Rectified Decacross

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

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.
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Alternate names

* rectified decacross (Acronym rake) (Jonathan Bowers)

Construction

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

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified 10-orthoplex, centered at the origin, edge length $\sqrt$ 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

Birectified 10-orthoplex

Alternate names

* Birectified decacross

Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 10-orthoplex, centered at the origin, edge length $\sqrt$ are all permutations of:
: (±1,±1,±1,0,0,0,0,0,0,0)

Images

Trirectified 10-orthoplex

Alternate names

* Trirectified decacross (Acronym trake) (Jonathan Bowers)

Cartesian coordinates

Cartesian coordinates for the vertices of a trirectified 10-orthoplex, centered at the origin, edge length $\sqrt$ are all permutations of:
: (±1,±1,±1,±1,0,0,0,0,0,0)

Images

Quadrirectified 10-orthoplex

Alternate names

* Quadrirectified decacross (Acronym brake) (Jonthan Bowers)Klitzing, (o3o3x3o3o3o3o3o3o4o - brake)

Cartesian coordinates

Cartesian coordinates for the vertices of a quadrirectified 10-orthoplex, centered at the origin, edge length $\sqrt$ are all permutations of:
: (±1,±1,±1,±1,±1,0,0,0,0,0)

Images

Notes

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,

*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', ath. Zeit. 46 (1940) 380-407, MR 2,10*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', ath. Zeit. 188 (1985) 559-591*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45* Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D. (1966)
* 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



{{Polytopes

10-polytopes