Rectified 10-orthoplexes

In ten-dimensional geometry, a rectified 10-orthoplex is a convex uniform 10-polytope, being a rectification of the regular 10-orthoplex.

10-orthoplex	Rectified 10-orthoplex	Birectified 10-orthoplex	Trirectified 10-orthoplex
Quadrirectified 10-orthoplex	Quadrirectified 10-cube	Trirectified 10-cube	Birectified 10-cube
Rectified 10-cube	10-cube
Orthogonal projections in A10 Coxeter plane

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 of 1023 uniform 10-polytopes with BC₁₀ symmetry.

Rectified 10-orthoplex

Rectified 10-orthoplex
Type	uniform 10-polytope
Schläfli symbol	t1{38,4}
Coxeter-Dynkin diagrams
9-faces
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges	2880
Vertices	180
Vertex figure	8-orthoplex prism
Petrie polygon	icosagon
Coxeter groups	C10, [4,38] D10, [37,1,1]
Properties	convex

In ten-dimensional geometry, a rectified 10-orthoplex is a 10-polytope, being a rectification of the regular 10-orthoplex.

The rectified 10-orthoplex is the vertex figure of the demidekeractic honeycomb.

or

Alternate names

• Rectified decacross (Acronym: rake) (Jonathan Bowers)^[1]

Construction

There are two Coxeter groups associated with the rectified 10-orthoplex, one with the C₁₀ or [4,3⁸] Coxeter group, and a lower symmetry with two copies of 9-orthoplex facets, alternating, with the D₁₀ or [3^7,1,1] Coxeter group.

Cartesian coordinates

Cartesian coordinates for the vertices of a rectified 10-orthoplex, centered at the origin, edge length ${\displaystyle {\sqrt {2}}}$ 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 hyperplanes, 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

Orthographic projections

B10	B9	B8
[20]	[18]	[16]
B7	B6	B5
[14]	[12]	[10]
B4	B3	B2
[8]	[6]	[4]
A9		A5
—		—
[10]		[6]
A7		A3
—		—
[8]		[4]

Birectified 10-orthoplex

Birectified 10-orthoplex
Type	uniform 10-polytope
Schläfli symbol	t2{38,4}
Coxeter-Dynkin diagrams
9-faces
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	C10, [4,38] D10, [37,1,1]
Properties	convex

Alternate names

• Birectified decacross (Acronym: brake) (Jonathan Bowers)^[2]

Cartesian coordinates

Cartesian coordinates for the vertices of a birectified 10-orthoplex, centered at the origin, edge length ${\displaystyle {\sqrt {2}}}$ are all permutations of:

(±1,±1,±1,0,0,0,0,0,0,0)

Images

Orthographic projections

B10	B9	B8
[20]	[18]	[16]
B7	B6	B5
[14]	[12]	[10]
B4	B3	B2
[8]	[6]	[4]
A9		A5
—		—
[10]		[6]
A7		A3
—		—
[8]		[4]

Trirectified 10-orthoplex

Trirectified 10-orthoplex
Type	uniform 10-polytope
Schläfli symbol	t3{38,4}
Coxeter-Dynkin diagrams
9-faces
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	C10, [4,38] D10, [37,1,1]
Properties	convex

Alternate names

• Trirectified decacross (Acronym: trake) (Jonathan Bowers)^[3]

Cartesian coordinates

Cartesian coordinates for the vertices of a trirectified 10-orthoplex, centered at the origin, edge length ${\displaystyle {\sqrt {2}}}$ are all permutations of:

(±1,±1,±1,±1,0,0,0,0,0,0)

Images

Orthographic projections

B10	B9	B8
[20]	[18]	[16]
B7	B6	B5
[14]	[12]	[10]
B4	B3	B2
[8]	[6]	[4]
A9		A5
—		—
[10]		[6]
A7		A3
—		—
[8]		[4]

Quadrirectified 10-orthoplex

Quadrirectified 10-orthoplex
Type	uniform 10-polytope
Schläfli symbol	t4{38,4}
Coxeter-Dynkin diagrams
9-faces
8-faces
7-faces
6-faces
5-faces
4-faces
Cells
Faces
Edges
Vertices
Vertex figure
Coxeter groups	C10, [4,38] D10, [37,1,1]
Properties	convex

Alternate names

• Quadrirectified decacross (Acronym: terake) (Jonthan Bowers)^[4]

Cartesian coordinates

Cartesian coordinates for the vertices of a quadrirectified 10-orthoplex, centered at the origin, edge length ${\displaystyle {\sqrt {2}}}$ are all permutations of:

(±1,±1,±1,±1,±1,0,0,0,0,0)

Images

Orthographic projections

B10	B9	B8
[20]	[18]	[16]
B7	B6	B5
[14]	[12]	[10]
B4	B3	B2
[8]	[6]	[4]
A9		A5
—		—
[10]		[6]
A7		A3
—		—
[8]		[4]