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| | Friends call him Royal Seyler. The occupation he's been occupying for many years is a messenger. The thing I adore most bottle tops gathering and now I have time to consider on new things. Delaware is the place I love most but I require to move for my family.<br><br>Here is my web page; extended car warranty - [http://Odw.my-hobbys.com/index.php?mod=users&action=view&id=966 More Help] - |
| !bgcolor=#e7dcc3 colspan=2|5-cubic honeycomb
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| |bgcolor=#ffffff align=center colspan=2|(no image)
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| |bgcolor=#e7dcc3|Type||[[List_of_regular_polytopes#Tessellations of Euclidean space|Regular 5-space honeycomb]]
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| |bgcolor=#e7dcc3|Family||[[Hypercube honeycomb]]
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]|| {4,3<sup>3</sup>,4}<BR>t<sub>0,5</sub>{4,3<sup>3</sup>,4}<BR>{4,3,3,3<sup>1,1</sup>}<BR>{4,3,4}x{∞}<BR>{4,3,4}x{4,4}<BR>{4,3,4}x{∞}<sup>2</sup><BR>{4,4}<sup>2</sup>x{∞}<BR>{∞}<sup>5</sup>
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]s||
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| {{CDD|node_1|4|node|3|node|3|node|3|node|4|node}}<BR>{{CDD|node_1|4|node|3|node|3|node|3|node|4|node_1}}<BR>{{CDD|node_1|4|node|3|node|3|node|split1|nodes}}<BR>{{CDD|node_1|4|node|3|node|3|node|4|node|2|node_1|infin|node}}<BR>{{CDD|node_1|4|node|3|node|3|node|4|node_1|2|node_1|infin|node}}
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| <BR>{{CDD|node_1|4|node|3|node|4|node|2|node_1|4|node|4|node}}<BR>{{CDD|node_1|4|node|3|node|4|node_1|2|node_1|4|node|4|node}}
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| <BR>{{CDD|node_1|4|node|3|node|4|node|2|node_1|infin|node|2|node_1|infin|node}}<BR>{{CDD|node_1|4|node|3|node|4|node_1|2|node_1|infin|node|2|node_1|infin|node}}
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| <BR>{{CDD|node_1|4|node|4|node|2|node_1|infin|node|2|node_1|infin|node|2|node_1|infin|node}}<BR>{{CDD|node_1|4|node|4|node_1|2|node_1|infin|node|2|node_1|infin|node|2|node_1|infin|node}}
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| <BR>{{CDD|node_1|4|node|4|node|2|node_1|4|node|4|node|2|node_1|infin|node}}
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| <BR>{{CDD|node_1|4|node|4|node_1|2|node_1|4|node|4|node|2|node_1|infin|node}}
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| <BR>{{CDD|node_1|4|node|4|node_1|2|node_1|4|node|4|node_1|2|node_1|infin|node}}
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| <BR>{{CDD|node_1|infin|node|2|node_1|infin|node|2|node_1|infin|node|2|node_1|infin|node|2|node_1|infin|node}}
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| |bgcolor=#e7dcc3|5-face type||[[5-cube|{4,3<sup>3</sup>}]]
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| |bgcolor=#e7dcc3|4-face type||[[tesseract|{4,3,3}]]
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| |bgcolor=#e7dcc3|Cell type||[[cube|{4,3}]]
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| |bgcolor=#e7dcc3|Face type||[[square (geometry)|{4}]]
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| |bgcolor=#e7dcc3|Face figure||[[cube|{4,3}]]<BR>([[octahedron]])
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| |bgcolor=#e7dcc3|Edge figure||8 [[tesseract|{4,3,3}]]<BR>([[16-cell]])
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| |bgcolor=#e7dcc3|Vertex figure||32 [[5-cube|{4,3<sup>3</sup>}]]<BR>([[5-orthoplex]])
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| |bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{C}}_5</math>, [4,3<sup>3</sup>,4]
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| |bgcolor=#e7dcc3|Dual||[[Self-dual polytope|self-dual]]
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| |bgcolor=#e7dcc3|Properties||[[vertex-transitive]], [[edge-transitive]], [[face-transitive]], [[cell-transitive]]
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| |}
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| The '''5-cubic honeycomb''' or '''penteractic honeycomb''' is the only regular space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 5-space. Four [[5-cube]]s meet at each cubic cell, and it is more explicitly called an ''order-4 penteractic honeycomb''. | |
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| It is analogous to the [[square tiling]] of the plane and to the [[cubic honeycomb]] of 3-space, and the [[tesseractic honeycomb]] of 4-space.
