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Concept
Trihexagonal tiling
Summary
In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. 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.
This pattern, and its place in the classification of uniform tilings, was already known to Johannes Kepler in his 1619 book Harmonices Mundi. 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).
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.
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.
Image:Snowshoe2.jpg|[[Inuit]] [[snowshoe]] Tagluk
File:Kagome lattice blue.svg|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.
The kagome lattice in this sense consists of the vertices and edges of the trihexagonal tiling.
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