Arrangement of Lines - Multigrids and Penrose Tilings

Multigrids and Penrose Tilings

The dual graph of a simple line arrangement may be represented geometrically as a collection of rhombi, one per vertex of the arrangement, with sides perpendicular to the lines that meet at that vertex. These rhombi may be joined together to form a tiling of a convex polygon in the case of an arrangement of finitely many lines, or of the entire plane in the case of a locally finite arrangement with infinitely many lines. de Bruijn (1981) investigated special cases of this construction in which the line arrangement consists of k sets of equally spaced parallel lines. For two perpendicular families of parallel lines this construction just gives the familiar square tiling of the plane, and for three families of lines at 120-degree angles from each other (themselves forming a trihexagonal tiling) this produces the rhombille tiling. However, for more families of lines this construction produces aperiodic tilings. In particular, for five families of lines at equal angles to each other (or, as de Bruijn calls this arrangement, a pentagrid) it produces a family of tilings that include the rhombic version of the Penrose tilings.

The tetrakis square tiling is an infinite arrangement of lines forming a periodic tiling that resembles a multigrid with four parallel families, but in which two of the families are more widely spaced than the other two, and in which the arrangement is simplicial rather than simple. Its dual is the truncated square tiling. Similarly, the triangular tiling is an infinite simplicial line arrangement with three parallel families, which has as its dual the hexagonal tiling, and the bisected hexagonal tiling is an infinite simplicial line arrangement with six parallel families and two line spacings, dual to the great rhombitrihexagonal tiling.