View source for Trihexagonal tiling

{{short description|Tiling of the plane by regular hexagons and equilateral triangles}}
In [[Geometry|geometry]], the '''trihexagonal tiling''' is one of 11 [[Uniform tiling|uniform tiling]]s of the [[Euclidean plane]] by regular polygons.<ref name="t+p">{{cite book
 | last1 = Grünbaum
 | first1 = Branko
 | last2 = Shephard
 | first2 = G. C.
 | isbn = 978-0-7167-1193-3
 | publisher = W. H. Freeman
 | title = Tilings and Patterns
 | year = 1987
 | url-access = registration
 | url = https://archive.org/details/isbn_0716711931
 }} See in particular Theorem 2.1.3, p. 59 (classification of uniform tilings); Figure 2.1.5, p.63 (illustration of this tiling), Theorem 2.9.1, p. 103 (classification of colored tilings), Figure 2.9.2, p. 105 (illustration of colored tilings), Figure 2.5.3(d), p. 83 (topologically equivalent star tiling), and Exercise 4.1.3, p. 171 (topological equivalence of trihexagonal and two-triangle tilings).</ref> It consists of [[Equilateral triangle|equilateral triangle]]s and regular hexagons, arranged so that each hexagon is surrounded by triangles and vice versa. The name derives from the fact that it combines a regular [[Hexagonal tiling|hexagonal tiling]] and a regular [[Triangular tiling|triangular tiling]]. Two hexagons and two triangles alternate around each [[Vertex (geometry)|vertex]], and its edges form an infinite [[Arrangement of lines|arrangement of lines]]. Its dual is the [[Rhombille tiling|rhombille tiling]].<ref>{{The Geometrical Foundation of Natural Structure (book)|page=38}}</ref>

This pattern, and its place in the classification of uniform tilings, was already known to [[Biography:Johannes Kepler|Johannes Kepler]] in his 1619 book ''[[Physics:Harmonices Mundi|Harmonices Mundi]]''.<ref>{{cite book|title=The Harmony of the World by Johannes Kepler|volume=209|series=Memoirs of the American Philosophical Society|editor1-first=E. J.|editor1-last=Aiton|editor2-first=Alistair Matheson|editor2-last=Duncan|editor3-first=Judith Veronica|editor3-last=Field|editor3-link= Judith V. Field |publisher=American Philosophical Society|year=1997|isbn=978-0-87169-209-2|pages=104–105|url=https://books.google.com/books?id=rEkLAAAAIAAJ&pg=PA104}}.</ref> The pattern has long been used in Japanese basketry, where it is called '''kagome'''. The Japanese term for this pattern has been taken up in physics, where it is called a '''kagome lattice'''. It occurs also in the crystal structures of certain minerals. [[Biography:John Horton Conway|Conway]] calls it a '''hexadeltille''', combining alternate elements from a hexagonal tiling (hextille) and triangular tiling (deltille).<ref>{{cite book
 | last2 = Burgiel | first2 = Heidi
 | last3 = Goodman-Strauss | first3 = Chaim
 | contribution = Chapter 21: Naming Archimedean and Catalan polyhedra and tilings; Euclidean plane tessellations
 | isbn = 978-1-56881-220-5
 | location = Wellesley, MA
 | mr = 2410150
 | page = 288
 | publisher = A K Peters, Ltd.
 | title = The Symmetries of Things
 | year = 2008}}</ref>

==Kagome ==
[[Image:p12fig1.jpg|thumb|left|Japanese basket showing the kagome pattern]]
'''Kagome''' ({{lang-ja|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.

The kagome pattern is common in bamboo weaving in East Asia. In 2022, archaeologists found bamboo weaving remains at the Dongsunba ruins in Chongqing, China, 200 BC. After 2200 years, the kagome pattern is still clear.<ref name="CCTV2022">
{{Cite web
|last=China Central Television
|first=CCTV-13 News Channel
|date=2022-03-25
|title=[News Live Room] Bamboo weaving products of Ba culture first appeared in Chongqing about 2200 years ago |url=https://tv.cctv.com/2022/03/25/VIDERnzB5HyTqn1IMNTHCtnL220325.shtml
|access-date=2023-03-20 |website=tv.cctv.com}}
</ref><ref>
{{cite journal
 | last = Yin | first = Jia-Xin
 | date = March 2023
 | doi = 10.7693/wl20230301
 | issue = 3
 | journal = Physics (物理)
 | pages = 157–165
 | title = Exploring hitherto unknown quantum phases in kagome crystals
 | volume = 52
 }}</ref>
<gallery>
Image:Snowshoe2.jpg|Inuit [[Engineering:Snowshoe|snowshoe]]  Tagluk
File:Kagome lattice blue.svg|Kagome pattern in detail
</gallery>

It is a woven [[Arrangement of lines|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 woven process gives the Kagome a chiral [[Wallpaper group|wallpaper group]] symmetry, [[Wallpaper group#Group p6|p6]] (632).

