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Stericated 5 simplexes

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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.

Contents

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:

    ( ± 1 ,   0 ,   0 ,   0 ,   0 ) ( 0 ,   ± 1 ,   0 ,   0 ,   0 ) ( 0 ,   0 ,   ± 1 ,   0 ,   0 ) ( ± 1 / 2 ,   0 ,   ± 1 / 2 ,   − 1 / 8 ,   − 3 / 8 ) ( ± 1 / 2 ,   0 ,   ± 1 / 2 ,   1 / 8 ,   3 / 8 ) ( 0 ,   ± 1 / 2 ,   ± 1 / 2 ,   − 1 / 8 ,   3 / 8 ) ( 0 ,   ± 1 / 2 ,   ± 1 / 2 ,   1 / 8 ,   − 3 / 8 ) ( ± 1 / 2 ,   ± 1 / 2 ,   0 ,   ± 1 / 2 ,   0 )

    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.

    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.

    Alternate names

  • Stericantellated hexateron
  • Celliprismatotruncated dodecateron (Acronym: captid) (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.

    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.

    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

    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

  • 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 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}, .

    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.

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

    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

    Stericated 5-simplexes Wikipedia
    (Text) CC BY-SA


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