Trihexagonal tiling
Trihexagonal tiling  

Type  Semiregular tiling 
Vertex configuration  (3.6)^{2} 
Schläfli symbol  r{6,3} h_{2}{6,3} 
Wythoff symbol  2  6 3 3 3  3 
Coxeter diagram  = 
Symmetry  p6m, [6,3], (*632) p3m1, [3^{[3]}, (*333) 
Rotation symmetry  p6, [6,3]^{+}, (632) p3, [3^{[3]}^{+}, (333) 
Bowers acronym  That 
Dual  Rhombille tiling 
Properties  Vertextransitive Edgetransitive 
Vertex figure: (3.6)^{2} 
In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. There are two triangles and two hexagons alternating on each vertex. It has Schläfli symbol of t_{1}{6,3}; its edges form an infinite arrangement of lines.^{1}^{2} It can also be constructed as a cantic hexagonal tiling, h_{2}{6,3}, if drawn by alternating two colors of triangles.
In physics as well as in Japanese basketry, the same pattern is called a Kagome lattice. Conway calls it a hexadeltille, combining alternate elements from a hexagonal tiling (hextille) and triangular tiling (deltille).^{3}
There are 3 regular and 8 semiregular tilings in the plane.
Contents
Kagome lattice
A kagome lattice originally referred to an arrangement of laths composed of interlaced triangles such that each point where two laths cross has four neighboring points, forming the pattern of a trihexagonal tiling. The name derives from the Japanese word kagome (籠目), a traditional woven bamboo pattern, composed from the words kago, meaning "basket", and me, literally meaning "eye(s)", referring to the pattern of holes in a woven basket. Although called a lattice, its crossing points do not form a mathematical lattice.
Some minerals, namely jarosites and herbertsmithite, contain layers with kagome lattice arrangement of atoms in their crystal structure. These minerals display novel physical properties connected with geometrically frustrated magnetism. The term is much in use nowadays in the scientific literature, especially by theorists studying the magnetic properties of a theoretical kagome lattice in two or three dimensions. The term "kagome lattice" in this context was coined by Japanese physicist Kōji Fushimi, who was working with Ichirō Shōji. The first paper^{4} on the subject appeared in 1951.^{5}
Uniform colorings
There are two distinct uniform colorings of a trihexagonal tiling. (Naming the colors by indices on the 4 faces around a vertex (3.6.3.6): 1212, 1232.)
Coloring  

Wythoff symbol  2  6 3  3 3  3 
CoxeterDynkin diagram  = 
Related polyhedra and tilings
A tiling with alternate large and small triangles is topologically identical to the trihexagonal tiling, but has a different symmetry group. The hexagons are distorted so 3 vertices are on the midedge of the larger triangles. As with the trihexagonal tiling, it has two uniform colorings:
The trihexagonal tiling is also one of eight uniform tilings that can be formed from the regular hexagonal tiling (or the dual triangular tiling) by a Wythoff construction. Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 which are topologically distinct. (The truncated triangular tiling is topologically identical to the hexagonal tiling.)
Symmetry: [6,3], (*632)  [6,3]^{+} (632) 
[1^{+},6,3] (*333) 
[6,3^{+} (3*3) 


{6,3}  t{6,3}  r{6,3} r{3^{[3]}} 
t{3,6} t{3^{[3]}} 
{3,6} {3^{[3]}} 
rr{6,3} s_{2}{6,3} 
tr{6,3}  sr{6,3}  h{6,3} {3^{[3]}} 
h_{2}{6,3} r{3^{[3]}} 
s{3,6} s{3^{[3]}} 
= 
= 
= 
= or 
= or 
= 

Uniform duals  
V6^{3}  V3.12^{2}  V(3.6)^{2}  V6^{3}  V3^{6}  V3.4.12.4  V.4.6.12  V3^{4}.6  V3^{6}  V(3.6)^{2}  V3^{6} 
Triangle symmetry 
Extended symmetry 
Extended diagram 
Extended order 
Honeycomb diagrams 

a1  [3^{[3]}  ×1  (None)  
i2  <[3^{[3]}> = [6,3] 
= 
×2  _{1}, _{2} 
r6  [3[3^{[3]}] = [6,3] 
= 
×6  _{3}, _{(1)} 
Wythoff  3  3 3  3 3  3  3  3 3  3 3  3  3  3 3  3 3  3  3 3 3    3 3 3 