==Kagome lattice==
The term '''kagome lattice''' was coined by Japanese physicist [[Biography:Kôdi Husimi|Kôdi Husimi]], and first appeared in a 1951 paper by his assistant Ichirō Shōji.<ref>{{cite journal
 | last = Mekata | first = Mamoru
 | date = February 2003
 | doi = 10.1063/1.1564329
 | issue = 2
 | journal = Physics Today
 | pages = 12–13
 | title = Kagome: The story of the basketweave lattice
 | volume = 56|bibcode = 2003PhT....56b..12M | doi-access = free
 }}</ref>
The kagome lattice in this sense consists of the vertices and edges of the trihexagonal tiling.
Despite the name, these crossing points do not form a [[Lattice (group)|mathematical lattice]].

A related three dimensional structure formed by the vertices and edges of the [[Quarter cubic honeycomb|quarter cubic honeycomb]], filling space by regular tetrahedra and truncated tetrahedra, has been called a ''hyper-kagome lattice''.<ref name=Lawler>{{cite journal
 | last1 = Lawler | first1 = Michael J.
 | last2 = Kee | first2 = Hae-Young
 | last3 = Kim | first3 = Yong Baek
 | last4 = Vishwanath | first4 = Ashvin
 | s2cid = 31984687
 | arxiv = 0705.0990
 | year = 2008
 | doi = 10.1103/physrevlett.100.227201
 | pmid = 18643453
 | issue = 22
 | page = 227201
 | journal = Physical Review Letters
 | title = Topological spin liquid on the hyperkagome lattice of Na<sub>4</sub>Ir<sub>3</sub>O<sub>8</sub>
 | volume = 100|bibcode = 2008PhRvL.100v7201L }}</ref> It is represented by the vertices and edges of the [[Quarter cubic honeycomb|quarter cubic honeycomb]], filling space by regular tetrahedra and truncated tetrahedra. It contains four sets of parallel planes of points and lines, each plane being a two dimensional kagome lattice. A second expression in three dimensions has parallel layers of two dimensional lattices and is called an ''orthorhombic-kagome lattice''.<ref name=Lawler /> The trihexagonal prismatic honeycomb represents its edges and vertices.

Some [[Earth:Mineral|mineral]]s, namely [[Chemistry:Jarosite|jarosite]]s and [[Chemistry:Herbertsmithite|herbertsmithite]], contain two-dimensional layers or three-dimensional kagome lattice arrangement of [[Physics:Atom|atom]]s in their [[Chemistry:Crystal structure|crystal structure]]. These minerals display novel physical properties connected with geometrically frustrated magnetism. For instance, the spin arrangement of the magnetic ions in Co<sub>3</sub>V<sub>2</sub>O<sub>8</sub> rests in a kagome lattice which exhibits fascinating magnetic behavior at low temperatures.<ref>{{Cite journal|author=Yen, F. |author2=Chaudhury, R. P. |author3=Galstyan, E. |author4=Lorenz, B. |author5=Wang, Y. Q. |author6=Sun, Y. Y. |author7=Chu, C. W. |s2cid=14958188|title=Magnetic phase diagrams of the Kagome staircase compound Co<sub>3</sub>V<sub>2</sub>O<sub>8</sub>|doi=10.1016/j.physb.2007.10.334|journal=Physica B: Condensed Matter|volume=403|issue=5–9|pages=1487–1489|year=2008|arxiv = 0710.1009 |bibcode = 2008PhyB..403.1487Y }}</ref> Quantum magnets realized on [[Physics:Kagome metal|Kagome metals]] have been discovered to exhibit many unexpected electronic and magnetic phenomena.<ref>{{Cite web|url=https://discovery.princeton.edu/2019/02/22/a-quantum-magnet-with-a-topological-twist/|title=A quantum magnet with a topological twist|date=2019-02-22|website=Discovery: Research at Princeton|language=en-US|access-date=2020-04-26}}</ref><ref>{{Cite journal|last1=Yin|first1=Jia-Xin|last2=Zhang|first2=Songtian S.|last3=Li|first3=Hang|last4=Jiang|first4=Kun|last5=Chang|first5=Guoqing|last6=Zhang|first6=Bingjing|last7=Lian|first7=Biao|last8=Xiang|first8=Cheng|last9=Belopolski |s2cid=205570556|date=2018|title=Giant and anisotropic many-body spin–orbit tunability in a strongly correlated kagome magnet|url=https://www.nature.com/articles/s41586-018-0502-7|journal=Nature |volume=562|issue=7725|pages=91–95|doi=10.1038/s41586-018-0502-7|pmid=30209398 |arxiv=1810.00218|bibcode=2018Natur.562...91Y}}</ref><ref>{{Cite journal|last1=Yin|first1=Jia-Xin|last2=Zhang|first2=Songtian S.|last3=Chang|first3=Guoqing|last4=Wang|first4=Qi|last5=Tsirkin|first5=Stepan S.|last6=Guguchia|first6=Zurab|last7=Lian|first7=Biao|last8=Zhou|first8=Huibin|last9=Jiang|first9=Kun|last10=Belopolski|first10=Ilya|last11=Shumiya |first11=Nana |s2cid=119363372|date=2019|title=Negative flat band magnetism in a spin–orbit-coupled correlated kagome magnet|url=https://www.nature.com/articles/s41567-019-0426-7|journal=Nature Physics |volume=15|issue=5|pages=443–8|doi=10.1038/s41567-019-0426-7 |arxiv=1901.04822|bibcode=2019NatPh..15..443Y}}</ref><ref>{{Cite journal|last=Yazyev|first=Oleg V.|s2cid=128299874|date=2019|title=An upside-down magnet|url=https://www.nature.com/articles/s41567-019-0451-6|journal=Nature Physics |volume=15|issue=5|pages=424–5|doi=10.1038/s41567-019-0451-6 |bibcode=2019NatPh..15..424Y}}</ref> It is also proposed that [[Physics:Sachdev–Ye–Kitaev model|SYK behavior]] can be observed in two dimensional kagome lattice with impurities.<ref>{{Cite journal|last1=Wei|first1=Chenan|last2=Sedrakyan|first2=Tigran|date=2021-01-29|title=Optical lattice platform for the Sachdev-Ye-Kitaev model|journal=Phys. Rev. A|volume=103|issue=1|page=013323|doi=10.1103/PhysRevA.103.013323|arxiv=2005.07640|bibcode=2021PhRvA.103a3323W|s2cid=234363891 }}</ref>

