In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex. As the name implies this tiling is constructed by a truncation operation applies to a hexagonal tiling, leaving dodecagons in place of the original hexagons, and new triangles at the original vertex locations. It is given an extended Schläfli symbol of t{6,3}.
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In geometry, the truncated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,4}. A secondary construction tr{6,6} is called a truncated hexahexagonal tiling with two colors of dodecagons.
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A regular skew dodecagon is vertex-transitive with equal edge lengths. In 3-dimensions it will be a zig- zag skew dodecagon and can be seen in the vertices and side edges of a hexagonal antiprism with the same D5d, [2+,10] symmetry, order 20. The dodecagrammic antiprism, s{2,24/5} and dodecagrammic crossed-antiprism, s{2,24/7} also have regular skew dodecagons.
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From 1966 to 1972 Ron Davis created geometric shaped, illusionistic paintings using polyester resins and fiberglass. About Davis' paintings of the late 1960s in an essay accompanying the Ronald Davis retrospective exhibition Forty Years of Abstraction, at the Butler Institute of American Art in 2002, the abstract painter Ronnie Landfield wrote: "the Dodecagons from 1968–69 remain among the most visually stunning, audacious and intellectually interesting bodies of work made by an abstract painter in the last half of the twentieth century."The Essence Of Abstraction, retrieved online May 9, 2008 In 1966 Davis was an instructor at the University of California, Irvine.
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Conway calls it a kisdeltille,John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table) constructed as a kis operation applied to a triangular tiling (deltille). In Japan the pattern is called asanoha for hemp leaf, although the name also applies to other triakis shapes like the triakis icosahedron and triakis octahedron. It is the dual tessellation of the truncated hexagonal tiling which has one triangle and two dodecagons at each vertex. :320px It is one of eight edge tessellations, tessellations generated by reflections across each edge of a prototile..
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A rectangle (equiangular quadrilateral) with integer side lengths may be tiled by unit squares, and an equiangular hexagon with integer side lengths may be tiled by unit equilateral triangles. Some but not all equilateral dodecagons may be tiled by a combination of unit squares and equilateral triangles; the rest may be tiled by these two shapes together with rhombi with 30 and 150 degree angles. A cyclic polygon is equiangular if and only if the alternate sides are equal (that is, sides 1, 3, 5, ... are equal and sides 2, 4, ... are equal). Thus if n is odd, a cyclic polygon is equiangular if and only if it is regular.De Villiers, Michael, "Equiangular cyclic and equilateral circumscribed polygons", Mathematical Gazette 95, March 2011, 102-107.
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