View source for Quarter cubic honeycomb

{| class="wikitable" align="right" style="margin-left:10px" width="340"
!bgcolor=#e7dcc3 colspan=2|Quarter cubic honeycomb
|-
|bgcolor=#ffffff align=center colspan=2|[[Image:Bitruncated alternated cubic tiling.png|200px]]&nbsp;[[File:HC A1-P1.png|120px]]
|-
|bgcolor=#e7dcc3|Type||[[Convex uniform honeycomb|Uniform honeycomb]]
|-
|bgcolor=#e7dcc3|Family||[[Truncated simplectic honeycomb]]<BR>[[Quarter hypercubic honeycomb]]
|-
|bgcolor=#e7dcc3|Indexing<ref>For cross-referencing, they are given with list indices from Andreini (1-22), Williams(1-2,9-19), Johnson (11-19, 21-25, 31-34, 41-49, 51-52, 61-65), and Grünbaum(1-28).</ref>
|J<sub>25,33</sub>, A<sub>13</sub><BR>W<sub>10</sub>, G<sub>6</sub>
|-
|bgcolor=#e7dcc3|[[Schläfli symbol]]||t<sub>0,1</sub>{3<sup>[4]</sup>} or q{4,3,4}
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||{{CDD|branch_11|3ab|branch}} = {{CDD|node_h1|4|node|split1|nodes_10lu}} = {{CDD|branch|4a4b|nodes_h1h1}}
|-
|bgcolor=#e7dcc3|Cell types||[[tetrahedron|{3,3}]] [[Image:Tetrahedron.png|25px]]<BR>[[truncated tetrahedron|(3.6.6)]] [[Image:Truncated tetrahedron.png|25px]]
|-
|bgcolor=#e7dcc3|Face types||[[triangle|{3}]], [[hexagon|{6}]]
|-
|bgcolor=#e7dcc3|Vertex figure||[[File:T01 quarter cubic honeycomb verf.png|80px]]<BR>(isosceles [[triangular antiprism]])
|-
|bgcolor=#e7dcc3|[[Space group]]||Fd{{overline|3}}m (227)
|-
|bgcolor=#e7dcc3|[[Coxeter group]]||<math>{\tilde{A}}_3</math>×2<sub>2</sub>, [[3<sup>[4]</sup>]]
|-
|bgcolor=#e7dcc3|[[Dual polytope|Dual]]||[[oblate cubille]]<BR>Cell: [[File:Oblate cubille cell.png|80px]]<BR>(1/4 of rhombic dodecahedron)
|-
|bgcolor=#e7dcc3|Properties||[[vertex-transitive]], [[edge-transitive]]
|}
The '''quarter cubic honeycomb''', '''quarter cubic cellulation''' or '''bitruncated alternated cubic honeycomb''' is a space-filling [[tessellation]] (or [[honeycomb (geometry)|honeycomb]]) in [[Euclidean 3-space]].  It is composed of [[tetrahedron|tetrahedra]] and [[truncated tetrahedron|truncated tetrahedra]] in a ratio of 1:1.  It is called "quarter-cubic" because its symmetry unit – the minimal block from which the pattern is developed by reflections – is four times that of the [[cubic honeycomb]].

It is [[vertex-transitive]] with 6 [[truncated tetrahedron|truncated tetrahedra]] and 2 [[Tetrahedron|tetrahedra]] around each vertex.

{{Honeycomb}}

It is one of the 28 [[convex uniform honeycomb]]s.

The faces of this honeycomb's cells form four families of parallel planes, each with a [[Trihexagonal tiling|3.6.3.6 tiling]].

Its [[vertex figure]] is an isosceles [[antiprism]]: two [[equilateral triangle]]s joined by six [[isosceles triangle]]s.

[[John Horton Conway]] calls this honeycomb a '''truncated tetrahedrille''', and its dual [[oblate cubille]].

The vertices and edges represent a '''[[Kagome lattice]]''' in three dimensions,<ref>{{cite web | title=Physics Today article on the word ''kagome''|url=http://scitation.aip.org/journals/doc/PHTOAD-ft/vol_56/iss_2/12_1.shtml}}</ref> which is the '''[[pyrochlore]]''' lattice.

