en>Malcolma: added Category:Topological vector spaces; removed {{uncategorized}} using HotCat

2014-02-03T17:02:10Z

added Category:Topological vector spaces; removed {{uncategorized}} using HotCat

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{| class="wikitable" align="right" style="margin-left:10px" width="320"
!bgcolor=#e7dcc3 colspan=2|{{PAGENAME}}
|-
|bgcolor=#ffffff align=center colspan=2|[[File:Hexakis cubic honeycomb.png|280px]]
|-
|bgcolor=#e7dcc3|Type||Dual uniform honeycomb
|-
|bgcolor=#e7dcc3|[[Coxeter–Dynkin diagram]]s||{{CDD|node|4|node_f1|3|node|4|node}}
|-
|bgcolor=#e7dcc3|[[Cell (geometry)|Cell]]
|[[Square bipyramid]]s [[File:Square_bipyramid.png|30px]]
|-
|bgcolor=#e7dcc3|[[Face]]s
|Triangles
|-
|bgcolor=#e7dcc3|[[Space group]] [[Fibrifold notation]]||Pm{{overline|3}}m (221) 4−:2
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{C}}_3</math>, [4,3,4]
|-
|bgcolor=#e7dcc3|[[vertex figure]]s
|[[File:hexahedron.png|30px]][[File:Rhombic dodecahedron.jpg|30px]] {{CDD|node_f1|3|node|4|node}}, {{CDD|node|4|node_f1|3|node}}
|-
|bgcolor=#e7dcc3|Dual||[[Rectified cubic honeycomb]]
|-
|bgcolor=#e7dcc3|Properties||[[Cell-transitive]], [[Face-transitive]]
|}
The '''square bipyramidal honeycomb''' is a uniform space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 3-space. [[John Horton Conway]] calls it an '''oblate octahedrille'''.

It can be seen as a [[cubic honeycomb]] with each cube subdivided by a center point into 6 elongated [[square pyramid]] cells. The the original cubic honeycomb walls are removed, joining pairs of square pyramids into elongated square bipyramids (octahedron). Its vertex and edge framework is identical to the [[hexakis cubic honeycomb]].

There is one type of plane with faces: a flattended [[triangular tiling]] with half of the triangles as ''holes''. These cut face-diagonally through the original cubes. There are also [[square tiling]] plane that exist as nonface ''holes'' passing through the centers of the octahedral cells.
{| class=wikitable
|- align=center
!Tiling plane
|[[File:Koushi.JPG|140px]] [[Square tiling]] "holes"
|[[File:Square bipyramidal honeycomb triangular plane.png|150px]] flattened [[triangular tiling]]
|- align=center
![[List of planar symmetry groups|Symmetry]]
|p4m, [4,4] (*442)
|pmm, [∞,2,∞] (*2222)
|}

== Related honeycombs==
It is dual to the [[rectified cubic honeycomb]] with octahedral and cuboctahedral cells:
:[[File:Rectified cubic honeycomb.png|160px]]
== See also==
*[[Architectonic and catoptric tessellation]]

== References ==
{{reflist}}
* [[John Horton Conway|John H. Conway]], Heidi Burgiel, Chaim Goodman-Strauss, (2008) ''The Symmetries of Things'', ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, pp 292–298, includes all the nonprismatic forms)
* [[Branko Grünbaum]], Uniform tilings of 3-space. [[Geombinatorics]] 4(1994), 49–56.

{{polychora-stub}}

[[Category:Honeycombs (geometry)]]

Webbed space - Revision history

en>Malcolma: added Category:Topological vector spaces; removed {{uncategorized}} using HotCat