View source for Rectified 24-cell honeycomb

{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|Rectified 24-cell honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|(No image)
|-
|bgcolor=#e7dcc3|Type||[[Uniform 5-polytope#Regular and uniform honeycombs|Uniform 4-honeycomb]]
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||r{3,4,3,3}<BR>rr{3,3,4,3}<BR>r2r{4,3,3,4}<BR>r2r{4,3,3<sup>1,1</sup>}
|-
|bgcolor=#e7dcc3|Coxeter-Dynkin diagrams||
{{CDD|node|3|node_1|4|node|3|node|3|node}}<BR>
{{CDD|node_1|3|node|3|node_1|4|node|3|node}}<BR>
{{CDD|node|4|node_1|3|node|3|node_1|4|node}}<BR>
{{CDD|nodes_11|split2|node|3|node_1|4|node}} = {{CDD|node_h0|4|node_1|3|node|3|node_1|4|node}}<BR>
{{CDD|nodes_11|split2|node|split1|nodes_11}} = {{CDD|node|3|node_1|4|node_g|3sg|node_g|3g|node_g}}<BR>
{{CDD|nodes_11|split2|node|split1|nodes_11}} = {{CDD|node_h0|4|node_1|3|node|3|node_1|4|node_h0}}
|-
|bgcolor=#e7dcc3|4-face type||[[Tesseract]] [[File:Schlegel wireframe 8-cell.png|40px]]<BR>[[Rectified 24-cell|Rectified 24-cell]] [[File:Schlegel half-solid cantellated 16-cell.png|40px]]
|-
|bgcolor=#e7dcc3|Cell type||[[Cube]] [[File:Hexahedron.png|20px]]<BR>[[Cuboctahedron|Cuboctahedron]] [[File:Cuboctahedron.png|20px]]
|-
|bgcolor=#e7dcc3|Face type||Square<BR>[[Triangle]]
|-
|bgcolor=#e7dcc3|[[Vertex figure]]||[[File:Rectified 24-cell honeycomb verf.png|80px]]<BR>[[Tetrahedral prism]]
|-
|bgcolor=#e7dcc3|[[Coxeter group]]s||<math>{\tilde{F}}_4</math>, [3,4,3,3]<BR><math>{\tilde{C}}_4</math>, [4,3,3,4]<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|Properties||Vertex transitive
|}
In [[Four-dimensional space|four-dimensional]] [[Euclidean geometry]], the '''rectified 24-cell honeycomb''' is a uniform space-filling [[Honeycomb (geometry)|honeycomb]]. It is constructed by a [[Rectification (geometry)|rectification]] of the regular [[24-cell honeycomb|24-cell honeycomb]], containing [[Tesseract|tesseract]] and [[Rectified 24-cell|rectified 24-cell]] cells.

== Alternate names ==
* Rectified icositetrachoric tetracomb
* Rectified icositetrachoric honeycomb
* Cantellated 16-cell honeycomb
* Bicantellated tesseractic honeycomb

== Symmetry constructions ==

There are five different symmetry constructions of this tessellation. Each symmetry can be represented by different arrangements of colored [[Rectified 24-cell|rectified 24-cell]] and [[Tesseract|tesseract]] facets. The [[Tetrahedral prism|tetrahedral prism]] [[Vertex figure|vertex figure]] contains 4 rectified 24-cells capped by two opposite tesseracts.

{| class="wikitable"
![[Coxeter group]]
!Coxeter<BR>diagram
![[Facet (geometry)|Facets]]
![[Vertex figure]]
!Vertex<BR>figure<BR>symmetry<BR>(order)
|- align=center
|rowspan=2|<math>{\tilde{F}}_4</math><BR>= [3,4,3,3]
|{{CDD|node|3|node_1|4|node|3|node|3|node}}
|'''4:''' {{CDD|node|3|node_1|4|node|3|node}}<BR>'''1:''' {{CDD|node_1|4|node|3|node|3|node}}
|[[File:Rectified 24-cell honeycomb verf.png|80px]]
|{{CDD|node|3|node|3|node|2|node}}, [3,3,2]<BR>(48)
|- align=center
|{{CDD|node_1|3|node|3|node_1|4|node|3|node}}
|'''3:''' {{CDD|node|3|node|4|node_1|3|node}}<BR>'''1:''' {{CDD|node|4|node_1|3|node|3|node_1}}<BR>'''1:''' {{CDD|node|4|node|3|node_1|2|node_1}}
|[[File:Rectified 24-cell honeycomb F4b verf.png|80px]]
|{{CDD|node|3|node|2|node}}, [3,2]<BR>(12)
|- align=center
|<math>{\tilde{C}}_4</math><BR>= [4,3,3,4]
|{{CDD|node|4|node_1|3|node|3|node_1|4|node}}
|'''2,2:''' {{CDD|node_1|3|node|3|node_1|4|node}}<BR>'''1:''' {{CDD|node|4|node_1|2|node_1|4|node}}
|[[File:Bicantellated tesseractic honeycomb verf.png|80px]]
|{{CDD|node|2|node|2|node}}, [2,2]<BR>(8)
|- align=center
|<math>{\tilde{B}}_4</math><BR>= [3<sup>1,1</sup>,3,4]
|{{CDD|nodes_11|split2|node|3|node_1|4|node}}
|'''1,1:''' {{CDD|node_1|3|node|3|node_1|4|node}}<BR>'''2:''' {{CDD|nodes_11|split2|node|3|node_1}}<BR>'''1:''' {{CDD|node_1|2|node_1|2|node_1|4|node}}
|[[File:Rectified 24-cell honeycomb B4 verf.png|80px]]
|{{CDD|node|2|node}}, [2]<BR>(4)
|- align=center
|<math>{\tilde{D}}_4</math><BR>= [3<sup>1,1,1,1</sup>]
|{{CDD|nodes_11|split2|node|split1|nodes_11}}
|'''1,1,1,1:'''<BR>{{CDD|nodes_11|split2|node|3|node_1}}<BR>'''1:''' {{CDD|node_1|2|node_1|2|node_1|2|node_1}}
|[[File:Rectified 24-cell honeycomb D4 verf.png|80px]]
|{{CDD|node}}, []<BR>(2)
|}

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

== References ==
* Coxeter, H.S.M. ''Regular Polytopes'', (3rd edition, 1973), Dover edition, {{isbn|0-486-61480-8}} p.&nbsp;296, Table II: Regular honeycombs
* '''Kaleidoscopes: Selected Writings of [[Biography:Harold Scott MacDonald Coxeter|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)'' Model 93
* {{KlitzingPolytopes|flat.htm|4D|Euclidean tesselations}}, o3o3o4x3o, o4x3o3x4o - ricot - O93
{{Honeycombs}}
[[Category:5-polytopes]]
[[Category:Honeycombs (geometry)]]

{{Sourceattribution|Rectified 24-cell honeycomb}}