Coxeter  
Image Vertex figure 
(3.3)^{3} 
3.6.3.6 
(3.3)^{3} 
3.6.3.6 
(3.3)^{3} 
3.6.3.6 
6.6.6 
3.3.3.3.3.3 
Symmetry *6n2 [n,6] 
Euclidean  Compact hyperbolic  Paracompact  Noncompact  

*632 [3,6] 
*642 [4,6] 
*652 [5,6] 
*662 [6,6] 
*762 [7,6] 
*862 [8,6]... 
*∞62 [∞,6] 
[iπ/λ,6] 

Coxeter  
Quasiregular figures configuration 
6.3.6.3 
6.4.6.4 
6.5.6.5 
6.6.6.6 
6.7.6.7 
6.8.6.8 
6.∞.6.∞ 
6.∞.6.∞ 
Dual figures  
Coxeter  
Dual (rhombic) figures configuration 
V6.3.6.3 
V6.4.6.4 
V6.5.6.5 
V6.6.6.6 
V6.7.6.7 
V6.8.6.8 
V6.∞.6.∞ 
Symmetry *n32 [1^{+},2n,3] = [(n,3,3)] 
Spherical  Planar  Compact Hyperbolic  Paracompact  

*332 [1^{+},4,3] T_{d} 
*333 [1^{+},6,3] P3m1 
*433 [1^{+},8,3] = [(4,3,3)] 
*533 [1^{+},10,3] = [(5,3,3)] 
*633 [1^{+},12,3]... = [(6,3,3)] 
*∞33 [1^{+},∞,3] = [(∞,3,3)] 

Cantic figure 
3.6.2.6 
3.6.3.6 
3.6.4.6 
3.6.5.6 
3.6.6.6 
3.6.∞.6 
Coxeter Schläfli 
h_{2}{4,3} = 
h_{2}{6,3} = 
h_{2}{8,3} = 
h_{2}{10,3} = 
h_{2}{12,3} = 
h_{2}{∞,3} = 
Dual figure  V3.6.2.6 
V3.6.3.6 
V3.6.4.6 
V3.6.5.6 
V3.6.6.6 
V3.6.∞.6 
Coxeter 
The trihexagonal tiling forms the case k = 6 in a sequence of quasiregular polyhedra and tilings, each of which has a vertex figure with two kgons and two triangles:
Symmetry *n32 [n,3] 
Spherical  Euclidean  Compact hyperbolic  Paracompact  Noncompact  

*332 [3,3] T_{d} 
*432 [4,3] O_{h} 
*532 [5,3] I_{h} 
*632 [6,3] p6m 
*732 [7,3] 
*832 [8,3]... 
*∞32 [∞,3] 
[iπ/λ,3] 

Quasiregular figures configuration 
3.3.3.3 
3.4.3.4 
3.5.3.5 
3.6.3.6 
3.7.3.7 
3.8.3.8 
3.∞.3.∞ 
3.∞.3.∞ 
Coxeter diagram  
Dual (rhombic) figures configuration 
V3.3.3.3 
V3.4.3.4 
V3.5.3.5 
V3.6.3.6 
V3.7.3.7 
V3.8.3.8 
V3.∞.3.∞ 

Coxeter diagram 
The subset of this sequence in which k is an even number has (*n33) reflectional symmetry.
See also
Wikimedia Commons has media related to Uniform tiling 3636. 
 Percolation threshold
 Star of David
 Truncated simplectic honeycomb
 Tilings of regular polygons
 List of uniform tilings
References
 ^ Grünbaum, Branko ; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0716711931. (Chapter 2.1: Regular and uniform tilings, p.5865)
 ^ Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 38. ISBN 048623729X.
 ^ John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, The Symmetries of Things 2008, ISBN 9781568812205 [1]
 ^ "I. Syôzi, Prog. Theor. Phys. 6, 306 (1951).".
 ^ "Physics Today article on the word kagome".
External links
Wikimedia Commons has media related to Kagome structures. 
 Richard Klitzing, 2D Euclidean tilings, o3x6o  that  O5
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