Stericated 5-simplexes

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

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.

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:

:$\left(\pm1,\ 0,\ 0,\ 0,\ 0\right)$
:$\left(0,\ \pm1,\ 0,\ 0,\ 0\right)$
:$\left(0,\ 0,\ \pm1,\ 0,\ 0\right)$
:$\left(\pm1/2,\ 0,\ \pm1/2,\ -\sqrt,\ -\sqrt\right)$
:$\left(\pm1/2,\ 0,\ \pm1/2,\ \sqrt,\ \sqrt\right)$
:$\left( 0,\ \pm1/2,\ \pm1/2,\ -\sqrt,\ \sqrt\right)$
:$\left( 0,\ \pm1/2,\ \pm1/2,\ \sqrt,\ -\sqrt\right)$
:$\left(\pm1/2,\ \pm1/2,\ 0,\ \pm\sqrt,\ 0\right)$

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

Steritruncated 5-simplex

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

Stericantellated 5-simplex

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.

Images

Stericantitruncated 5-simplex

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

Steriruncitruncated 5-simplex

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.

Images

Omnitruncated 5-simplex

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)Klitizing, (x3x3x3x3x - gocad)

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, .

Images

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.

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 .

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 ⁺, 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 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

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,

*** (Paper 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', ath. Zeit. 46 (1940) 380-407, MR 2,10*** (Paper 23) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes II'', ath. Zeit. 188 (1985) 559-591*** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', ath. Zeit. 200 (1988) 3-45* Norman Johnson ''Uniform Polytopes'', Manuscript (1991)
** N.W. Johnson: ''The Theory of Uniform Polytopes and Honeycombs'', Ph.D.
* x3o3o3o3x - scad, x3x3o3o3x - cappix, x3o3x3o3x - card, x3x3x3o3x - cograx, x3x3o3x3x - captid, x3x3x3x3x - gocad

External links

*

Polytopes of Various Dimensions



{{Polytopes

5-polytopes