Zonotiles are centrally-symmetrical convex polygons (zonogons) tiled by smaller zonogons. 

A zonotile
They are related to quasicrystals and Penrose tilings. However, a zonotile may be strongly symmetrical, as in the image above, or those below. The two zonotiles below are of special interest, because they contain patches of semi-regular Archimedean tessellations.

An "Archimedean" zonotile with its corresponding line arrangement.

An "Archimedean" zonotile with its corresponding line arrangement.

Although zonotiles may exhibit strong n-fold rotational symmetry, one may obtain patches which exhibit quasiperiodic order, as in the image below.


The above images were made using the software Mathematica. I adapted an algorithm written in PostScript, in which what is called the generalized dual method is implemented. In this method, sets of parallel lines are constructed, typically, parallel to the sides of some regular polygon. The line sets intersect one another in many places. If some k of the lines intersect at a single point, a 2k-gon is made; then again, if some k of the lines bound a given region, k zonogons will meet there. However, we are not exactly constructing the dual of the line arrangment. Suppose there are 5 sets of 7 lines, 35 lines altogether. The lines are expressed in the form {x,y,d}, where {x,y} is the normal, d, the distance from the origin. For each of the 35 lines, we associate that line with every possible pair of lines, and find the sign of the determinant of these three 3-vectors. We obtain a list of "sign vectors" composed of 35 -1's, 0's, and 1's. These provide a code used to draw the actual tiles.

Download the Mathematica notebook.

Tiling Gallery

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