The term is much in use nowadays in the scientific literature, especially by theorists studying the magnetic properties of a theoretical kagome lattice.

See also:  [[Kagome crest]]s.

== Symmetry==
[[File:Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg|150px|thumb|30-60-90 triangle [[Fundamental domain|fundamental domain]]s of p6m (*632) symmetry]]
The trihexagonal tiling has [[Schläfli symbol]] of r{6,3}, or Coxeter diagram, {{CDD|node|6|node_1|3|node}}, symbolizing the fact that it is a [[Rectification (geometry)|rectified]] [[Hexagonal tiling|hexagonal tiling]], {6,3}. Its [[Symmetry|symmetries]] can be described by the [[Wallpaper group|wallpaper group]] p6mm, (*632),<ref>{{cite book|title=Crystallography of Quasicrystals: Concepts, Methods and Structures|volume=126|series=Springer Series in Materials Science|first1=Walter|last1=Steurer|first2=Sofia|last2=Deloudi|publisher=Springer|year=2009|isbn=978-3-642-01899-2|page=20|url=https://books.google.com/books?id=nVx-tu596twC&pg=PA20}}</ref> and the tiling can be derived as a [[Wythoff construction]] within the reflectional [[Fundamental domain|fundamental domain]]s of this group.  The trihexagonal tiling is a quasiregular tiling, alternating two types of polygons, with [[Vertex configuration|vertex configuration]] (3.6)<sup>2</sup>. It is also a [[Uniform tiling|uniform tiling]], one of eight derived from the regular hexagonal tiling.
{{clear}}
=== Uniform colorings ===
There are two distinct [[Uniform coloring|uniform coloring]]s of a trihexagonal tiling. Naming the colors by indices on the 4 faces around a vertex (3.6.3.6): 1212, 1232.<ref name="t+p"/> The second is called a '''cantic [[Hexagonal tiling|hexagonal tiling]]''', h<sub>2</sub>{6,3}, with two colors of triangles, existing in [[Wallpaper group#Group p3m1|p3m1]] (*333) symmetry.

