o3x3o3o3o3o3o3o4o - riv, o3o3x3o3o3o3o3o4o - brav, o3o3o3x3o3o3o3o4o - tarv, o3o3o3o3x3o3o3o4o - nav, o3o3o3o3o3x3o3o4o - tarn, o3o3o3o3o3o3x3o4o - barn, o3o3o3o3o3o3o3x4o - ren, o3o3o3o3o3o3o3o4x - enne

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 / E9 / E10 / 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