Quasi-Periodic tilings

A program to calculate and display tilings of arbitrary rotational symmetry using a minimum number of different rhombi. The 5-fold symmetric patterns are known as Penrose patterns, so I like to call it generalized Penrose patterns.

The tilings are produced from editable parameters in dual space. In this way an infinite number of different patterns can be produced.

The construction of these patterns using the dual method starts with a grid of lines, defined by a star of vectors. Each family of lines is perpendicular to a vector, its density proportional to the vector's length. To every open space between the gridlines corresponds a vertex of a rhombus as follows:

If the area lies between the i-th and i+1 th line of the j-th family then one finds a corresponding point of the tiling by adding i steps along the j-th starvector. This is to be repeated for every family of gridlines. The intersection of two gridlines thus corresponds to a rhomb in the tiling as there are four spaces around it defining four points in the pattern separated by single steps in one or both of two grid vectors.

By varying number, length and orientation of the star vectors an infinite number of tilings can be generated. Symmetry of vectors and phases of the gridlines determine the symmetry of the result.

Penrose tilings may be decomposed into patterns of rhombi as generated by the symmetric 5-fold star. Many of the rhomb-patterns, however, cannot be transformed into kites and darts, only if no more than two thin rombi meet alongside.