Rectified 9-orthoplex

Orthogonal projections in A₉ Coxeter plane
9-orthoplex	Rectified 9-orthoplex	Birectified 9-orthoplex	Trirectified 9-orthoplex	Quadrirectified 9-cube
Trirectified 9-cube	Birectified 9-cube	Rectified 9-cube Rectified 9-cube In nine-dimensional geometry, a rectified 9-cube is a convex uniform 9-polytope, being a rectification of the regular 9-cube.There are 9 rectifications of the 9-cube. The zeroth is the 9-cube itself, and the 8th is the dual 9-orthoplex. Vertices of the rectified 9-cube are located at the...	9-cube

In nine-dimensional geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification

Rectification (geometry)

In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points...

of the regular 9-orthoplex.

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

Tetrahedron

In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

cell centers of the 9-orthoplex.

These polytopes are part of a family 511 uniform 9-polytopes with BC₉ symmetry.

Rectified 9-orthoplex
Type	uniform 9-polytope
Schläfli symbol	t₁{3,3,3,3,3,3,3,4}
Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... s
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	2016
Vertices	144
Vertex figure Vertex figure In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...	7-orthoplex prism
Petrie polygon Petrie polygon In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon such that every consecutive sides belong to one of the facets...	octakaidecagon
Coxeter group 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 mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example... s	C₉, [3,3,3,3,3,3,3,4] D₉, [3^6,1,1]
Properties	convex Convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set of points in the n-dimensional space Rn...

The rectified 9-orthoplex is the vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

for the demienneractic honeycomb.

Construction

There are two Coxeter group

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 mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...

s associated with the rectified 9-orthoplex, one with the C₉ or [4,3,3,3,3,3,3,3] Coxeter group, and a lower symmetry with two copies of heptcross facets, alternating, with the D₉ or [3^6,1,1] Coxeter group.

Cartesian coordinates

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

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

Simple Lie group

In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself...

D₉. When combined with the 18 vertices of the 9-orthoplex, these vertices represent the 162 root vectors of the simple Lie group B₉.

Alternate names

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

External links

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