Rectified 10-orthoplexes

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 A10 Coxeter plane

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

There are 10 rectifications of the 10-orthoplex. Vertices of the rectified 10-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 10-orthoplex are located in the triangular face centers of the 10-orthoplex. Vertices of the trirectified 10-orthoplex are located in the tetrahedral cell centers of the 10-orthoplex.

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

Rectified 10-orthoplex

Rectified 10-orthoplex
Type	uniform 10-polytope
Schläfli symbol	t1{3^8,4}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	2880
Vertices	180
Vertex figure	8-orthoplex prism
Petrie polygon	icosagon
Coxeter groups	C10, [4,3^8] D10, [3^7,1,1]
Properties	convex

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

Rectified 10-orthoplex

The rectified 10-orthoplex is the vertex figure for the demidekeractic honeycomb.

or

Alternate names

• rectified decacross (Acronym rake) (Jonathan Bowers)^[1]

Construction

There are two Coxeter groups associated with the rectified 10-orthoplex, one with the C₁₀ or [4,3⁸] Coxeter group, and a lower symmetry with two copies of 9-orthoplex facets, alternating, with the D₁₀ or [3^7,1,1] Coxeter group.

Cartesian coordinates

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

(±1,±1,0,0,0,0,0,0,0,0)

Root vectors

Its 180 vertices represent the root vectors of the simple Lie group D₁₀. The vertices can be seen in 3 hyperplanes, with the 45 vertices rectified 9-simplices facets on opposite sides, and 90 vertices of an expanded 9-simplex passing through the center. When combined with the 20 vertices of the 9-orthoplex, these vertices represent the 200 root vectors of the simple Lie group B₁₀.

Images

Orthographic projections

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

Birectified 10-orthoplex

Birectified 10-orthoplex
Type	uniform 10-polytope
Schläfli symbol	t2{3^8,4}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	C10, [4,3^8] D10, [3^7,1,1]
Properties	convex

Alternate names

• Birectified decacross

Cartesian coordinates

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

(±1,±1,±1,0,0,0,0,0,0,0)

Images

Orthographic projections

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

Trirectified 10-orthoplex

Trirectified 10-orthoplex
Type	uniform 10-polytope
Schläfli symbol	t3{3^8,4}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	C10, [4,3^8] D10, [3^7,1,1]
Properties	convex

Alternate names

• Trirectified decacross (Acronym trake) (Jonathan Bowers)^[2]

Cartesian coordinates

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

(±1,±1,±1,±1,0,0,0,0,0,0)

Images

Orthographic projections

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

Quadrirectified 10-orthoplex

Quadrirectified 10-orthoplex
Type	uniform 10-polytope
Schläfli symbol	t4{3^8,4}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	C10, [4,3^8] D10, [3^7,1,1]
Properties	convex

Alternate names

• Quadrirectified decacross (Acronym brake) (Jonthan Bowers)^[3]

Cartesian coordinates

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

(±1,±1,±1,±1,±1,0,0,0,0,0)

Images

Orthographic projections

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

Notes

1. Klitzing, (o3x3o3o3o3o3o3o3o4o - rake)
2. Klitzing, (o3o3o3x3o3o3o3o3o4o - trake)
3. Klitzing, (o3o3x3o3o3o3o3o3o4o - brake)

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

• 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