Quarter cubic honeycomb
The quarter cubic honeycomb, quarter cubic cellulation or bitruncated alternated cubic honeycomb is a spacefilling tessellation (or honeycomb) in Euclidean 3space. It is composed of tetrahedra and truncated tetrahedra in a ratio of 1:1. It is called "quartercubic" because its symmetry unit – the minimal block from which the pattern is developed by reflections – consists of four such units of the cubic honeycomb.
Quarter cubic honeycomb  

Type  Uniform honeycomb 
Family  Truncated simplectic honeycomb Quarter hypercubic honeycomb 
Indexing[1]  J_{25,33}, A_{13} W_{10}, G_{6} 
Schläfli symbol  t_{0,1}{3^{[4]}} or q{4,3,4} 
CoxeterDynkin diagram  
Cell types  {3,3} (3.6.6) 
Face types  {3}, {6} 
Vertex figure  (isosceles triangular antiprism) 
Space group  Fd3m (227) 
Coxeter group  ×2_{2}, [[3^{[4]}]] 
Dual  oblate cubille Cell: (1/4 of rhombic dodecahedron) 
Properties  vertextransitive, edgetransitive 
It is vertextransitive with 6 truncated tetrahedra and 2 tetrahedra around each vertex.
A geometric honeycomb is a spacefilling of polyhedral or higherdimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
Honeycombs are usually constructed in ordinary Euclidean ("flat") space, like the convex uniform honeycombs. They may also be constructed in nonEuclidean spaces, such as hyperbolic uniform honeycombs. Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space.
It is one of the 28 convex uniform honeycombs.
The faces of this honeycomb's cells form four families of parallel planes, each with a 3.6.3.6 tiling.
Its vertex figure is an isosceles antiprism: two equilateral triangles joined by six isosceles triangles.
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.[2]
Construction
The quarter cubic honeycomb can be constructed in slab layers of truncated tetrahedra and tetrahedral cells, seen as two trihexagonal tilings. 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.
trihexagonal tiling: 
Symmetry
Cells can be shown in two different symmetries. The reflection generated form represented by its CoxeterDynkin diagram has two colors of truncated cuboctahedra. The symmetry can be doubled by relating the pairs of ringed and unringed nodes of the CoxeterDynkin diagram, which can be shown with one colored tetrahedral and truncated tetrahedral cells.
Symmetry  , [3^{[4]}]  ×2, [[3^{[4]}]] 

Space group  F43m (216)  Fd3m (227) 
Coloring  
Vertex figure  
Vertex figure symmetry 
C_{3v} [3] (*33) order 6 
D_{3d} [2^{+},6] (2*3) order 12 
Related polyhedra
The subset of hexagonal faces of this honeycomb contains a regular skew apeirohedron {6,63}. 
Four sets of parallel planes of trihexagonal tilings exist throughout this honeycomb. 
This honeycomb is one of five distinct uniform honeycombs[3] constructed by the Coxeter group. The symmetry can be multiplied by the symmetry of rings in the Coxeter–Dynkin diagrams:
A3 honeycombs  

Space group 
Fibrifold  Square symmetry 
Extended symmetry 
Extended diagram 
Extended group 
Honeycomb diagrams 
F43m (216) 
1^{o}:2  a1 
[3^{[4]}]  (None)  
Fm3m (225) 
2^{−}:2  d2 
<[3^{[4]}]> ↔ [4,3^{1,1}] 
↔ 
×2_{1} ↔ 

Fd3m (227) 
2^{+}:2  g2 
[[3^{[4]}]] or [2^{+}[3^{[4]}]] 
↔ 
×2_{2}  
Pm3m (221) 
4^{−}:2  d4 
<2[3^{[4]}]> ↔ [4,3,4] 
↔ 
×4_{1} ↔ 

