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{3^2,2} = ${\displaystyle 2r\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}}$
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

• 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:

${\displaystyle \left(\pm 1,\ 0,\ 0,\ 0,\ 0\right)}$ 

${\displaystyle \left(0,\ \pm 1,\ 0,\ 0,\ 0\right)}$ 

${\displaystyle \left(0,\ 0,\ \pm 1,\ 0,\ 0\right)}$ 

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

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

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

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

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

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

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

• Steritruncated hexateron
• Celliprismated hexateron (Acronym: cappix) (Jonathan Bowers)^[2]

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)^[3]

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

• Stericantitruncated hexateron
• Celligreatorhombated hexateron (Acronym: cograx) (Jonathan Bowers)^[4]

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{3^2,2}
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)^[5]

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{3^2,2}
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 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)^[6]

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

orthographic projections

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

 

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	${\displaystyle {\tilde {I}}_{1}}$	${\displaystyle {\tilde {A}}_{2}}$	${\displaystyle {\tilde {A}}_{3}}$	${\displaystyle {\tilde {A}}_{4}}$	${\displaystyle {\tilde {A}}_{5}}$
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)

A5 polytopes
t0	t1	t2	t0,1	t0,2	t1,2	t0,3
t1,3	t0,4	t0,1,2	t0,1,3	t0,2,3	t1,2,3	t0,1,4
t0,2,4	t0,1,2,3	t0,1,2,4	t0,1,3,4	t0,1,2,3,4

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

• 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)". x3o3o3o3x - scad, x3x3o3o3x - cappix, x3o3x3o3x - card, x3x3x3o3x - cograx, x3x3o3x3x - captid, x3x3x3x3x - gocad

External links

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

v t e Fundamental convex regular and uniform polytopes in dimensions 2–10
Family	An	Bn	I2(p) / Dn	E6 / E7 / E8 / F4 / G2	Hn
Regular polygon	Triangle	Square	p-gon	Hexagon	Pentagon
Uniform polyhedron	Tetrahedron	Octahedron • Cube	Demicube		Dodecahedron • Icosahedron
Uniform polychoron	Pentachoron	16-cell • Tesseract	Demitesseract	24-cell	120-cell • 600-cell
Uniform 5-polytope	5-simplex	5-orthoplex • 5-cube	5-demicube
Uniform 6-polytope	6-simplex	6-orthoplex • 6-cube	6-demicube	122 • 221
Uniform 7-polytope	7-simplex	7-orthoplex • 7-cube	7-demicube	132 • 231 • 321
Uniform 8-polytope	8-simplex	8-orthoplex • 8-cube	8-demicube	142 • 241 • 421
Uniform 9-polytope	9-simplex	9-orthoplex • 9-cube	9-demicube
Uniform 10-polytope	10-simplex	10-orthoplex • 10-cube	10-demicube
Uniform n-polytope	n-simplex	n-orthoplex • n-cube	n-demicube	1k2 • 2k1 • k21	n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds