Cantellated 5-simplexes

The cantellated 5-simplex has 60 vertices, 240 edges, 290 faces (200 triangles and 90 squares), 135 cells (30 tetrahedra, 30 octahedra, 15 cuboctahedra and 60 triangular prisms), and 27 4-faces (6 cantellated 5-cell, 6 rectified 5-cells, and 15 tetrahedral prisms).

• Cantellated hexateron
• Small rhombated hexateron (Acronym: sarx) (Jonathan Bowers)Klitizing, (x3o3x3o3o - sarx)

The vertices of the cantellated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,1,2) or of (0,1,1,2,2,2). These represent positive orthant facets of the cantellated hexacross and bicantellated hexeract respectively.

{{5-simplex Coxeter plane graphs|t02|150}}

• Bicantellated hexateron
• Small birhombated {{not a typo|dodecateron}} (Acronym: {{not a typo|sibrid}}) (Jonathan Bowers)Klitizing, (o3x3o3x3o - {{not a typo|sibrid}})

The coordinates can be made in 6-space, as 90 permutations of:

: (0,0,1,1,2,2)

This construction exists as one of 64 orthant facets of the bicantellated 6-orthoplex.

{{5-simplex2 Coxeter plane graphs|t13|150}}

• Cantitruncated hexateron
• Great rhombated hexateron (Acronym: {{not a typo|garx}}) (Jonathan Bowers)Klitizing, (x3x3x3o3o - {{not a typo|garx}})

The vertices of the cantitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,3) or of (0,1,2,3,3,3). These construction can be seen as facets of the cantitruncated 6-orthoplex or bicantitruncated 6-cube respectively.

{{5-simplex Coxeter plane graphs|t012|150}}

• Bicantitruncated hexateron
• Great birhombated {{not a typo|dodecateron}}(Acronym: {{not a typo|gibrid}}) (Jonathan Bowers)Klitizing, (o3x3x3x3o - {{not a typo|gibrid}})

The coordinates can be made in 6-space, as 180 permutations of:

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

This construction exists as one of 64 orthant facets of the bicantitruncated 6-orthoplex.

{{5-simplex2 Coxeter plane graphs|t123|150}}

The cantellated 5-simplex is one 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)}} x3o3x3o3o - sarx, o3x3o3x3o - {{not a typo|sibrid}}, x3x3x3o3o - {{not a typo|garx}}, o3x3x3x3o - {{not a typo|gibrid}}

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

{{Polytopes}}

Category:5-polytopes