I3 (204) 
8^{−o}  r8 
[4[3^{[4]}]]^{+} ↔ [[4,3<sup>+</sup>,4]] 
↔ 
½×8 ↔ ½×2 

Im3m (229) 
8^{o}:2  [4[3^{[4]}]] ↔ [[4,3,4]] 
×8 ↔ ×2 
C3 honeycombs  

Space group 
Fibrifold  Extended symmetry 
Extended diagram 
Order  Honeycombs 
Pm3m (221) 
4^{−}:2  [4,3,4]  ×1  
Fm3m (225) 
2^{−}:2  [1^{+},4,3,4] ↔ [4,3^{1,1}] 
↔ 
Half  
I43m (217) 
4^{o}:2  [[(4,3,4,2^{+})]]  Half × 2  
Fd3m (227) 
2^{+}:2  [[1^{+},4,3,4,1^{+}]] ↔ [[3^{[4]}]] 
↔ 
Quarter × 2  
Im3m (229) 
8^{o}:2  [[4,3,4]]  ×2 
The Quarter cubic honeycomb is related to a matrix of 3dimensional honeycombs: q{2p,4,2q}
Euclidean/hyperbolic(paracompact/noncompact) quarter honeycombs q{p,3,q}  

p \ q  4  6  8  ... ∞  
4  q{4,3,4} 
q{4,3,6} 
q{4,3,8} 
q{4,3,∞}  
6  q{6,3,4} 
q{6,3,6} 
q{6,3,8} 
q{6,3,∞}  
8  q{8,3,4} 
q{8,3,6} 
q{8,3,8} 
q{8,3,∞}  
... ∞  q{∞,3,4} 
q{∞,3,6} 
q{∞,3,8} 
q{∞,3,∞} 
See also
Wikimedia Commons has media related to Quarter cubic honeycomb. 
References
 For crossreferencing, they are given with list indices from Andreini (122), Williams(12,919), Johnson (1119, 2125, 3134, 4149, 5152, 6165), and Grünbaum(128).
 "Physics Today article on the word kagome".
 , OEIS sequence A000029 61 cases, skipping one with zero marks
 John H. Conway, Heidi Burgiel, Chaim GoodmanStrauss, (2008) The Symmetries of Things, ISBN 9781568812205 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292298, 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 3space. Geombinatorics 4(1994), 49  56.
 Norman Johnson Uniform Polytopes, Manuscript (1991)
 Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 048623729X.
 Critchlow, Keith (1970). Order in Space: A design source book. Viking Press. ISBN 0500340331.
 Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, WileyInterscience Publication, 1995, ISBN 9780471010036
 (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380407, MR 2,10] (1.9 Uniform spacefillings)
 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.
 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
 Klitzing, Richard. "3D Euclidean Honeycombs x3x3o3o3*a  batatoh  O27".
 Uniform Honeycombs in 3Space: 15Batatoh
Fundamental convex regular and uniform honeycombs in dimensions 29  

Space  Family  / /  
E^{2}  Uniform tiling  {3^{[3]}}  δ_{3}  hδ_{3}  qδ_{3}  Hexagonal 
E^{3}  Uniform convex honeycomb  {3^{[4]}}  δ_{4}  hδ_{4}  qδ_{4}  
E^{4}  Uniform 4honeycomb  {3^{[5]}}  δ_{5}  hδ_{5}  qδ_{5}  24cell honeycomb 
E^{5}  Uniform 5honeycomb  {3^{[6]}}  δ_{6}  hδ_{6}  qδ_{6}  
E^{6}  Uniform 6honeycomb  {3^{[7]}}  δ_{7}  hδ_{7}  qδ_{7}  2_{22} 
E^{7}  Uniform 7honeycomb  {3^{[8]}}  δ_{8}  hδ_{8}  qδ_{8}  1_{33} • 3_{31} 
E^{8}  Uniform 8honeycomb  {3^{[9]}}  δ_{9}  hδ_{9}  qδ_{9}  1_{52} • 2_{51} • 5_{21} 
E^{9}  Uniform 9honeycomb  {3^{[10]}}  δ_{10}  hδ_{10}  qδ_{10}  
E^{n1}  Uniform (n1)honeycomb  {3^{[n]}}  δ_{n}  hδ_{n}  qδ_{n}  1_{k2} • 2_{k1} • k_{21} 