Trihexagonal tiling | |
---|---|

| |

Type | Semiregular tiling |

Vertex configuration | (3.6) ^{2} |

Schläfli symbol | r{6,3} or h _{2}{6,3} |

Wythoff symbol | 2 | 6 3 3 3 | 3 |

Coxeter diagram | = |

Symmetry | p6m, [6,3], (*632) |

Rotation symmetry | p6, [6,3]^{+}, (632)p3, [3 ^{[3]}]^{+}, (333) |

Bowers acronym | That |

Dual | Rhombille tiling |

Properties | Vertex-transitive Edge-transitive |

In geometry, the **trihexagonal tiling** is one of 11 uniform tilings of the Euclidean plane by regular polygons.^{ [1] } It consists of equilateral triangles 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 and a regular triangular tiling. Two hexagons and two triangles alternate around each vertex, and its edges form an infinite arrangement of lines. Its dual is the rhombille tiling.^{ [2] }

- Kagome
- Kagome lattice
- Symmetry
- Uniform colorings
- Circle packing
- Topologically equivalent tilings
- Related quasiregular tilings
- Related regular complex apeirogons
- See also
- References
- Further reading

This pattern, and its place in the classification of uniform tilings, was already known to Johannes Kepler in his 1619 book * Harmonices Mundi *.^{ [3] } 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. Conway calls it a **hexadeltille**, combining alternate elements from a hexagonal tiling (hextille) and triangular tiling (deltille).^{ [4] }

**Kagome** (Japanese : 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.^{ [5] }^{ [6] }

- kagome pattern in detail

It is a woven 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 symmetry, p6, (632).

The term **kagome lattice** was coined by Japanese physicist Kôdi Husimi, and first appeared in a 1951 paper by his assistant Ichirō Shōji.^{ [7] } 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 mathematical lattice.

A related three dimensional structure formed by the vertices and edges of the quarter cubic honeycomb, filling space by regular tetrahedra and truncated tetrahedra, has been called a *hyper-kagome lattice*.^{ [8] } It is represented by the vertices and edges of the 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*.^{ [8] } The trihexagonal prismatic honeycomb represents its edges and vertices.

Some minerals, namely jarosites and herbertsmithite, contain two-dimensional layers or three-dimensional kagome lattice arrangement of atoms in their crystal structure. These minerals display novel physical properties connected with geometrically frustrated magnetism. For instance, the spin arrangement of the magnetic ions in Co_{3}V_{2}O_{8} rests in a kagome lattice which exhibits fascinating magnetic behavior at low temperatures.^{ [9] } Quantum magnets realized on Kagome metals have been discovered to exhibit many unexpected electronic and magnetic phenomena.^{ [10] }^{ [11] }^{ [12] }^{ [13] } It is also proposed that SYK behavior can be observed in two dimensional kagome lattice with impurities.^{ [14] }

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 crests.

The trihexagonal tiling has Schläfli symbol of r{6,3}, or Coxeter diagram, , symbolizing the fact that it is a rectified hexagonal tiling, {6,3}. Its symmetries can be described by the wallpaper group p6mm, (*632),^{ [15] } and the tiling can be derived as a Wythoff construction within the reflectional fundamental domains of this group. The trihexagonal tiling is a quasiregular tiling, alternating two types of polygons, with vertex configuration (3.6)^{2}. It is also a uniform tiling, one of eight derived from the regular hexagonal tiling.

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.^{ [1] } The second is called a **cantic hexagonal tiling **, h_{2}{6,3}, with two colors of triangles, existing in p3m1 (*333) symmetry.

Symmetry | p6m, (*632) | p3m, (*333) |
---|---|---|

Coloring | ||

fundamental domain | ||

Wythoff | 2 | 6 3 | 3 3 | 3 |

Coxeter | = | |

Schläfli | r{6,3} | r{3^{[3]}} = h_{2}{6,3} |

The trihexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point.^{ [16] } Every circle is in contact with 4 other circles in the packing (kissing number).

