en>BG19bot: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9876)

2014-01-23T06:22:52Z

WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9876)

New page

{| 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_f1|4|node_f1|3|node|4|node}}
|-
|bgcolor=#e7dcc3|[[Cell (geometry)|Cell]]
|Elongated [[square pyramid]] [[File:square pyramid.png|30px]]
|-
|bgcolor=#e7dcc3|[[Face]]s
|Triangle square
|-
|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||[[Truncated cubic honeycomb]]
|-
|bgcolor=#e7dcc3|Properties||[[Cell-transitive]]
|}
The '''hexakis cubic honeycomb''' is a uniform space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in Euclidean 3-space. It can be seen as a [[cubic honeycomb]] with each cube subdivided by a center point into 6 elongated [[square pyramid]] cells.

[[John Horton Conway]] calls this honeycomb a '''pyramidille'''.

There are two types of planes of faces: one as a [[square tiling]], and flattened [[triangular tiling]] with half of the triangles removed as ''holes''.
{| class=wikitable
|- align=center
!Tiling plane
|[[File:Square_tiling_uniform_coloring_1.png|140px]]
|[[File:Hexakis cubic honeycomb triangular plane.png|150px]]
|- align=center
![[List of planar symmetry groups|Symmetry]]
|p4m, [4,4] (*442)
|pmm, [∞,2,∞] (*2222)
|}
== Related honeycombs==
It is dual to the [[truncated cubic honeycomb]] with octahedral and truncated cubic cells:
:[[File:Truncated cubic honeycomb.png|160px]]

If the square pyramids of the '''pyramidille''' are [[Conway join operator|joined]] on their bases, another honeycomb is created with identical vertices and edges, called an [[square bipyramidal honeycomb]] or ''oblate octahedrille'', or the dual of the [[rectified cubic honeycomb]].

It is analogous to the 2-dimensional [[tetrakis square tiling]]:
:[[File:Tiling Dual Semiregular V4-8-8 Tetrakis Square.svg|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)]]

Peter Keevash - Revision history

en>BG19bot: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (9876)