Stericated 5-simplexes


5-simplex

Stericated 5-simplex

Steritruncated 5-simplex

Stericantellated 5-simplex

Stericantitruncated 5-simplex

Steriruncitruncated 5-simplex

Steriruncicantitruncated 5-simplex
(Omnitruncated 5-simplex)
Orthogonal projections in A5 and A4 Coxeter planes

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}
2r{32,2} = 2r\left\{\begin{array}{l}3, 3\\3 ,3\end{array}\right\}
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

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(\pm1,\ 0,\ 0,\ 0,\ 0\right)
\left(0,\ \pm1,\ 0,\ 0,\ 0\right)
\left(0,\ 0,\ \pm1,\ 0,\ 0\right)
\left(\pm1/2,\      0,\ \pm1/2,\ -\sqrt{1/8},\ -\sqrt{3/8}\right)
\left(\pm1/2,\      0,\ \pm1/2,\  \sqrt{1/8},\  \sqrt{3/8}\right)
\left(     0,\ \pm1/2,\ \pm1/2,\ -\sqrt{1/8},\  \sqrt{3/8}\right)
\left(     0,\ \pm1/2,\ \pm1/2,\  \sqrt{1/8},\ -\sqrt{3/8}\right)
\left(\pm1/2,\ \pm1/2,\ 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
Ak
Coxeter plane
A5 A4
Graph
Dihedral symmetry [6] [[5]]=[10]
Ak
Coxeter plane
A3 A2
Graph
Dihedral symmetry [4] [[3]]=[6]

orthogonal projection with [6] symmetry

Steritruncated 5-simplex

Steritruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,1,4{3,3,3,3}
Coxeter-Dynkin diagram
4-faces 62 6 t{3,3,3}
15 {}×t{3,3}
20 {3}×{6}
15 {}×{3,3}
6 t0,3{3,3,3}
Cells 330
Faces 570
Edges 420
Vertices 120
Vertex figure
Coxeter group A5 [3,3,3,3], order 720
Properties convex, isogonal

Alternate names

Coordinates

The coordinates can be made in 6-space, as 180 permutations of:

(0,1,1,1,2,3)

This construction exists as one of 64 orthant facets of the steritruncated 6-orthoplex.

Images

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

Stericantellated 5-simplex

Stericantellated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,2,4{3,3,3,3}
Coxeter-Dynkin diagram
or
4-faces 62 12 rr{3,3,3}
30 rr{3,3}x{}
20 {3}×{3}
Cells 420 60 rr{3,3}
240 {}×{3}
90 {}×{}×{}
30 r{3,3}
Faces 900 360 {3}
540 {4}
Edges 720
Vertices 180
Vertex figure
Coxeter group A5×2, 3,3,3,3, order 1440
Properties convex, isogonal

Alternate names

Coordinates

The coordinates can be made in 6-space, as permutations of:

(0,1,1,2,2,3)

This construction exists as one of 64 orthant facets of the stericantellated 6-orthoplex.

Images

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

Stericantitruncated 5-simplex

Stericantitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,1,2,4{3,3,3,3}
Coxeter-Dynkin diagram
4-faces 62
Cells 480
Faces 1140
Edges 1080
Vertices 360
Vertex figure
Coxeter group A5 [3,3,3,3], order 720
Properties convex, isogonal

Alternate names

Coordinates

The coordinates can be made in 6-space, as 360 permutations of:

(0,1,1,2,3,4)

This construction exists as one of 64 orthant facets of the stericantitruncated 6-orthoplex.

Images

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

Steriruncitruncated 5-simplex

Steriruncitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,1,3,4{3,3,3,3}
2t{32,2}
Coxeter-Dynkin diagram
or
4-faces 62 12 t0,1,3{3,3,3}
30 {}×t{3,3}
20 {6}×{6}
Cells 450
Faces 1110
Edges 1080
Vertices 360
Vertex figure
Coxeter group A5×2, 3,3,3,3, order 1440
Properties convex, isogonal

Alternate names

Coordinates

The coordinates can be made in 6-space, as 360 permutations of:

(0,1,2,2,3,4)

This construction exists as one of 64 orthant facets of the steriruncitruncated 6-orthoplex.

Images

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

Omnitruncated 5-simplex

Omnitruncated 5-simplex
Type Uniform 5-polytope
Schläfli symbol t0,1,2,3,4{3,3,3,3}
2tr{32,2}
Coxeter-Dynkin
diagram

or
4-faces 62 12 t0,1,2,3{3,3,3}
30 {}×tr{3,3}
20 {6}×{6}
Cells 540 360 t{3,4}
90 {4,3}
90 {}×{6}
Faces 1560 480 {6}
1080 {4}
Edges 1800
Vertices 720
Vertex figure
Irregular 5-cell
Coxeter group A5×2, 3,3,3,3, order 1440
Properties convex, isogonal, zonotope

The omnitruncated 5-simplex has 720 vertices, 1800 edges, 1560 faces (480 hexagons and 1080 squares), 540 cells (360 truncated octahedrons, 90 cubes, and 90 hexagonal prisms), and 62 4-faces (12 omnitruncated 5-cells, 30 truncated octahedral prisms, and 20 6-6 duoprisms).

Alternate names

Coordinates

The vertices of the truncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,1,2,3,4,5). These coordinates come from the positive orthant facet of the steriruncicantitruncated 6-orthoplex, t0,1,2,3,4{34,4}, .

Images

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

Permutohedron

The omnitruncated 5-simplex is the permutohedron of order 6. It is also a zonotope, the Minkowski sum of six line segments parallel to the six lines through the origin and the six vertices of the 5-simplex.


Orthogonal projection, vertices labeled as a permutohedron.

Related honeycomb

The omnitruncated 5-simplex honeycomb is constructed by omnitruncated 5-simplex facets with 3 facets around each ridge. It has Coxeter-Dynkin diagram of .

Coxeter group {\tilde{I}}_{1} {\tilde{A}}_{2} {\tilde{A}}_{3} {\tilde{A}}_{4} {\tilde{A}}_{5}
Coxeter-Dynkin
Picture
Name Apeirogon Hextille Omnitruncated
3-simplex
honeycomb
Omnitruncated
4-simplex
honeycomb
Omnitruncated
5-simplex
honeycomb
Facets

Related uniform polytopes

These polytopes are a part of 19 uniform 5-polytopes based on the [3,3,3,3] Coxeter group, all shown here in A5 Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)

Notes

  1. Klitizing, (x3o3o3o3x - scad)
  2. Klitizing, (x3x3o3o3x - cappix)
  3. Klitizing, (x3o3x3o3x - card)
  4. Klitizing, (x3x3x3o3x - cograx)
  5. Klitizing, (x3x3o3x3x - captid)
  6. Klitizing, (x3x3x3x3x - gocad)

References

External links

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