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 A and ACoxeter 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
Type	Uniform 5-polytope
Schläfli symbol	2r2r{3,3,3,3}2r{32,2} = ${\displaystyle 2r\left\{{\begin{array}{l}3,3\\3,3\end{array}}\right\}}$
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	A×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).

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

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 orthantfacet 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:

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

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

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

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

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

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

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

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

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

Orthographic projections

ACoxeter plane	A	A
Graph
Dihedral symmetry	[6]	[[5]]=[10]
ACoxeter plane	A	A
Graph
Dihedral symmetry	[4]	[[3]]=[6]

Orthogonal projection with [6] symmetry

Steritruncated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	t{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 t{3,3,3}
Cells	330
Faces	570
Edges	420
Vertices	120
Vertex figure
Coxeter group	A [3,3,3,3], order 720
Properties	convex, isogonal

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

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 orthantfacets of the steritruncated 6-orthoplex.

Orthographic projections

ACoxeter plane	A	A
Graph
Dihedral symmetry	[6]	[5]
ACoxeter plane	A	A
Graph
Dihedral symmetry	[4]	[3]

Stericantellated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	t{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	A×2, [[3,3,3,3]], order 1440
Properties	convex, isogonal

• Stericantellated hexateron
• Cellirhombated dodecateron (Acronym: card) (Jonathan Bowers)^[3]

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

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

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

Orthographic projections

ACoxeter plane	A	A
Graph
Dihedral symmetry	[6]	[[5]]=[10]
ACoxeter plane	A	A
Graph
Dihedral symmetry	[4]	[[3]]=[6]

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

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

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 orthantfacets of the stericantitruncated 6-orthoplex.

Orthographic projections

ACoxeter plane	A	A
Graph
Dihedral symmetry	[6]	[5]
ACoxeter plane	A	A
Graph
Dihedral symmetry	[4]	[3]

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

• Steriruncitruncated hexateron
• Celliprismatotruncated dodecateron (Acronym: captid) (Jonathan Bowers)^[5]

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 orthantfacets of the steriruncitruncated 6-orthoplex.

Orthographic projections

ACoxeter plane	A	A
Graph
Dihedral symmetry	[6]	[[5]]=[10]
ACoxeter plane	A	A
Graph
Dihedral symmetry	[4]	[[3]]=[6]

Omnitruncated 5-simplex
Type	Uniform 5-polytope
Schläfli symbol	t{3,3,3,3}2tr{32,2}
Coxeter-Dynkindiagram	or
4-faces	62	12 t{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	A×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).

• Steriruncicantitruncated 5-simplex (Full description of omnitruncation for 5-polytopes by Johnson)
• Omnitruncated hexateron
• Great cellated dodecateron (Acronym: gocad) (Jonathan Bowers)^[6]

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 orthantfacet of the steriruncicantitruncated 6-orthoplex, t{3⁴,4}, .

Orthographic projections

ACoxeter plane	A	A
Graph
Dihedral symmetry	[6]	[[5]]=[10]
ACoxeter plane	A	A
Graph
Dihedral symmetry	[4]	[[3]]=[6]

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.

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	Omnitruncated3-simplexhoneycomb	Omnitruncated4-simplexhoneycomb	Omnitruncated5-simplexhoneycomb
Facets

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.

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

A5 polytopes
t	t	t	t	t	t	t
t	t	t	t	t	t	t
t	t	t	t	t

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