Trihexagonal

Trihexagonal describes patterns or tilings that combine triangles and hexagons in a regular arrangement. The most studied example is the trihexagonal tiling, a uniform tessellation of the plane formed by equilateral triangles and regular hexagons of the same edge length.

In this tiling, the vertex figure is 3.6.3.6, meaning that at each vertex a triangle and a hexagon alternate around the point. The tiling possesses hexagonal symmetry, with sixfold rotational symmetry and mirror lines, and is classified as an Archimedean tiling, one of the 11 uniform tilings of the plane.

Construction and appearance: the pattern can be produced by inserting a triangle into each gap between adjacent hexagons in a hexagonal tiling, or by overlaying a triangular grid on a hexagonal lattice so that triangles fill the interstitial spaces. All tiles share the same edge length, and the plane is tiled without gaps or overlaps.

Applications and related concepts: the trihexagonal tiling is used in art, design, and architectural tiling patterns. In mathematics and physics, related structures based on this geometry appear as the Kagome lattice, a network of corner-sharing triangles that arises in various models of magnetism and electronic structure.

See also: Archimedean tilings, triangular tiling, hexagonal tiling, Kagome lattice.