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)
• Richard Klitzing, 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 at Glossary for Hyperspace.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary