Omnitruncated 5-simplex honeycomb
{{short description|Five dimensional space-filling tessellation}}
| class="wikitable" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Omnitruncated 5-simplex honeycomb | |
| bgcolor=#ffffff align=center colspan=2|(No image) | |
| bgcolor=#e7dcc3|Type | Uniform honeycomb |
| bgcolor=#e7dcc3|Family | Omnitruncated simplectic honeycomb |
| bgcolor=#e7dcc3|Schläfli symbol | t012345{3[6]} |
| bgcolor=#e7dcc3|Coxeter–Dynkin diagram | {{CDD|node_1|split1|nodes_11|3ab|nodes_11|split2|node_1}} |
| bgcolor=#e7dcc3|5-face types | t01234{3,3,3,3} 40px |
| bgcolor=#e7dcc3|4-face types | t0123{3,3,3}25px {}×t012{3,3}25px {6}×{6}25px |
| bgcolor=#e7dcc3|Cell types | t012{3,3}25px {4,3}25px {}x{6}25px |
| bgcolor=#e7dcc3|Face types | {4} {6} |
| bgcolor=#e7dcc3|Vertex figure | 62px Irr. 5-simplex |
| bgcolor=#e7dcc3|Symmetry | ×12, [6[3[6]]] |
| bgcolor=#e7dcc3|Properties | vertex-transitive |
In five-dimensional Euclidean geometry, the omnitruncated 5-simplex honeycomb or omnitruncated hexateric honeycomb is a space-filling tessellation (or honeycomb). It is composed entirely of omnitruncated 5-simplex facets.
The facets of all omnitruncated simplectic honeycombs are called permutahedra and can be positioned in n+1 space with integral coordinates, permutations of the whole numbers (0,1,..,n).
A<sub>5</sub><sup>*</sup> lattice
The A{{sup sub|*|5}} lattice (also called A{{sup sub|6|5}}) is the union of six A5 lattices, and is the dual vertex arrangement to the omnitruncated 5-simplex honeycomb, and therefore the Voronoi cell of this lattice is an omnitruncated 5-simplex.
:
{{CDD|node_1|split1|nodes|3ab|nodes|split2|node}} ∪
{{CDD|node|split1|nodes_10lur|3ab|nodes|split2|node}} ∪
{{CDD|node|split1|nodes_01lr|3ab|nodes|split2|node}} ∪
{{CDD|node|split1|nodes|3ab|nodes_10lru|split2|node}} ∪
{{CDD|node|split1|nodes|3ab|nodes_01lr|split2|node}} ∪
{{CDD|node|split1|nodes|3ab|nodes|split2|node_1}} = dual of {{CDD|node_1|split1|nodes_11|3ab|nodes_11|split2|node_1}}
Related polytopes and honeycombs
{{5-simplex honeycomb family}}
= Projection by folding =
The omnitruncated 5-simplex honeycomb can be projected into the 3-dimensional omnitruncated cubic honeycomb by a geometric folding operation that maps two pairs of mirrors into each other, sharing the same 3-space vertex arrangement:
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|{{CDD|node_1|split1|nodes_11|3ab|nodes_11|split2|node_1}} |
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|{{CDD|node_1|4|node_1|3|node_1|4|node_1}} |
See also
Regular and uniform honeycombs in 5-space:
Notes
{{reflist}}
References
- Norman Johnson Uniform Polytopes, Manuscript (1991)
- 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] (1.9 Uniform space-fillings)
- (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
{{Honeycombs}}