Stericated 5-simplexes

120px 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} = [math]\displaystyle{ 2r\left\{\begin{array}{l}3, 3\\3 ,3\end{array}\right\} }[/math]
Coxeter-Dynkin diagram	or
4-faces	62	6+6 {3,3,3} 15+15 {}×{3,3}25px 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)

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.

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:

[math]\displaystyle{ \left(\pm1,\ 0,\ 0,\ 0,\ 0\right) }[/math]

[math]\displaystyle{ \left(0,\ \pm1,\ 0,\ 0,\ 0\right) }[/math]

[math]\displaystyle{ \left(0,\ 0,\ \pm1,\ 0,\ 0\right) }[/math]

[math]\displaystyle{ \left(\pm1/2,\ 0,\ \pm1/2,\ -\sqrt{1/8},\ -\sqrt{3/8}\right) }[/math]

[math]\displaystyle{ \left(\pm1/2,\ 0,\ \pm1/2,\ \sqrt{1/8},\ \sqrt{3/8}\right) }[/math]

[math]\displaystyle{ \left( 0,\ \pm1/2,\ \pm1/2,\ -\sqrt{1/8},\ \sqrt{3/8}\right) }[/math]

[math]\displaystyle{ \left( 0,\ \pm1/2,\ \pm1/2,\ \sqrt{1/8},\ -\sqrt{3/8}\right) }[/math]

[math]\displaystyle{ \left(\pm1/2,\ \pm1/2,\ 0,\ \pm\sqrt{1/2},\ 0\right) }[/math]

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

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

• Steritruncated hexateron
• Celliprismated hexateron (Acronym: cappix) (Jonathan Bowers)

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

• Stericantellated hexateron
• Cellirhombated dodecateron (Acronym: card) (Jonathan Bowers)

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.

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

• Stericantitruncated hexateron
• Celligreatorhombated hexateron (Acronym: cograx) (Jonathan Bowers)

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

• Steriruncitruncated hexateron
• Celliprismatotruncated dodecateron (Acronym: captid) (Jonathan Bowers)

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.

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}25px 20 {6}×{6}
Cells	540	360 t{3,4} 90 {4,3}25px 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 octahedra, 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

• Steriruncicantitruncated 5-simplex (Full description of omnitruncation for 5-polytopes by Johnson)
• Omnitruncated hexateron
• Great cellated dodecateron (Acronym: gocad) (Jonathan Bowers)

Coordinates

The vertices of the omnitruncated 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, t_0,1,2,3,4{3⁴,4}, .

Images

Stereographic projection

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	[math]\displaystyle{ {\tilde{I}}_{1} }[/math]	[math]\displaystyle{ {\tilde{A}}_{2} }[/math]	[math]\displaystyle{ {\tilde{A}}_{3} }[/math]	[math]\displaystyle{ {\tilde{A}}_{4} }[/math]	[math]\displaystyle{ {\tilde{A}}_{5} }[/math]
Coxeter-Dynkin
Picture
Name	Apeirogon	Hextille	Omnitruncated 3-simplex honeycomb	Omnitruncated 4-simplex honeycomb	Omnitruncated 5-simplex honeycomb
Facets

Full snub 5-simplex

The full snub 5-simplex or omnisnub 5-simplex, defined as an alternation of the omnitruncated 5-simplex is not uniform, but it can be given Coxeter diagram and symmetry [[3,3,3,3]]⁺, and constructed from 12 snub 5-cells, 30 snub tetrahedral antiprisms, 20 3-3 duoantiprisms, and 360 irregular 5-cells filling the gaps at the deleted vertices.

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)

References

• H.S.M. Coxeter:
  • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [1]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D.
• Klitzing, Richard. "5D uniform polytopes (polytera)". https://bendwavy.org/klitzing/dimensions/polytera.htm.  x3o3o3o3x - scad, x3x3o3o3x - cappix, x3o3x3o3x - card, x3x3x3o3x - cograx, x3x3o3x3x - captid, x3x3x3x3x - gocad

External links

• Glossary for hyperspace, George Olshevsky.
• Polytopes of Various Dimensions
• Multi-dimensional Glossary