Stericated 5-simplexes#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).

• Expanded 5-simplex
• Stericated hexateron
• Small cellated dodecateron (Acronym: scad) (Jonathan Bowers)Klitizing, (x3o3o3o3x - scad)

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:

:$\backslash left(\backslash pm1,\backslash \; 0,\backslash \; 0,\backslash \; 0,\backslash \; 0\backslash right)$

:$\backslash left(0,\backslash \; \backslash pm1,\backslash \; 0,\backslash \; 0,\backslash \; 0\backslash right)$

:$\backslash left(0,\backslash \; 0,\backslash \; \backslash pm1,\backslash \; 0,\backslash \; 0\backslash right)$

:$\backslash left(\backslash pm1/2,\backslash \; 0,\backslash \; \backslash pm1/2,\backslash \; -\backslash sqrt\{1/8\},\backslash \; -\backslash sqrt\{3/8\}\backslash right)$

:$\backslash left(\backslash pm1/2,\backslash \; 0,\backslash \; \backslash pm1/2,\backslash \; \backslash sqrt\{1/8\},\backslash \; \backslash sqrt\{3/8\}\backslash right)$

:$\backslash left(\; 0,\backslash \; \backslash pm1/2,\backslash \; \backslash pm1/2,\backslash \; -\backslash sqrt\{1/8\},\backslash \; \backslash sqrt\{3/8\}\backslash right)$

:$\backslash left(\; 0,\backslash \; \backslash pm1/2,\backslash \; \backslash pm1/2,\backslash \; \backslash sqrt\{1/8\},\backslash \; -\backslash sqrt\{3/8\}\backslash right)$

:$\backslash left(\backslash pm1/2,\backslash \; \backslash pm1/2,\backslash \; 0,\backslash \; \backslash pm\backslash sqrt\{1/2\},\backslash \; 0\backslash right)$

Its 30 vertices represent the root vectors of the simple Lie group A₅. It is also the vertex figure of the 5-simplex honeycomb.

{{5-simplex2 Coxeter plane graphs|t04|100}}

• Steritruncated hexateron
• Celliprismated hexateron (Acronym: cappix) (Jonathan Bowers)Klitizing, (x3x3o3o3x - cappix)

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.

{{5-simplex Coxeter plane graphs|t014|100}}

• Stericantellated hexateron
• Cellirhombated dodecateron (Acronym: card) (Jonathan Bowers)Klitizing, (x3o3x3o3x - card)

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.

{{5-simplex2 Coxeter plane graphs|t024|100}}

• Stericantitruncated hexateron
• Celligreatorhombated hexateron (Acronym: cograx) (Jonathan Bowers)Klitizing, (x3x3x3o3x - cograx)

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.

{{5-simplex Coxeter plane graphs|t0124|100}}

• Steriruncitruncated hexateron
• Celliprismatotruncated dodecateron (Acronym: captid) (Jonathan Bowers)Klitizing, (x3x3o3x3x - captid)

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.

{{5-simplex2 Coxeter plane graphs|t0134|100}}

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)Klitizing, (x3x3x3x3x - gocad)

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}, {{CDD|node|4|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}.

{{5-simplex2 Coxeter plane graphs|t01234|120}}

Image:Omnitruncated Hexateron.png]]

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 {{CDD|branch_11|3ab|nodes_11|3ab|branch_11}}.

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 {{CDD|node_h|3|node_h|3|node_h|3|node_h|3|node_h}} and symmetry {{brackets|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 A₅ Coxeter plane orthographic projections. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)

{{Hexateron family}}

{{reflist}}

• 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}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html]
• (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.
• {{KlitzingPolytopes|polytera.htm|5D|uniform polytopes (polytera)}} x3o3o3o3x - scad, x3x3o3o3x - cappix, x3o3x3o3x - card, x3x3x3o3x - cograx, x3x3o3x3x - captid, x3x3x3x3x - gocad

• {{PolyCell | urlname = glossary.html#simplex| title = Glossary for hyperspace}}
• [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions]
• [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

Category:5-polytopes