Rectified 10-cubes

In ten-dimensional geometry, a rectified 10-cube is a convex uniform 10-polytope, being a rectification of the regular 10-cube.

10-orthoplex	Rectified 10-orthoplex	Birectified 10-orthoplex	Trirectified 10-orthoplex
Quadirectified 10-orthoplex	Quadrirectified 10-cube	Trirectified 10-cube	Birectified 10-cube
Rectified 10-cube	10-cube
Orthogonal projections in BC10 Coxeter plane

There are 10 rectifications of the 10-cube, with the zeroth being the 10-cube itself. Vertices of the rectified 10-cube are located at the edge-centers of the 10-cube. Vertices of the birectified 10-cube are located in the square face centers of the 10-cube. Vertices of the trirectified 10-cube are located in the cubic cell centers of the 10-cube. The others are more simply constructed relative to the 10-cube dual polytope, the 10-orthoplex.

These polytopes are part of a family 1023 uniform 10-polytopes with BC₁₀ symmetry.

Rectified 10-cube

Rectified 10-orthoplex
Type	uniform 10-polytope
Schläfli symbol	t1{38,4}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	46080
Vertices	5120
Vertex figure	8-simplex prism
Coxeter groups	C10, [4,38] D10, [37,1,1]
Properties	convex

Alternate names

• Rectified dekeract (Acronym rade) (Jonathan Bowers)^[1]

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified 10-cube, centered at the origin, edge length ${\displaystyle {\sqrt {2}}}$ are all permutations of:

(±1,±1,±1,±1,±1,±1,±1,±1,±1,0)

Images

Orthographic projections

B10	B9	B8
[20]	[18]	[16]
B7	B6	B5
[14]	[12]	[10]
B4	B3	B2
[8]	[6]	[4]
A9		A5
—		—
[10]		[6]
A7		A3
—		—
[8]		[4]

Birectified 10-cube

Birectified 10-orthoplex
Type	uniform 10-polytope
Coxeter symbol	0711
Schläfli symbol	t2{38,4}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	184320
Vertices	11520
Vertex figure	{4}x{36}
Coxeter groups	C10, [4,38] D10, [37,1,1]
Properties	convex

Alternate names

• Birectified dekeract (Acronym brade) (Jonathan Bowers)^[2]

Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 10-cube, centered at the origin, edge length ${\displaystyle {\sqrt {2}}}$ are all permutations of:

(±1,±1,±1,±1,±1,±1,±1,±1,0,0)

Images

Orthographic projections

B10	B9	B8
[20]	[18]	[16]
B7	B6	B5
[14]	[12]	[10]
B4	B3	B2
[8]	[6]	[4]
A9		A5
—		—
[10]		[6]
A7		A3
—		—
[8]		[4]

Trirectified 10-cube

Trirectified 10-orthoplex
Type	uniform 10-polytope
Schläfli symbol	t3{38,4}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	322560
Vertices	15360
Vertex figure	{4,3}x{35}
Coxeter groups	C10, [4,38] D10, [37,1,1]
Properties	convex

Alternate names

• Tririrectified dekeract (Acronym trade) (Jonathan Bowers)^[3]

Cartesian coordinates

Cartesian coordinates for the vertices of a triirectified 10-cube, centered at the origin, edge length ${\displaystyle {\sqrt {2}}}$ are all permutations of:

(±1,±1,±1,±1,±1,±1,±1,0,0,0)

Images

Orthographic projections

B10	B9	B8
[20]	[18]	[16]
B7	B6	B5
[14]	[12]	[10]
B4	B3	B2
[8]	[6]	[4]
A9		A5
—		—
[10]		[6]
A7		A3
—		—
[8]		[4]

Quadrirectified 10-cube

Quadrirectified 10-orthoplex
Type	uniform 10-polytope
Schläfli symbol	t4{38,4}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	322560
Vertices	13440
Vertex figure	{4,3,3}x{34}
Coxeter groups	C10, [4,38] D10, [37,1,1]
Properties	convex

Alternate names

• Quadrirectified dekeract
• Quadrirectified decacross (Acronym terade) (Jonathan Bowers)^[4]

Cartesian coordinates

Cartesian coordinates for the vertices of a quadrirectified 10-cube, centered at the origin, edge length ${\displaystyle {\sqrt {2}}}$ are all permutations of:

(±1,±1,±1,±1,±1,±1,0,0,0,0)

Images

Orthographic projections

B10	B9	B8
[20]	[18]	[16]
B7	B6	B5
[14]	[12]	[10]
B4	B3	B2
[8]	[6]	[4]
A9		A5
—		—
[10]		[6]
A7		A3
—		—
[8]		[4]

Notes

1. Klitzing, (o3o3o3o3o3o3o3o3x4o - rade)
2. Klitzing, (o3o3o3o3o3o3o3x3o4o - brade)
3. Klitzing, (o3o3o3o3o3o3x3o3o4o - trade)
4. Klitzing, (o3o3o3o3o3x3o3o3o4o - terade)

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. "10D uniform polytopes (polyxenna)". x3o3o3o3o3o3o3o3o4o - ka, o3x3o3o3o3o3o3o3o4o - rake, o3o3x3o3o3o3o3o3o4o - brake, o3o3o3x3o3o3o3o3o4o - trake, o3o3o3o3x3o3o3o3o4o - terake, o3o3o3o3o3x3o3o3o4o - terade, o3o3o3o3o3o3x3o3o4o - trade, o3o3o3o3o3o3o3x3o4o - brade, o3o3o3o3o3o3o3o3x4o - rade, o3o3o3o3o3o3o3o3o4x - deker

External links

• Polytopes of Various Dimensions
• Multi-dimensional Glossary

v t e 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 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	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds