Trihexagonal tiling

Trihexagonal tiling

Type	Semiregular tiling
Vertex configuration	(3.6)²
Schläfli symbol	r{6,3} or ${\begin{Bmatrix}6\\3\end{Bmatrix}}$ h₂{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]

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]

^ 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.

[t+p-1] 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).

[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] 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..

[4] 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.

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