Hexagonal tiling honeycomb - Revision history

en>Tomruen at 21:34, 22 May 2014

2014-05-22T21:34:41Z

← Older revision		Revision as of 23:34, 22 May 2014
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en>Tomruen at 02:16, 4 March 2014

2014-03-04T02:16:28Z

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en>Tomruen: /* References */

2014-01-12T07:20:06Z

References

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In [[geometry]], the '''quarter hypercube honeycomb''' (or '''quarter n-cubic honeycomb''') is a dimensional infinite series of [[Honeycomb (geometry)|honeycomb]]s, based on the [[hypercube honeycomb]]. It is given a [[Schläfli symbol]] q{4,3...3,4} or Coxeter symbol qδ4 representing the regular form with quarter the vertices removed and containing the symmetry of [[Coxeter_diagram#Infinite_Coxeter_groups|Coxeter group]] <math>{\tilde{D}}_{n-1}</math> for n ≥ 5, with <math>{\tilde{D}}_4</math> = <math>{\tilde{A}}_4</math> and for quarter n-cubic honeycombs <math>{\tilde{D}}_5</math> = <math>{\tilde{B}}_5</math>.<ref>Coxeter, 1988</ref>

{| class="wikitable"
!qδn
!Name
![[Schläfli symbol|Schläfli symbol]]
![[Coxeter diagram]]
!colspan=3|Facets
!Vertex figure
|- align=center
| qδ3
| [[File:Square_tiling_uniform_coloring_1.png|80px]] ''quarter [[square tiling]]''
| q{4,4}
|{{CDD|nodes_11|iaib|nodes}} {{CDD|node_h1|4|node|4|node_h1}}
|h{4}={2}
|{ }×{ }
|
|[[File:Regular_polygon_4_annotated.svg|40px]] { }×{ }
|- align=center
| qδ4
| [[File:Bitruncated alternated cubic tiling.png|120px]] ''[[quarter cubic honeycomb]]''
| q{4,3,4}
|{{CDD|branch_10r|3ab|branch_10l}} {{CDD|node_h1|4|node|3|node|4|node_h1}}
|[[File:Tetrahedron.png|60px]] [[Tetrahedron|h{4,3}]] ||[[File:Truncated tetrahedron.png|60px]] [[Truncated tetrahedron|h2{4,3}]]
||
|[[File:T01 quarter cubic honeycomb verf.png|60px]] Elongated [[triangular antiprism]]
|- align=center
| qδ5
| ''[[quarter tesseractic honeycomb]]''
| q{4,32,4}
|{{CDD|nodes_10ru|split2|node|split1|nodes_10lu}} = {{CDD|nodes|split2|node|3|node_1|4|node}} {{CDD|node_h1|4|node|3|node|3|node|4|node_h1}}
|[[File:Schlegel_wireframe_16-cell.png|60px]] [[16-cell|h{4,32}]] ||[[File:Schlegel_half-solid_rectified_8-cell.png|60px]] [[Rectified tesseract|h3{4,32}]]
||
||[[File:Rectified_tesseractic_honeycomb_verf.png|60px]] [[Octahedral prism|{3,4}×{}]]
|- align=center
| qδ6
| ''[[quarter 5-cubic honeycomb]]''
| q{4,33,4}
|{{CDD|nodes_10ru|split2|node|3|node|split1|nodes_10lu}} {{CDD|node_h1|4|node|3|node|3|node|3|node|4|node_h1}}
|[[File:Demipenteract graph ortho.svg|60px]] [[5-demicube|h{4,33}]] ||[[File:5-demicube t03 D5.svg|60px]] [[Runcinated 5-demicube|h4{4,33}]]
|
|[[File:Quarter_5-cubic_honeycomb_verf.png|60px]] [[Rectified 5-cell]] antiprism
|- align=center
| qδ7
| ''[[quarter 6-cubic honeycomb]]''
| q{4,34,4}
|{{CDD|nodes_10ru|split2|node|3|node|3|node|split1|nodes_10lu}} {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|4|node_h1}}
|[[File:Demihexeract_ortho_petrie.svg|60px]] [[6-demicube|h{4,34}]] ||[[File:6-demicube t04 D6.svg|60px]] [[Stericated 6-demicube|h5{4,34}]]
|{3,3}×{3,3}
|- align=center
| qδ8
| ''[[quarter 7-cubic honeycomb]]''
| q{4,35,4}
|{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|split1|nodes_10lu}} {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node|4|node_h1}}
|[[File:Demihepteract_ortho_petrie.svg|60px]] [[7-demicube|h{4,35}]] ||[[File:7-demicube t05 D7.svg|60px]] [[Pentellated 7-demicube|h6{4,35}]]
|{3,3}×{3,31,1}
|- align=center
| qδ9
| ''[[quarter 8-cubic honeycomb]]''
|q{4,36,4}
|{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|split1|nodes_10lu}} {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node|3|node|4|node_h1}}
|[[File:Demiocteract_ortho_petrie.svg|60px]] [[8-demicube|h{4,36}]] ||[[File:8-demicube t06 D8.svg|60px]] [[Hexellated 8-demicube|h7{4,36}]]
|{3,3}×{3,32,1} {3,31,1}×{3,31,1}
|- align=center
|colspan=7| 
|- align=center
| qδn
| ''quarter n-cubic honeycomb''
|q{4,3n-3,4}
|...
|h{4,3n-2}
|hn-2{4,3n-2}
|...
|}

== See also==
* [[Hypercubic honeycomb]]
* [[Alternated hypercubic honeycomb]]
* [[Simplectic honeycomb]]
* [[Truncated simplectic honeycomb]]
* [[Omnitruncated simplectic honeycomb]]

== References ==
{{reflist}}
* [[Coxeter|Coxeter, H.S.M.]] ''[[Regular Polytopes (book)|Regular Polytopes]]'', (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8
*# pp. 122-123, 1973. (The lattice of hypercubes γn form the ''cubic honeycombs'', δn+1)
*# pp. 154-156: Partial truncation or alternation, represented by ''q'' prefix
*# p. 296, Table II: Regular honeycombs, δn+1
* '''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] See p318 [http://books.google.com/books?id=fUm5Mwfx8rAC&lpg=PA318&ots=dnT1LYgmij&dq=%22quarter%20cubic%20honeycomb%22%20q%7B4%2C3%2C4%7D&pg=PA318#v=onepage&q=%22quarter%20cubic%20honeycomb%22%20q%7B4,3,4%7D&f=false]
* {{KlitzingPolytopes|flat.htm|1D-8D|Euclidean tesselations}}

{{Honeycombs}}

[[Category:Honeycombs (geometry)]]
[[Category:Polytopes]]