Quasicrystal

Quasicrystals are aperiodic structures that still have some regularities. Both the spatial and frequency domain representations of quasicrystals are quite beautiful. See also this more recent post for more quasicrystal entertainment. Here is the classic diffraction pattern from a synthesized icosahedral quasicrystal

You can create two-dimensional quasicrystals by summing together more than 3 plane waves. For example, here are some spatial domain two-dimensional quasicrystals :

4-fold :
5-fold :
7-fold :

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The picture at the beginning of this post is an x-ray diffraction pattern, which is a little bit like the Fourier transform of a quasicrystal lattice. Here is a simulated 2D Fourier spectrum from a 7-fold 2D quasicrystal :

You can create your own images like this by rendering large two-dimension quasicrystals as described here or here. The frequency domain images can be rendered using ImageJ, or your own personal 2d FFT code. In ImageJ, open an image, and then create a frequency domain image by clicking Process→FFT→FFT. If you rendered a N-fold quasicrystal, you should see 2N points arranged in a circle around the origin, corresponding to your N plane waves. If you applied some nonlinear image operation, like contrast enhancement or thresholding, to the spatial domain image, you will have created some harmonics and overtones of your original N plane waves, which should appear as a constellation of other points that themselves are arranges on a quasi-crystal lattice.

Quasicrystals can be related to more abstract mathematical concepts. For instance, this article finds that some quasicrystals are related to sorting algoritms :

Current research in abstract tiling theory examines tilings of high rotational symmetry in collaboration with Remy Mosseri and co-workers. Possibly the limit of high rotational symmetry may prove easier to analyze than specific finite symmetries (10-fold, for example) of direct physical interest. Surprisingly, rhombus tilings are related to algorithms for sorting of lists. Counting the number of distinct tilings enumerates simultaneously the number of equivalance classes of sorting algorithms, a problem previously considered by computer scientist D.E. Knuth. Our random tiling theory implies an upper bound of log(2) for the tiling entropy per vertex, consistent with a conjecture by Knuth. Click here for a preprint on this research.