Trihexagonal tiling

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Trihexagonal tiling
Trihexagonal tiling
Type Semiregular tiling
Vertex configuration (3.6)2
Schläfli symbol r{6,3}
h2{6,3}
Wythoff symbol 2 | 6 3
3 3 | 3
Coxeter diagram CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel branch 10ru.pngCDel split2.pngCDel node 1.png = CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
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 Vertex-transitive Edge-transitive
Trihexagonal tiling
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 t1{6,3}; its edges form an infinite arrangement of lines.[1][2] It can also be constructed as a cantic hexagonal tiling, h2{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.

Kagome lattice[edit]

Japanese basket showing the kagome pattern

Kagome (籠目) is a traditional Japanese woven bamboo pattern; its name is composed from the words kago, meaning "basket", and me, meaning "eye(s)", referring to the pattern of holes in a woven basket. A kagome lattice is 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. Despite the name, these 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[edit]

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 Uniform polyhedron-63-t1.png Uniform tiling 333-t01.png
Wythoff symbol 2 | 6 3 3 3 | 3
Coxeter-Dynkin diagram CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel branch 10ru.pngCDel split2.pngCDel node 1.png = CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
CDel branch 11.pngCDel split2.pngCDel node.png = CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png

Related polyhedra and tilings[edit]

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 mid-edge of the larger triangles. As with the trihexagonal tiling, it has two uniform colorings:

Distorted trihexagonal tiling.png Distorted trihexagonal tiling2.png

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.)

Uniform hexagonal/triangular tilings
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}
s2{6,3}
tr{6,3} sr{6,3} h{6,3}
{3[3]}
h2{6,3}
r{3[3]}
s{3,6}
s{3[3]}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel branch 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel branch 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel branch.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png =
CDel branch 10ru.pngCDel split2.pngCDel node.png or CDel branch 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png =
CDel branch 10ru.pngCDel split2.pngCDel node 1.png or CDel branch 01rd.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
= CDel branch hh.pngCDel split2.pngCDel node h.png
Uniform tiling 63-t0.png Uniform tiling 63-t01.png Uniform tiling 63-t1.png
Uniform tiling 333-t01.png
Uniform tiling 63-t12.png
Uniform tiling 333-t012.png
Uniform tiling 63-t2.png
Uniform tiling 333-t2.png
Uniform tiling 63-t02.png
Rhombitrihexagonal tiling snub edge coloring.png
Uniform tiling 63-t012.png Uniform tiling 63-snub.png Uniform tiling 333-t0.pngUniform tiling 333-t1.png Uniform tiling 333-t02.pngUniform tiling 333-t12.png Uniform tiling 63-h12.png
Uniform tiling 333-snub.png
Uniform duals
V63 V3.122 V(3.6)2 V63 V36 V3.4.12.4 V.4.6.12 V34.6 V36 V(3.6)2 V36
CDel node f1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 6.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node fh.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Uniform tiling 63-t2.png Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg Rhombic star tiling.png Uniform tiling 63-t2.png Uniform tiling 63-t0.png Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg Uniform tiling 63-t0.png Rhombic star tiling.png Uniform tiling 63-t0.png


Triangle
symmetry
Extended
symmetry
Extended
diagram
Extended
order
Honeycomb diagrams
a1 [3[3]] CDel node.pngCDel split1.pngCDel branch.png ×1 (None)
i2 <[3[3]]>
= [6,3]
CDel node c1.pngCDel split1.pngCDel branch c2.png
= CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 6.pngCDel node.png
×2 CDel node 1.pngCDel split1.pngCDel branch.png 1, CDel node.pngCDel split1.pngCDel branch 11.png 2
r6 [3[3[3]]]
= [6,3]
CDel node c1.pngCDel split1.pngCDel branch c1.png
= CDel node c1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
×6 CDel node 1.pngCDel split1.pngCDel branch 11.png 3, CDel node h.pngCDel split1.pngCDel branch hh.png (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 CDel node 1.pngCDel split1.pngCDel branch.png CDel node 1.pngCDel split1.pngCDel branch 10l.png CDel node.pngCDel split1.pngCDel branch 10l.png CDel node.pngCDel split1.pngCDel branch 11.png CDel node.pngCDel split1.pngCDel branch 01l.png CDel node 1.pngCDel split1.pngCDel branch 01l.png CDel node 1.pngCDel split1.pngCDel branch 11.png CDel node h.pngCDel split1.pngCDel branch hh.png
Image
Vertex figure
Uniform tiling 333-t0.png
(3.3)3
Uniform tiling 333-t01.png
3.6.3.6
Uniform tiling 333-t1.png
(3.3)3
Uniform tiling 333-t12.png
3.6.3.6
Uniform tiling 333-t2.png
(3.3)3
Uniform tiling 333-t02.png
3.6.3.6
Uniform tiling 333-t012.png
6.6.6
Uniform tiling 333-snub.png
3.3.3.3.3.3
Dimensional family of quasiregular polyhedra and tilings: 6.n.6.n
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 CDel node.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 6.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 6.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 6.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 6.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 6.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 6.pngCDel node.png CDel node.pngCDel ultra.pngCDel node 1.pngCDel 6.pngCDel node.png
Quasiregular
figures
configuration
Uniform polyhedron-63-t1.png
6.3.6.3
H2 tiling 246-2.png
6.4.6.4
H2 tiling 256-2.png
6.5.6.5
H2 tiling 266-2.png
6.6.6.6
H2 tiling 267-2.png
6.7.6.7
H2 tiling 268-2.png
6.8.6.8
H2 tiling 26i-2.png
6.∞.6.∞