The *trihexagonal tiling* can be geometrically distorted into topologically equivalent tilings of lower symmetry.^{ [1] } In these variants of the tiling, the edges do not necessarily line up to form straight lines.

p3m1, (*333) | p3, (333) | p31m, (3*3) | cmm, (2*22) | |||
---|---|---|---|---|---|---|

The *trihexagonal tiling* exists in a sequence of symmetries of quasiregular tilings with vertex configurations (3.*n*)^{2}, progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With orbifold notation symmetry of **n*32 all of these tilings are wythoff construction within a fundamental domain of symmetry, with generator points at the right angle corner of the domain.^{ [17] }^{ [18] }

*n32 orbifold symmetries of quasiregular tilings: (3.n)^{2} | |||||||
---|---|---|---|---|---|---|---|

Construction | Spherical | Euclidean | Hyperbolic | ||||

*332 | *432 | *532 | *632 | *732 | *832... | *∞32 | |

Quasiregular figures | |||||||

Vertex | (3.3)^{2} | (3.4)^{2} | (3.5)^{2} | (3.6)^{2} | (3.7)^{2} | (3.8)^{2} | (3.∞)^{2} |

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, and vertex figures are *r*-gonal.^{ [19] }

The first is made of triangular edges, two around every vertex, second has hexagonal edges, two around every vertex.

3{12}2 or | 6{6}2 or |
---|

Wikimedia Commons has media related to Kagome structures .

A **cuboctahedron** is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral.

In geometry, an **octahedron** is a polyhedron with eight faces. The term is most commonly used to refer to the **regular octahedron**, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

In four-dimensional geometry, a **runcinated 5-cell** is a convex uniform 4-polytope, being a runcination of the regular 5-cell.

In geometry, the **hexagonal tiling** or **hexagonal tessellation** is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or *t*{3,6} .

In geometry, the **square tiling**, **square tessellation** or **square grid** is a regular tiling of the Euclidean plane. It has Schläfli symbol of {4,4}, meaning it has 4 squares around every vertex. Conway called it a **quadrille**.

In geometry, the **triangular tiling** or **triangular tessellation** is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of {3,6}.

In geometry, the **truncated trihexagonal tiling** is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of *tr*{3,6}.

In geometry, the **rhombitrihexagonal tiling** is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr{3,6}.

The **cubic honeycomb** or **cubic cellulation** is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a **cubille**.

The **tetrahedral-octahedral honeycomb**, **alternated cubic honeycomb** is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.

The **quarter cubic honeycomb**, **quarter cubic cellulation** or **bitruncated alternated cubic honeycomb** is a space-filling tessellation in Euclidean 3-space. It is composed of tetrahedra and 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.

The **triangular prismatic honeycomb** or **triangular prismatic cellulation** is a space-filling tessellation in Euclidean 3-space. It is composed entirely of triangular prisms.

In geometry, a **quasiregular polyhedron** is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

In geometry, a **uniform tiling** is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.

In geometry, the **order-3 snub octagonal tiling** is a semiregular tiling of the hyperbolic plane. There are four triangles, one octagon on each vertex. It has Schläfli symbol of *sr{8,3}*.

In the field of hyperbolic geometry, the **hexagonal tiling honeycomb** is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is *paracompact* because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere, a surface in hyperbolic space that approaches a single ideal point at infinity.

In the field of hyperbolic geometry, the **order-4 hexagonal tiling honeycomb** arises as one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is *paracompact* because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.

In the field of hyperbolic geometry, the **order-6 hexagonal tiling honeycomb** is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is *paracompact* because it has cells with an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.

The **triangular tiling honeycomb** is one of 11 paracompact regular space-filling tessellations in hyperbolic 3-space. It is called *paracompact* because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. It has Schläfli symbol {3,6,3}, being composed of triangular tiling cells. Each edge of the honeycomb is surrounded by three cells, and each vertex is ideal with infinitely many cells meeting there. Its vertex figure is a hexagonal tiling.