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| == Constructions==
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| There are many different [[Wythoff construction]]s of this honeycomb. The most symmetric form is [[Regular polytope|regular]], with [[Schläfli symbol]] {4,3<sup>3</sup>,4}. Another form has two alternating [[5-cube]] facets (like a checkerboard) with Schläfli symbol {4,3,3,3<sup>1,1</sup>}. The lowest symmetry Wythoff construction has 32 types of facets around each vertex and a prismatic product Schläfli symbol {∞}<sup>5</sup>.
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| == Related polytopes and honeycombs ==
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| The [4,3<sup>3</sup>,4], {{CDD|node|4|node|3|node|3|node|3|node|4|node}}, Coxeter group generates 63 permutations of uniform tessellations, 35 with unique symmetry and 34 with unique geometry. The [[Expansion (geometry)|expanded]] 5-cubic honeycomb is geometrically identical to the 5-cubic honeycomb.
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| The ''5-cubic honeycomb'' can be [[Alternation (geometry)|alternated]] into the [[5-demicubic honeycomb]], replacing the 5-cubes with [[5-demicube]]s, and the alternated gaps are filled by [[5-orthoplex]] facets.
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| It is also related to the regular [[6-cube]] which exists in 6-space with ''3'' ''5''-cubes on each cell. This could be considered as a tessellation on the [[N-sphere|5-sphere]], an ''order-3 penteractic honeycomb'', {4,3<sup>4</sup>}.
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| === Tritruncated 5-cubic honeycomb ===
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| A '''tritruncated 5-cubic honeycomb''', {{CDD|branch_11|3ab|nodes|4a4b|nodes}}, containins all [[bitruncated 5-orthoplex]] facets and is the [[Voronoi tessellation]] of the [[5-demicube honeycomb#D5 lattice|D<sub>5</sub><sup>*</sup> lattice]]. Facets can be identically colored from a doubled <math>{\tilde{C}}_5</math>×2, <nowiki>[[</nowiki>4,3<sup>3</sup>,4]] symmetry, alternately colored from <math>{\tilde{C}}_5</math>, [4,3<sup>3</sup>,4] symmetry, three colors from <math>{\tilde{B}}_5</math>, [4,3,3,3<sup>1,1</sup>] symmetry, and 4 colors from <math>{\tilde{D}}_5</math>, [3<sup>1,1</sup>,3,3<sup>1,1</sup>] symmetry.
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| ==See also==
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| *[[List of regular polytopes]]
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| Regular and uniform honeycombs in 5-space:
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| *[[5-demicubic honeycomb]]
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| *[[5-simplex honeycomb]]
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| *[[Truncated 5-simplex honeycomb]]
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| *[[Omnitruncated 5-simplex honeycomb]]
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| == References ==
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| * [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
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| * '''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]
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| ** (Paper 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
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| {{Honeycombs}}
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| [[Category:Honeycombs (geometry)]]
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| [[Category:6-polytopes]]
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Friends call him Royal Seyler. The occupation he's been occupying for many years is a messenger. The thing I adore most bottle tops gathering and now I have time to consider on new things. Delaware is the place I love most but I require to move for my family.
Here is my web page; extended car warranty - More Help -