User:Tomruen/Stericated 5-simplexes

Orthogonal projections in A₅ and A₄ Coxeter planes
5-simplex	Stericated 5-simplex
Steritruncated 5-simplex	Stericantellated 5-simplex
Stericantitruncated 5-simplex	Steriruncitruncated 5-simplex
Steriruncicantitruncated 5-simplex (Omnitruncated 5-simplex)

In five-dimensional geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-simplex.

There are six unique sterications of the 5-simplex, including permutations of truncations, cantellations, and runcinations. The simplest stericated 5-simplex is also called an expanded 5-simplex, with the first and last nodes ringed, for being constructible by an expansion operation applied to the regular 5-simplex. The highest form, the steriruncicantitruncated 5-simplex is more simply called an omnitruncated 5-simplex with all of the nodes ringed.

Stericated 5-simplex

Stericated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	2r2r{3,3,3,3}
Coxeter-Dynkin diagram	or
4-faces	62	6 6 {3,3,3} 15 15 {}×{3,3} 20 {3}×{3}
Cells	180	60 {3,3} 120 {}×{3}
Faces	210	120 {3} 90 {4}
Edges	120
Vertices	30
Vertex figure	Tetrahedral antiprism
Coxeter group	A₅×2, [[3,3,3,3]], order 1440
Properties	convex, isogonal, isotoxal

A stericated 5-simplex can be constructed by an expansion operation applied to the regular 5-simplex, and thus is also sometimes called an expanded 5-simplex. It has 30 vertices, 120 edges, 210 faces (120 triangles and 90 squares), 180 cells (60 tetrahedra and 120 triangular prisms) and 62 4-faces (12 5-cells, 30 tetrahedral prisms and 20 3-3 duoprisms).

Alternate names

Expanded 5-simplex
Stericated hexateron
Small cellated dodecateron (Acronym: scad) (Jonathan Bowers)^[1]

Cross-sections

The maximal cross-section of the stericated hexateron with a 4-dimensional hyperplane is a runcinated 5-cell. This cross-section divides the stericated hexateron into two pentachoral hypercupolas consisting of 6 5-cells, 15 tetrahedral prisms and 10 3-3 duoprisms each.

Coordinates

The vertices of the stericated 5-simplex can be constructed on a hyperplane in 6-space as permutations of (0,1,1,1,1,2). This represents the positive orthant facet of the stericated 6-orthoplex.

A second construction in 6-space, from the center of a rectified 6-orthoplex is given by coordinate permutations of:

(1,-1,0,0,0,0)

The Cartesian coordinates in 5-space for the normalized vertices of an origin-centered stericated hexateron are:

\left(\pm 1,\ 0,\ 0,\ 0,\ 0\right)

\left(0,\ \pm 1,\ 0,\ 0,\ 0\right)

\left(0,\ 0,\ \pm 1,\ 0,\ 0\right)

\left(\pm 1/2,\ 0,\ \pm 1/2,\ -{\sqrt {1/8}},\ -{\sqrt {3/8}}\right)

\left(\pm 1/2,\ 0,\ \pm 1/2,\ {\sqrt {1/8}},\ {\sqrt {3/8}}\right)

\left(0,\ \pm 1/2,\ \pm 1/2,\ -{\sqrt {1/8}},\ {\sqrt {3/8}}\right)

\left(0,\ \pm 1/2,\ \pm 1/2,\ {\sqrt {1/8}},\ -{\sqrt {3/8}}\right)

\left(\pm 1/2,\ \pm 1/2,\ 0,\ \pm {\sqrt {1/2}},\ 0\right)

Root system

Its 30 vertices represent the root vectors of the simple Lie group A₅. It is also the vertex figure of the 5-simplex honeycomb.

Images

orthographic projections
A_k Coxeter plane	A₅	A₄
Graph
Dihedral symmetry	[6]	[[5]]=[10]
A_k Coxeter plane	A₃	A₂
Graph
Dihedral symmetry	[4]	[[3]]=[6]

orthogonal projection with [6] symmetry