- 1 2 3 Grünbaum, Branko; Shephard, G. C. (1987).
*Tilings and Patterns*. W. H. Freeman. ISBN 978-0-7167-1193-3. 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). - ↑ Williams, Robert (1979).
*The Geometrical Foundation of Natural Structure: A Source Book of Design*. Dover Publications, Inc. p. 38. ISBN 0-486-23729-X. - ↑ Aiton, E. J.; Duncan, Alistair Matheson; Field, Judith Veronica, eds. (1997).
*The Harmony of the World by Johannes Kepler*. Memoirs of the American Philosophical Society. Vol. 209. American Philosophical Society. pp. 104–105. ISBN 978-0-87169-209-2.. - ↑ Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). "Chapter 21: Naming Archimedean and Catalan polyhedra and tilings; Euclidean plane tessellations".
*The Symmetries of Things*. Wellesley, MA: A K Peters, Ltd. p. 288. ISBN 978-1-56881-220-5. MR 2410150. - ↑ China Central Television, CCTV-13 News Channel (2022-03-25). "[News Live Room] Bamboo weaving products of Ba culture first appeared in Chongqing about 2200 years ago".
*tv.cctv.com*. Retrieved 2023-03-20. - ↑ Yin, Jia-Xin (March 2023). "Exploring hitherto unknown quantum phases in kagome crystals".
*Physics (物理)*.**52**(3): 157–165. doi:10.7693/wl20230301. - ↑ Mekata, Mamoru (February 2003). "Kagome: The story of the basketweave lattice".
*Physics Today*.**56**(2): 12–13. Bibcode:2003PhT....56b..12M. doi: 10.1063/1.1564329 . - 1 2 Lawler, Michael J.; Kee, Hae-Young; Kim, Yong Baek; Vishwanath, Ashvin (2008). "Topological spin liquid on the hyperkagome lattice of Na
_{4}Ir_{3}O_{8}".*Physical Review Letters*.**100**(22): 227201. arXiv: 0705.0990 . Bibcode:2008PhRvL.100v7201L. doi:10.1103/physrevlett.100.227201. PMID 18643453. S2CID 31984687. - ↑ Yen, F.; Chaudhury, R. P.; Galstyan, E.; Lorenz, B.; Wang, Y. Q.; Sun, Y. Y.; Chu, C. W. (2008). "Magnetic phase diagrams of the Kagome staircase compound Co
_{3}V_{2}O_{8}".*Physica B: Condensed Matter*.**403**(5–9): 1487–1489. arXiv: 0710.1009 . Bibcode:2008PhyB..403.1487Y. doi:10.1016/j.physb.2007.10.334. S2CID 14958188. - ↑ "A quantum magnet with a topological twist".
*Discovery: Research at Princeton*. 2019-02-22. Retrieved 2020-04-26. - ↑ Yin, Jia-Xin; Zhang, Songtian S.; Li, Hang; Jiang, Kun; Chang, Guoqing; Zhang, Bingjing; Lian, Biao; Xiang, Cheng; Belopolski (2018). "Giant and anisotropic many-body spin–orbit tunability in a strongly correlated kagome magnet".
*Nature*.**562**(7725): 91–95. arXiv: 1810.00218 . Bibcode:2018Natur.562...91Y. doi:10.1038/s41586-018-0502-7. PMID 30209398. S2CID 205570556. - ↑ Yin, Jia-Xin; Zhang, Songtian S.; Chang, Guoqing; Wang, Qi; Tsirkin, Stepan S.; Guguchia, Zurab; Lian, Biao; Zhou, Huibin; Jiang, Kun; Belopolski, Ilya; Shumiya, Nana (2019). "Negative flat band magnetism in a spin–orbit-coupled correlated kagome magnet".
*Nature Physics*.**15**(5): 443–8. arXiv: 1901.04822 . Bibcode:2019NatPh..15..443Y. doi:10.1038/s41567-019-0426-7. S2CID 119363372. - ↑ Yazyev, Oleg V. (2019). "An upside-down magnet".
*Nature Physics*.**15**(5): 424–5. Bibcode:2019NatPh..15..424Y. doi:10.1038/s41567-019-0451-6. S2CID 128299874. - ↑ Wei, Chenan; Sedrakyan, Tigran (2021-01-29). "Optical lattice platform for the Sachdev-Ye-Kitaev model".
*Phys. Rev. A*.**103**(1): 013323. arXiv: 2005.07640 . Bibcode:2021PhRvA.103a3323W. doi:10.1103/PhysRevA.103.013323. S2CID 234363891. - ↑ Steurer, Walter; Deloudi, Sofia (2009).
*Crystallography of Quasicrystals: Concepts, Methods and Structures*. Springer Series in Materials Science. Vol. 126. Springer. p. 20. ISBN 978-3-642-01899-2. - ↑ Critchlow, Keith (2000) [1969]. "pattern G".
*Order in Space: A design source book*. Thames & Hudson. pp. 74–75. ISBN 978-0-500-34033-2. - ↑ Coxeter, H.S.M. (1973). "V. The Kaleidoscope, §5.7 Wythoff's construction".
*Regular Polytopes*(3rd ed.). Dover. ISBN 0-486-61480-8. - ↑ Huson, Daniel H. "Two Dimensional symmetry Mutations". CiteSeerX 10.1.1.30.8536 .
- ↑ Coxeter, H.S.M. (1991).
*Regular Complex Polytopes*(2nd ed.). Cambridge University Press. pp. 111–2, 136. ISBN 978-0-521-39490-1.

- Seymour, Dale; Britton, Jill (1989).
*Introduction to Tessellations*. pp. 50–56. ISBN 978-0-86651-461-3.

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