Copyright 2005 S.Monnier

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This image displays an aperiodic tiling (the term "aperiodic" is explained here). It is composed of four types of tile : triangles (white), parallelograms (black), trapezoids (blue) and hexagons (red).

Such tilings are produced by "deflation". First one gives a prescription explaining how to break each type of tile into tiles twice smaller. Then, start wth just one shape, break it into smaller tiles following the prescription and iterate the process. It is possible to cover any finite region of the plane with this procedure. To check that the tiling can really cover the whole plane, one have to show that the limiting procedure is well-defined, and it's the case.

Of course such images cannot be considered as fractal. Yet I wouldn't be surprised that these tilings have some very interesting scaling properties. For instance look for diagonal strings of red hexagons. You have a very long one crossing the whole image. But there are several shorter ones besides, too. It could be interesting to look at the number of strings of a given length as a function of the length, (on a larger portion of the tiling, of course). I'm sure one would get some power law frequency distribution, like in FBM patterns..