== Construction==

The quarter cubic honeycomb can be constructed in slab layers of truncated tetrahedra and tetrahedral cells, seen as two [[trihexagonal tiling]]s. Two tetrahedra are stacked by a vertex and a [[central inversion]]. In each [[trihexagonal tiling]], half of the triangles belong to tetrahedra, and half belong to truncated tetrahedra. These slab layers must be stacked with tetrahedra triangles to truncated tetrahedral triangles to construct the uniform ''quarter cubic honeycomb''. Slab layers of hexagonal prisms and triangular prisms can be alternated for ''elongated'' honeycombs, but these are also not uniform.
{| class=wikitable
|[[File:Tetrahedral-truncated tetrahedral honeycomb slab.png|240px]]
||[[File:Uniform polyhedron-63-t1-1.svg|120px]]<BR>trihexagonal tiling: {{CDD|node|split1|branch_11}}
|}

== Symmetry ==

Cells can be shown in two different symmetries. The reflection generated form represented by its [[Coxeter-Dynkin diagram]] has two colors of [[truncated cuboctahedron|truncated cuboctahedra]]. The symmetry can be doubled by relating the pairs of ringed and unringed nodes of the Coxeter-Dynkin diagram, which can be shown with one colored tetrahedral and truncated tetrahedral cells.
{| class="wikitable"
|+ Two uniform colorings
|-
![[Coxeter notation|Symmetry]]
!<math>{\tilde{A}}_3</math>, [3<sup>[4]</sup>]
!<math>{\tilde{A}}_3</math>×2, [[3<sup>[4]</sup>]]
|-
![[Space group]]
!F{{overline|4}}3m (216)
!Fd{{overline|3}}m (227)
|- align=center
!Coloring
|[[Image:Quarter cubic honeycomb.png|100px]]
|[[Image:Quarter cubic honeycomb2.png|100px]]
|- align=center
![[Vertex figure]]
|[[File:T01 quarter cubic honeycomb verf.png|80px]]
|[[File:T01 quarter cubic honeycomb verf2.png|80px]]
|- align=center
!Vertex<BR>figure<BR>[[List of spherical symmetry groups|symmetry]]
|C<sub>3v</sub><BR>[3]<BR>(*33)<BR>order 6
|D<sub>3d</sub><BR>[2<sup>+</sup>,6]<BR>(2*3)<BR>order 12
|}

== Related polyhedra ==

{| class=wikitable width=480
|[[File:Mutetrahedron.png|240px]]<BR>The subset of hexagonal faces of this honeycomb contains a [[regular skew apeirohedron]] {6,6|3}.
|[[File:Tiling Semiregular 3-6-3-6 Trihexagonal.svg|240px]]<BR>Four sets of parallel planes of [[trihexagonal tiling]]s exist throughout this honeycomb.
|}

This honeycomb is one of five distinct uniform honeycombs<ref>[http://mathworld.wolfram.com/Necklace.html], {{OEIS el|1=A000029}} 6-1 cases, skipping one with zero marks</ref> constructed by the <math>{\tilde{A}}_3</math> [[Coxeter group]]. The symmetry can be multiplied by the symmetry of rings in the [[Coxeter–Dynkin diagram]]s: {{A3 honeycombs}}

{{C3 honeycombs}}

The Quarter cubic honeycomb is related to a matrix of 3-dimensional honeycombs: q{2p,4,2q}
{{Quarter hyperbolic honeycomb table}}

== See also ==
{{Commons category|Quarter cubic honeycomb}}
* [[Truncated simplectic honeycomb]]
* [[Triakis truncated tetrahedral honeycomb]]
* [[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, p 292-298, includes all the nonprismatic forms)
* George Olshevsky, ''Uniform Panoploid Tetracombs'', Manuscript (2006) ''(Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs)''
* [[Branko Grünbaum]], Uniform tilings of 3-space. [[Geombinatorics]] 4(1994), 49 - 56.
* [[Norman Johnson (mathematician)|Norman Johnson]] ''Uniform Polytopes'', Manuscript (1991)
* {{The Geometrical Foundation of Natural Structure (book)}}
* {{cite book | first=Keith | last=Critchlow | authorlink=Keith Critchlow  | title=Order in Space: A design source book | publisher=Viking Press| year=1970 | isbn=0-500-34033-1 }}
* '''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 22) H.S.M. Coxeter, ''Regular and Semi Regular Polytopes I'', [Math. Zeit. 46 (1940) 380-407, MR 2,10] (1.9 Uniform space-fillings)
* [[Alfredo Andreini|A. Andreini]], ''Sulle reti di poliedri regolari e semiregolari e sulle corrispondenti reti correlative'' (On the regular and semiregular nets of polyhedra and on the corresponding correlative nets), Mem. Società Italiana della Scienze, Ser.3, 14 (1905) 75–129.
* [[Duncan MacLaren Young Sommerville|D. M. Y. Sommerville]], ''An Introduction to the Geometry of '''n''' Dimensions.'' New York, E. P. Dutton, 1930. 196 pp. (Dover Publications edition, 1958) Chapter X: The Regular Polytopes
* {{KlitzingPolytopes|flat.htm|3D Euclidean Honeycombs|x3x3o3o3*a - batatoh - O27}}
* [http://www.doskey.com/polyhedra/UniformHoneycombs.html Uniform Honeycombs in 3-Space: 15-Batatoh]
{{Honeycombs}}

[[Category:3-honeycombs]]