# Rectified 10-cubes

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 Orthogonal projections in BC10 Coxeter plane 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

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

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 polytpoe, 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]

## 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]

## 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]

## 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 trade) (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]

## 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 [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)
• 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