In ten-dimensional geometry, a rectified 10-orthoplex is a convex

uniform 10-polytope In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge. A uniform 10-polytope is one which is vertex-transitive, and construct ...

, being a

rectification Rectification has the following technical meanings: Mathematics * Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points * Rectifiable curve, in mathematics * Recti ...

of the regular

10-orthoplex In geometry, a 10-orthoplex or 10-cross polytope, is a regular 10-polytope with 20 vertices, 180 edges, 960 triangle faces, 3360 octahedron cells, 8064 5-cells ''4-faces'', 13440 ''5-faces'', 15360 ''6-faces'', 11520 ''7-faces'', 5120 ''8-faces'', ...

. 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

s with BC₁₀ symmetry.

Rectified 10-orthoplex

In ten-dimensional geometry, a rectified 10-orthoplex is a

10-polytope In ten-dimensional geometry, a 10-polytope is a 10-dimensional polytope whose boundary consists of 9-polytope facets, exactly two such facets meeting at each 8-polytope ridge. A uniform 10-polytope is one which is vertex-transitive, and construct ...

, being a

of the regular

Rectified 10-orthoplex

The ''rectified 10-orthoplex'' is the vertex figure for the

demidekeractic honeycomb In geometry, the alternated hypercube honeycomb (or demicubic honeycomb) is a dimensional infinite series of honeycombs, based on the hypercube honeycomb with an alternation operation. It is given a Schläfli symbol h representing the regular fo ...

. : or

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 A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

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

hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyper ...

s, 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

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

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

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 Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

: ** 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