Rectified 9-cubes

Orthogonal projections in BC₉ Coxeter plane
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

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

There are 9 rectifications of the 9-cube. The zeroth is the 9-cube itself, and the 8th is the dual 9-orthoplex. Vertices of the rectified 9-cube are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 9-cube are located in the square face centers of the 9-cube. Vertices of the trirectified 9-orthoplex are located in the cube cell centers of the 9-cube. Vertices of the quadrirectified 9-cube are located in the tesseract centers of the 9-cube.

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

Rectified 9-cube

Alternate names

Rectified enneract (Acronym ren) (Jonathan Bowers)^[1]

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]

Birectified 9-cube

Alternate names

Birectified enneract (Acronym barn) (Jonathan Bowers)^[2]

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]

Trirectified 9-cube

Alternate names

Trirectified enneract (Acronym tarn) (Jonathan Bowers)^[3]

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]

Quadrirectified 9-cube

Alternate names

Quadrirectified enneract (Acronym nav) (Jonathan Bowers)^[4]

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]

Notes

^ Klitzing (o3o3o3o3o3o3o3x4o - ren)
^ Klitzing (o3o3o3o3o3o3x3o4o - barn)
^ Klitzing (o3o3o3o3o3x3o3o4o - tarn)
^ Klitzing (o3o3o3o3x3o3o3o4o - nav)

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 [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 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

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	A_n	B_n	I₂(p) / D_n	E₆ / E₇ / E₈ / F₄ / G₂	H_n
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	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	1_k2 • 2_k1 • k₂₁	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds

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