In nine-dimensional

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, a rectified 9-simplex is a convex uniform 9-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

9-orthoplex In geometry, a 9-orthoplex or 9-cross polytope, is a regular 9-polytope with 18 vertices, 144 edges, 672 triangle faces, 2016 tetrahedron cells, 4032 5-cells ''4-faces'', 5376 5-simplex ''5-faces'', 4608 6-simplex ''6-faces'', 2304 7-simplex '' ...

. There are 9 rectifications of the 9-orthoplex. Vertices of the rectified 9-orthoplex are located at the edge-centers of the 9-orthoplex. Vertices of the birectified 9-orthoplex are located in the triangular face centers of the 9-orthoplex. Vertices of the trirectified 9-orthoplex are located in the

tetrahedral In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ...

cell centers of the 9-orthoplex. These polytopes are part of a family 511 uniform 9-polytopes with BC₉ symmetry.

Rectified 9-orthoplex

The ''rectified 9-orthoplex'' is the

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw lines ...

for the

demienneractic 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 enneacross (Acronym riv) (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 9-orthoplex'', one with the C₉ or ,3⁷Coxeter group, and a lower symmetry with two copies of 8-orthoplex facets, alternating, with the D₉ or ^6,1,1Coxeter 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 i ...

for the vertices of a rectified 9-orthoplex, centered at the origin, edge length

\sqrt

are all permutations of: : (±1,±1,0,0,0,0,0,0,0)

Root vectors

Its 144 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 symme ...

D₉. 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 hype ...

s, with the 36 vertices

rectified 8-simplex In eight-dimensional geometry, a rectified 8-simplex is a convex uniform 8-polytope, being a rectification of the regular 8-simplex. There are unique 3 degrees of rectifications in regular 8-polytopes. Vertices of the rectified 8-simplex are locat ...

s cells on opposite sides, and 72 vertices of an

expanded 8-simplex In eight-dimensional geometry, a heptellated 8-simplex is a convex uniform 8-polytope, including 7th-order truncations (heptellation) from the regular 8-simplex. There are 35 unique heptellations for the 8-simplex, including all permutations of ...

passing through the center. When combined with the 18 vertices of the 9-orthoplex, these vertices represent the 162 root vectors of the B₉ and C₉ simple Lie groups.

Images

Birectified 9-orthoplex

Alternate names

* Rectified 9-demicube * Birectified enneacross (Acronym brav) (Jonathan Bowers)

Images

Trirectified 9-orthoplex

Alternate names

* trirectified enneacross (Acronym tarv) (Jonathan Bowers)Klitzing (o3o3o3x3o3o3o3o4o - tarv)

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) * x3o3o3o3o3o3o3o4o - vee, o3x3o3o3o3o3o3o4o - riv, o3o3x3o3o3o3o3o4o - brav, o3o3o3x3o3o3o3o4o - tarv, o3o3o3o3x3o3o3o4o - nav, o3o3o3o3o3x3o3o4o - tarn, o3o3o3o3o3o3x3o4o - barn, o3o3o3o3o3o3o3x4o - ren, o3o3o3o3o3o3o3o4x - enne

External links

Polytopes of Various Dimensions

{{Polytopes 9-polytopes