Rectified 9-orthoplexes

In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-orthoplex.

9-orthoplex	Rectified 9-orthoplex	Birectified 9-orthoplex	Trirectified 9-orthoplex	Quadrirectified 9-cube
Trirectified 9-cube	Birectified 9-cube	Rectified 9-cube	9-cube
Orthogonal projections in A9 Coxeter plane

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 cell centers of the 9-orthoplex.

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

Rectified 9-orthoplex

Rectified 9-orthoplex
Type	uniform 9-polytope
Schläfli symbol	t1{37,4}
Coxeter-Dynkin diagrams
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	2016
Vertices	144
Vertex figure	7-orthoplex prism
Petrie polygon	octakaidecagon
Coxeter groups	C9, [4,37] D9, [36,1,1]
Properties	convex

The rectified 9-orthoplex is the vertex figure for the demienneractic honeycomb.

or

Alternate names

• rectified enneacross (Acronym riv) (Jonathan Bowers)^[1]

Construction

There are two Coxeter groups associated with the rectified 9-orthoplex, one with the C₉ or [4,3⁷] Coxeter group, and a lower symmetry with two copies of 8-orthoplex 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 ${\displaystyle {\sqrt {2}}}$ 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 D₉. The vertices can be seen in 3 hyperplanes, with the 36 vertices rectified 8-simplexs cells on opposite sides, and 72 vertices of an expanded 8-simplex 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

orthographic projections

B9		B8		B7
[18]		[16]		[14]
B6			B5
[12]			[10]
B4		B3		B2
[8]		[6]		[4]
A7		A5		A3
—		—		—
[8]		[6]		[4]

Birectified 9-orthoplex

Alternate names

• Rectified 9-demicube
• Birectified enneacross (Acronym brav) (Jonathan Bowers)^[2]

Images

orthographic projections

B9		B8		B7
[18]		[16]		[14]
B6			B5
[12]			[10]
B4		B3		B2
[8]		[6]		[4]
A7		A5		A3
—		—		—
[8]		[6]		[4]

Trirectified 9-orthoplex

Alternate names

• trirectified enneacross (Acronym tarv) (Jonathan Bowers)^[3]

Images

orthographic projections

B9		B8		B7
[18]		[16]		[14]
B6			B5
[12]			[10]
B4		B3		B2
[8]		[6]		[4]
A7		A5		A3
—		—		—
[8]		[6]		[4]