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The fractal pentaplexity tiling
This picture shows a patch of the fractal pentaplexity tiling, together with its set of substitution rules. This is similar to the well-known Penrose tilings of the plane, the difference being that the tiles involved are fractals. The picture comes from the paper Fractal dual substitution tilings by Natalie Priebe Frank, Samuel B.G. Webster and Michael F. Whittaker (http://arxiv.org/abs/1410.4708).
A substitution tiling is a tiling produced from an initial cluster of tiles by iteratively substituting clusters of tiles for single tiles, and then scaling so that the individual tiles in the new tiling have the same size as the original tiles. In this particular case, there are six types of tiles, which are distinguished in the picture by colour, although the yellowish-green and bluish-green tiles differ only in size. By repeatedly applying the six substitution rules (at the bottom of the picture) simultaneously to all the tiles in a pattern, one can produce larger and larger fractal tilings. If everything is set up carefully, this leads to a tiling of the whole plane by fractal tiles, and the top part of the picture shows a patch of the resulting infinite tiling.
As the authors discuss in Section A.2 of the appendix of their paper, the tiling shown here is closely related to the “pentaplexity” tiling that Roger Penrose described in the late 1970s. In fact, one of the main results of the paper is a description of a method to construct infinitely many new substitution tilings from a given substitution tiling, in such a way that the new tilings have fractal-shaped tiles. The new tilings are locally derivable to the original tiling, which means that the old tilings and new tilings can be converted into each other by applying fairly simple local rules.
Although in some ways, the fractal tilings are similar to their non-fractal equivalents, the fractal tilings have some advantages. One of these is that the fractal tilings can be used to construct tilings with border-forcing substitution rules. Roughly speaking, what this means is that a reasonably big patch of tiles can be extended outwards in only one way.
The word dual in the title of the paper refers to the concept of a combinatorially dual tiling. Two tilings T1 and T2 are said to be combinatorially dual if the vertices of T1 correspond to the tiles of T2 (and vice versa) and the edges of T1 correspond to the edges of T2. The authors show that in many cases, the combinatorial duals of their tiling are also substitution tilings.
A distinctive property of fractals is that they can have dimensions that are not integers. Fractals such as the well known Koch snowflake or the tiles in this picture have boundaries whose dimension is more than that of a line, but less than that of a plane. For example, the dimension of the Koch snowflake turns out to be given by log 4/log 3, which is about 1.26. Theorem 7.2 of the paper gives a precise value for the Hausdorff dimension of the boundary of a tiling, which is also given by a ratio of logarithms.
The mathematics in the paper is rather sophisticated and includes, for example, a method to compute the Čech cohomology of the tilings. However, the authors also give many other illustrated examples of fractal substitution tilings like the one here, including the octagonal Ammann–Beenker tiling and the so-called chair tiling.
Penrose tiling: http://en.wikipedia.org/wiki/Penrose_tiling
Penrose’s paper on pentaplexity, originally published in 1978: http://world.mathigon.org/resources/Polygons_and_Polyhedra/Pentaplexity.pdf
Koch snowflake: http://en.wikipedia.org/wiki/Koch_snowflake
Slides from a talk about pentaplexity by Adam Brunell and Daniel Sherwood: http://web.williams.edu/Mathematics/sjmiller/public_html/hudson/BrunellSherwood_FractalsChaos.pdf
Another recent post by me about substitution tilings: https://plus.google.com/101584889282878921052/posts/JnpG7CA12tL
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