{| class=wikitable
![[List of planar symmetry groups#Wallpaper groups|Symmetry]]
!p6m, (*632)
!p3m, (*333)
|-
!Coloring
|[[File:Uniform polyhedron-63-t1.png|150px]]
|[[File:Uniform tiling 333-t12.png|150px]]
|- align=center
![[Fundamental domain|fundamental<br />domain]]
|[[File:632 fundamental domain t1.png|80px]]
|[[File:333 fundamental domain t01.png|80px]]
|- align=center
![[Wythoff symbol|Wythoff]]
|2 &#124; 6 3
|3 3 &#124; 3
|- align=center
!Coxeter
|{{CDD|node|6|node_1|3|node}}
|{{CDD|branch_10ru|split2|node_1}} = {{CDD|node_h1|6|node|3|node_1}}
|- align=center
![[Schläfli symbol|Schläfli]]
| r{6,3}
| r{3<sup>[3]</sup>} = h<sub>2</sub>{6,3}
|}

=== Circle packing ===
The trihexagonal tiling  can be used as a [[Circle packing|circle packing]], placing equal diameter circles at the center of every point.<ref name=Critchlow>{{cite book |title=Order in Space: A design source book |first=Keith |last=Critchlow |pages=74–75 |chapter=pattern G |year=2000 |orig-date=1969 |publisher=Thames & Hudson |isbn=978-0-500-34033-2 }}</ref> Every circle is in contact with 4 other circles in the packing ([[Kissing number|kissing number]]). <!--The packing density is ... % coverage.)-->
:[[File:1-uniform-7-circlepack.svg|200px]]

== Topologically equivalent tilings==
The ''trihexagonal tiling'' can be geometrically distorted into topologically equivalent tilings of lower symmetry.<ref name="t+p"/> In these variants of the tiling, the edges do not necessarily line up to form straight lines.

{| class=wikitable width=300
!colspan=3|p3m1, (*333)
!p3, (333)
!colspan=2|p31m, (3*3)
!cmm, (2*22)
|-
|[[File:Trihexagonal tiling unequal.png|100px]]
|[[File:3-9-star-tiling.png|100px]]
|[[File:Hex-hexstar-tiling.svg|100px]]
|[[File:Trihexagonal tiling unequal2.svg|100px]]
|[[File:Distorted trihexagonal tiling.png|100px]]
|[[File:Triangle and triangular star tiling.png|100px]]
|[[File:Trihexagonal tiling in square tiling.svg|100px]]
|}

== Related quasiregular tilings==
The ''trihexagonal tiling'' exists in a sequence of symmetries of quasiregular tilings with [[Vertex configuration|vertex configuration]]s (3.''n'')<sup>2</sup>, progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With [[Orbifold notation|orbifold notation]] symmetry of *''n''32 all of these tilings are [[Wythoff construction|wythoff construction]] within a [[Fundamental domain|fundamental domain]] of symmetry, with generator points at the right angle corner of the domain.<ref>{{cite book |last=Coxeter |first=H.S.M. |title-link=Regular Polytopes (book) |title=Regular Polytopes |edition=3rd |year=1973 |publisher=Dover |isbn=0-486-61480-8 |chapter=V. The Kaleidoscope, §5.7 Wythoff's construction}}</ref><ref>{{cite CiteSeerX |title=Two Dimensional symmetry Mutations |first=Daniel H. |last=Huson |citeseerx=10.1.1.30.8536}}</ref>

== Related regular complex apeirogons ==

There are 2 regular complex apeirogons, sharing the vertices of the trihexagonal tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices. Regular apeirogons ''p''{''q''}''r'' are constrained by: 1/''p'' + 2/''q'' + 1/''r'' = 1. Edges have ''p'' vertices arranged like a [[Regular polygon|regular polygon]], and [[Vertex figure|vertex figure]]s are ''r''-gonal.<ref>{{cite book |last=Coxeter |first=H.S.M. |title=Regular Complex Polytopes |pages=111–2, 136 |isbn=978-0-521-39490-1 |publisher=Cambridge University Press |year=1991 |edition=2nd}}</ref>

The first is made of triangular edges, two around every vertex, second has hexagonal edges, two around every vertex.
{| class=wikitable
|-
|[[File:Complex apeirogon 3-12-2.png|160px]]
|[[File:Complex apeirogon 6-6-2.png|160px]]
|-
!3{12}2 or {{CDD|3node_1|12|node}}
!6{6}2 or {{CDD|6node_1|6|node}}
|}

== See also ==
* [[Percolation threshold]]
* [[Kagome crest]]
* [[Star of David]]
* Trihexagonal prismatic honeycomb
* [[Cyclotruncated simplectic honeycomb]]
* List of uniform tilings

== References ==
{{reflist}}

==Further reading==
* {{cite book |first1=Dale |last1=Seymour |first2=Jill |last2=Britton |title=Introduction to Tessellations |year=1989 |isbn=978-0-86651-461-3 |pages=50–56}}

{{Tessellation}}

[[Category:Euclidean tilings]]
[[Category:Isogonal tilings]]
[[Category:Isotoxal tilings]]
[[Category:Semiregular tilings]]
[[Category:Quasiregular polyhedra]]
[[Category:Crystallography]]

{{Sourceattribution|Trihexagonal tiling}}