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| | I'm Siobhan and I live with my husband and our two children in Mollands, in the south part. My hobbies are Gardening, Gaming and Cubing.<br><br>My webpage; [http://tinyurl.com/o8ux5y6 http://tinyurl.com/o8ux5y6] |
| !bgcolor=#e7dcc3 colspan=2|Birectified 16-cell honeycomb
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| |bgcolor=#ffffff align=center colspan=2|(No image)
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| |bgcolor=#e7dcc3|Type||Uniform honeycomb
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| |bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>2</sub>{3,3,4,3}
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| |bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|node|3|node|3|node_1|4|node|3|node}}<BR>{{CDD|node|4|node_1|3|node|split1|nodes_10lu}} = {{CDD|node|4|node_1|3|node|3|node|4|node_h}}<BR>{{CDD|node|3|node|splitsplit1|branch3_11|node_1}}
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| |-
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| |bgcolor=#e7dcc3|4-face type||[[Rectified tesseract]] [[File:Schlegel half-solid rectified 8-cell.png|40px]]<BR>[[Rectified 24-cell]] [[File:Schlegel half-solid cantellated 16-cell.png|40px]]
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| |-
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| |bgcolor=#e7dcc3|Cell type||[[Cube]] [[File:Hexahedron.png|20px]]<BR>[[Cuboctahedron]] [[File:Cuboctahedron.png|20px]]<BR>[[Tetrahedron]] [[File:Tetrahedron.png|20px]]
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| |-
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| |bgcolor=#e7dcc3|Face type||{3}, {4}
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| |bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Birectified_16-cell_honeycomb_verf.png|120px]]<BR>{3}×{3} [[duoprism]]
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| |bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{F}}_4</math> = [3,3,4,3]<BR><math>{\tilde{B}}_4</math> = [4,3,3<sup>1,1</sup>]<BR><math>{\tilde{D}}_4</math> = [3<sup>1,1,1,1</sup>]
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| |bgcolor=#e7dcc3|Dual||?
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| |bgcolor=#e7dcc3|Properties||[[vertex-transitive]]
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| |}
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| In [[Four-dimensional space|four-dimensional]] [[Euclidean geometry]], the '''birectified 16-cell honeycomb''' (or '''runcic tesseractic honeycomb''') is a uniform space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 4-space.
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| == Symmetry constructions==
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| There are 3 different symmetry constructions, all with 3-3 [[duoprism]] vertex figures. The <math>{\tilde{B}}_4</math> symmetry doubles on <math>{\tilde{D}}_4</math> in three possible ways, while <math>{\tilde{F}}_4</math> contains the highest symmetry.
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| {| class=wikitable
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| |- align=center
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| !Affine [[Coxeter group]]
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| !<math>{\tilde{F}}_4</math><BR>[3,3,4,3]
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| !<math>{\tilde{B}}_4</math><BR>[4,3,3<sup>1,1</sup>]
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| !<math>{\tilde{D}}_4</math><BR>[3<sup>1,1,1,1</sup>]
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| |- align=center
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| ![[Coxeter diagram]]
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| |{{CDD|node|3|node|3|node_1|4|node|3|node}}||{{CDD|node|4|node_1|3|node|split1|nodes_10lu}}||{{CDD|node|3|node|splitsplit1|branch3_11|node_1}}
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| |- align=center
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| ![[Vertex figure]]
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| |[[File:Birectified_16-cell_honeycomb_verf.png|100px]]||[[File:Birectified_16-cell_honeycomb_verf2.png|100px]]||[[File:Birectified_16-cell_honeycomb_verf3.png|100px]]
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| |- align=center
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| !Vertex figure<BR>symmetry
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| |[3,2,3]<BR>(order 36)
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| |[3,2]<BR>(order 12)
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| |[3]<BR>(order 6)
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| |- align=center
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| !4-faces
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| |{{CDD|node|3|node|3|node_1|4|node}}<BR>{{CDD|node|3|node_1|4|node|3|node}}
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| |{{CDD|node_1|3|node|split1|nodes_10lu}}<BR>{{CDD|node|4|node_1|3|node|3|node_1}}<BR>{{CDD|node|4|node_1|3|node|3|node}}
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| |{{CDD|node|splitsplit1|branch3_11|node_1}}<BR>{{CDD|node|3|node|split1|nodes_11}}
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| |- align=center
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| !Cells
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| |{{CDD|node|3|node_1|4|node}}<BR>{{CDD|node_1|4|node|3|node}}<BR>{{CDD|node|3|node|3|node_1}}
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| |{{CDD|node_1|3|node|3|node_1}}<BR>{{CDD|node|4|node_1|2|node_1}}<BR>{{CDD|node_1|3|node|3|node}}<BR>{{CDD|node|4|node_1|3|node}}
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| |{{CDD|node_1|3|node|3|node_1}}<BR>{{CDD|node_1|2|node_1|2|node_1}}<BR>{{CDD|node_1|3|node|3|node}}
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| |}
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| == Related honeycombs==
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| {{F4 honeycombs}}
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| {{B4 honeycombs}}
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| {{D4 honeycombs}}
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| == See also ==
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| Regular and uniform honeycombs in 4-space:
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| *[[Tesseractic honeycomb]]
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| * [[16-cell honeycomb]]
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| *[[24-cell honeycomb]]
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| *[[Rectified 24-cell honeycomb]]
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| *[[Truncated 24-cell honeycomb]]
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| *[[Snub 24-cell honeycomb]]
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| *[[5-cell honeycomb]]
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| *[[Truncated 5-cell honeycomb]]
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| *[[Omnitruncated 5-cell honeycomb]]
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| ==Notes==
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| {{reflist}}
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| == References ==
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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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| * [[George Olshevsky]], ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)''
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| * {{KlitzingPolytopes|flat.htm|4D|Euclidean tesselations}} x3o3x *b3x *b3o, x3o3o *b3x4o, o3o3x4o3o - bricot - O106
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| {{Honeycombs}}
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| [[Category:Honeycombs (geometry)]]
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| [[Category:5-polytopes]]
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I'm Siobhan and I live with my husband and our two children in Mollands, in the south part. My hobbies are Gardening, Gaming and Cubing.
My webpage; http://tinyurl.com/o8ux5y6