In ten-dimensional

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a rectified 10-orthoplex is a convex uniform 10-polytope, 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 tetrahedron cells, 8064 5-cell ''4-faces'', 13440 ''5-faces'', 15360 ''6-faces'', 11520 ''7-faces'', 5120 ''8-face ...

. 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 In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

cell centers of the 10-orthoplex. These polytopes are part of a family of 1023 uniform 10-polytopes with BC₁₀ symmetry.

Rectified 10-orthoplex

In ten-dimensional

, 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 Facet (mathematics), facets, exactly two such facets meeting at each 8-polytope Ridge (geometry), ridge. A uniform 10-polytope is one wh ...

, being a

of the regular

. The rectified 10-orthoplex is the

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

of the demidekeractic honeycomb. : or

Alternate names

* Rectified decacross (Acronym: rake) (Jonathan Bowers)

Construction

There are two

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean ref ...

s 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 In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

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 In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...

D₁₀. The vertices can be seen in 3

hyperplane In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is ...

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 In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ...

B₁₀.

Images

Birectified 10-orthoplex

Alternate names

* Birectified decacross (Acronym: brake) (Jonathan Bowers)

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: terake) (Jonthan Bowers)Klitzing, (o3o3o3o3x3o3o3o3o4o - terake)

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-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

: ** 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
wiley.com
*** (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