View source for Birectified 16-cell honeycomb

{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Birectified 16-cell honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|(No image)
|-
|bgcolor=#e7dcc3|Type||Uniform honeycomb
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>2</sub>{3,3,4,3}
|-
|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_h1}}<BR>{{CDD|node|3|node|splitsplit1|branch3_11|node_1}}
|-
|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]]
|-
|bgcolor=#e7dcc3|Cell type||[[Cube]] [[File:Hexahedron.png|20px]]<BR>[[Cuboctahedron]] [[File:Cuboctahedron.png|20px]]<BR>[[Tetrahedron]] [[File:Tetrahedron.png|20px]]
|-
|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]]
|-
|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>]
|-
|bgcolor=#e7dcc3|Dual||?
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|bgcolor=#e7dcc3|Properties||[[vertex-transitive]]
|}
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.

== Symmetry constructions==
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.

{| class=wikitable
|- align=center
!Affine [[Coxeter group]]
!<math>{\tilde{F}}_4</math><BR>[3,3,4,3]
!<math>{\tilde{B}}_4</math><BR>[4,3,3<sup>1,1</sup>]
!<math>{\tilde{D}}_4</math><BR>[3<sup>1,1,1,1</sup>]
|- align=center
![[Coxeter diagram]]
|{{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}}
|- align=center
![[Vertex figure]]
|[[File:Birectified_16-cell_honeycomb_verf.png|100px]]||[[File:Birectified_16-cell_honeycomb_verf2.png|100px]]||[[File:Birectified_16-cell_honeycomb_verf3.png|100px]]
|- align=center
!Vertex figure<BR>symmetry
|[3,2,3]<BR>(order 36)
|[3,2]<BR>(order 12)
|[3]<BR>(order 6)
|- align=center
!4-faces
|{{CDD|node|3|node|3|node_1|4|node}}<BR>{{CDD|node|3|node_1|4|node|3|node}}
|{{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}}
|{{CDD|node|splitsplit1|branch3_11|node_1}}<BR>{{CDD|node|3|node|split1|nodes_11}}
|- align=center
!Cells
|{{CDD|node|3|node_1|4|node}}<BR>{{CDD|node_1|4|node|3|node}}<BR>{{CDD|node|3|node|3|node_1}}
|{{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}}
|{{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}}
|}

== Related honeycombs==
{{F4 honeycombs}}

{{B4 honeycombs}}

{{D4 honeycombs}}

== See also ==
Regular and uniform honeycombs in 4-space:
*[[Tesseractic honeycomb]]
* [[16-cell honeycomb]]
*[[24-cell honeycomb]]
*[[Rectified 24-cell honeycomb]]
*[[Truncated 24-cell honeycomb]]
*[[Snub 24-cell honeycomb]]
*[[5-cell honeycomb]]
*[[Truncated 5-cell honeycomb]]
*[[Omnitruncated 5-cell honeycomb]]

==Notes==
{{reflist}}

== References ==
* '''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 24) H.S.M. Coxeter, ''Regular and Semi-Regular Polytopes III'', [Math. Zeit. 200 (1988) 3-45]
* George Olshevsky, ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)''
* {{KlitzingPolytopes|flat.htm|4D|Euclidean tesselations}} x3o3x *b3x *b3o, x3o3o *b3x4o, o3o3x4o3o - bricot - O106

{{Honeycombs}}

[[Category:Honeycombs (geometry)]]
[[Category:5-polytopes]]