6.∞.6.∞
Dual figures
Coxeter CDel node.pngCDel 3.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node.pngCDel ultra.pngCDel node f1.pngCDel 6.pngCDel node.png
Dual
(rhombic)
figures
configuration
Rhombic star tiling.png
V6.3.6.3
H2chess 246a.png
V6.4.6.4
Order-6-5 quasiregular rhombic tiling.png
V6.5.6.5
H2 tiling 246-4.png
V6.6.6.6

V6.7.6.7
H2chess 268a.png
V6.8.6.8
H2chess 26ia.png
V6.∞.6.∞
Dimensional family of cantic polyhedra and tilings: 3.6.n.6
Symmetry
*n32
[1+,2n,3]
= [(n,3,3)]
Spherical Planar Compact Hyperbolic Paracompact
*332
[1+,4,3]
Td
*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
Uniform polyhedron-33-t12.png
3.6.2.6
Uniform tiling 333-t12.png
3.6.3.6
H2 tiling 334-6.png
3.6.4.6
H2 tiling 335-6.png
3.6.5.6
H2 tiling 336-6.png
3.6.6.6
H2 tiling 33i-6.png
3.6.∞.6
Coxeter
Schläfli
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
h2{4,3}
= CDel nodes 10ru.pngCDel split2.pngCDel node 1.png
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
h2{6,3}
= CDel branch 10ru.pngCDel split2.pngCDel node 1.png
CDel node h1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node 1.png
h2{8,3}
= CDel label4.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png
CDel node h1.pngCDel 10.pngCDel node.pngCDel 3.pngCDel node 1.png
h2{10,3}
= CDel label5.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png
CDel node h1.pngCDel 12.pngCDel node.pngCDel 3.pngCDel node 1.png
h2{12,3}
= CDel label6.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png
CDel node h1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node 1.png
h2{∞,3}
= CDel labelinfin.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png
Dual figure Triakistetrahedron.jpg
V3.6.2.6
Rhombic star tiling.png
V3.6.3.6
Uniform dual tiling 433-t12.png
V3.6.4.6

V3.6.5.6

V3.6.6.6

V3.6.∞.6
Coxeter CDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 10.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 12.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node f1.png

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 k-gons and two triangles:

Dimensional family of quasiregular polyhedra and tilings: 3.n.3.n
Symmetry
*n32
[n,3]
Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
p6m
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
 
[iπ/λ,3]
Quasiregular
figures
configuration
Uniform tiling 332-t1-1-.png
3.3.3.3
Uniform tiling 432-t1.png
3.4.3.4
Uniform tiling 532-t1.png
3.5.3.5
Uniform tiling 63-t1.png
3.6.3.6
Uniform tiling 73-t1.png
3.7.3.7
Uniform tiling 83-t1.png
3.8.3.8
H2 tiling 23i-2.png
3.∞.3.∞
3.∞.3.∞
Coxeter diagram CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel ultra.pngCDel node 1.pngCDel 3.pngCDel node.png
Dual
(rhombic)
figures
configuration
Hexahedron.svg
V3.3.3.3
Rhombicdodecahedron.jpg
V3.4.3.4
Rhombictriacontahedron.svg
V3.5.3.5
Rhombic star tiling.png
V3.6.3.6
Order73 qreg rhombic til.png
V3.7.3.7
Uniform dual tiling 433-t01-yellow.png
V3.8.3.8
Ord3infin qreg rhombic til.png
V3.∞.3.∞
Coxeter diagram CDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 7.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 8.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel ultra.pngCDel node f1.pngCDel 3.pngCDel node.png

The subset of this sequence in which k is an even number has (*n33) reflectional symmetry.

Rhombille tiling[edit]

The rhombille tiling
Main article: Rhombille tiling

The rhombille tiling,[6] also known as tumbling blocks, reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 60° and two 120° angles; rhombi with this shape are sometimes also called diamonds. Sets of three rhombi meet at their 120° angles and sets of six rhombi meet at their 60° angles. It is the dual tiling of the trihexagonal tiling.

See also[edit]

References[edit]

  1. ^ Grünbaum, Branko ; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1.  (Chapter 2.1: Regular and uniform tilings, p.58-65)
  2. ^ Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 38. ISBN 0-486-23729-X. 
  3. ^ John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]
  4. ^ "I. Syôzi, Prog. Theor. Phys. 6, 306 (1951).". 
  5. ^ "Physics Today article on the word kagome". 
  6. ^ Conway, John; Burgiel, Heidi; Goodman-Strass, Chaim (2008), "Chapter 21: Naming Archimedean and Catalan polyhedra and tilings", The Symmetries of Things, AK Peters, p. 288, ISBN 978-1-56881-220-5 .

External links[edit]