Truncated 5-simplexes#Truncated 5-simplex
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| colspan=3|Orthogonal projections in A5 Coxeter plane |
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In five-dimensional geometry, a truncated 5-simplex is a convex uniform 5-polytope, being a truncation of the regular 5-simplex.
There are unique 2 degrees of truncation. Vertices of the truncation 5-simplex are located as pairs on the edge of the 5-simplex. Vertices of the bitruncation 5-simplex are located on the triangular faces of the 5-simplex.
Truncated 5-simplex
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| bgcolor=#e7dcc3 align=center colspan=3|Truncated 5-simplex | |||
| bgcolor=#e7dcc3|Type
|colspan=2|Uniform 5-polytope | |||
| bgcolor=#e7dcc3|Schläfli symbol
|colspan=2| t{3,3,3,3} | |||
| bgcolor=#e7dcc3|Coxeter-Dynkin diagram
|colspan=2|{{CDD | node_1|3|node_1|3|node | 3|node|3|node}} {{CDD | branch_11|3b|nodeb|3b|nodeb|3b|nodeb}} |
| bgcolor=#e7dcc3|4-faces
|12 | |||
| bgcolor=#e7dcc3|Cells
|45 | |||
| bgcolor=#e7dcc3|Faces
|80 | |||
| bgcolor=#e7dcc3|Edges
|colspan=2|75 | |||
| bgcolor=#e7dcc3|Vertices
|colspan=2|30 | |||
| bgcolor=#e7dcc3|Vertex figure | |||
| bgcolor=#e7dcc3|Coxeter group
|colspan=2| A5 [3,3,3,3], order 720 | |||
| bgcolor=#e7dcc3|Properties
|colspan=2|convex |
The truncated 5-simplex has 30 vertices, 75 edges, 80 triangular faces, 45 cells (15 tetrahedral, and 30 truncated tetrahedron), and 12 4-faces (6 5-cell and 6 truncated 5-cells).
= Alternate names =
- Truncated hexateron (Acronym: tix) (Jonathan Bowers)Klitizing, (x3x3o3o3o - tix)
= Coordinates =
The vertices of the truncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,0,1,2) or of (0,1,2,2,2,2). These coordinates come from facets of the truncated 6-orthoplex and bitruncated 6-cube respectively.
= Images =
{{5-simplex Coxeter plane graphs|t01|150}}
Bitruncated 5-simplex
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| bgcolor=#e7dcc3 align=center colspan=3|bitruncated 5-simplex | ||
| bgcolor=#e7dcc3|Type
|colspan=2|Uniform 5-polytope | ||
| bgcolor=#e7dcc3|Schläfli symbol
|colspan=2| 2t{3,3,3,3} | ||
| bgcolor=#e7dcc3|Coxeter-Dynkin diagram
|colspan=2|{{CDD | node|3|node_1|3|node_1|3|node|3|node}} {{CDD | branch_11|3ab|nodes|3b|nodeb}} |
| bgcolor=#e7dcc3|4-faces
|12 | ||
| bgcolor=#e7dcc3|Cells
|60 | ||
| bgcolor=#e7dcc3|Faces
|140 | ||
| bgcolor=#e7dcc3|Edges
|colspan=2|150 | ||
| bgcolor=#e7dcc3|Vertices
|colspan=2|60 | ||
| bgcolor=#e7dcc3|Vertex figure | ||
| bgcolor=#e7dcc3|Coxeter group
|colspan=2| A5 [3,3,3,3], order 720 | ||
| bgcolor=#e7dcc3|Properties
|colspan=2|convex |
= Alternate names =
- Bitruncated hexateron (Acronym: bittix) (Jonathan Bowers)Klitizing, (o3x3x3o3o - bittix)
= Coordinates =
The vertices of the bitruncated 5-simplex can be most simply constructed on a hyperplane in 6-space as permutations of (0,0,0,1,2,2) or of (0,0,1,2,2,2). These represent positive orthant facets of the bitruncated 6-orthoplex, and the tritruncated 6-cube respectively.
= Images =
{{5-simplex Coxeter plane graphs|t12|150}}
Related uniform 5-polytopes
The truncated 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}}
Notes
{{Reflist}}
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, {{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)}} x3x3o3o3o - tix, o3x3x3o3o - bittix
External links
- {{PolyCell | urlname = glossary.html#simplex| title = Glossary for hyperspace}}
- [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions], Jonathan Bowers
- [http://www.polytope.net/hedrondude/truncates5.htm Truncated uniform polytera] (tix), Jonathan Bowers
- [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]
{{Polytopes}}