Hexagonal lattice
The hexagonal lattice or triangular lattice is one of the five 2D lattice types.
Three nearby points form an equilateral triangle. In images, four orientations of such a triangle are by far the most common. They can conveniently be referred to by viewing the triangle as an arrow, as pointing up, down, to the left, or to the right; although in each case they could also be considered to point into two oblique directions.
Two orientations of an image of the lattice are by far the most common. They can conveniently be referred to as "hexagonal lattice with horizontal rows" (like in the figure below), with triangles pointing up and down, and "hexagonal lattice with vertical rows", with triangles pointing left and right. They differ by an angle of 30°.
The hexagonal lattice with horizontal rows is a special case of a centered rectangular (i.e. rhombic) grid, with rectangles which are √3 times as high as wide. Of course for the other orientation the rectangles are √3 times as wide as high.
Its symmetry category is wallpaper group p6m. A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself.
For an image of a honeycomb structure, again two orientations are by far the most common. They can conveniently be referred to as "honeycomb structure with horizontal rows", with hexagons with two vertical sides, and "honeycomb structure with vertical rows", with hexagons with two horizontal sides. They differ by an angle of 90°, or equivalently 30°.
A honeycomb structure is in two ways related to a hexagonal lattice:
The ratio of the number of vertices and the number of hexagons is 2, so together with the centers the ratio is 3, the reciprocal of the square of the scale factor.
The term honeycomb lattice could mean a corresponding hexagonal lattice, or a structure which is not a lattice in the group sense, but e.g. one in the sense of a lattice model. A set of points forming the vertices of a honeycomb (without points at the centers) shows the honeycomb structure. It can be seen as the union of two offset triangular lattices, shown here red and blue.
A triangular lattice itself can be divided into 3 offset triangular lattices, shown above in red, green and blue. A triangular lattice is also called an A2 lattice, A2, and the union of three triangular lattices is A*2.
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