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I spent a little time looking around the Wikipedia and Wolfram articles on Penrose Tiling, the Domino Problem, Wang Tiles, etc., but I'm having a little trouble getting into them. A lot of these resources are fairly non-linear in that they often have circular dependencies of concepts that you need to know.

For example, I can tell that aperiodicity is important somehow, but I don't know why and I'm not quite sure about the actual definition.

I keep seeing tiling linked in unexpectedly (e.g. L-systems, the Entscheidungsproblem) or in non-mathematical places (games, geometric art... even Neal Stephenson's Anathem) that it seems important and I'd really like to know more.

So, with that said: does anyone have any resources they'd recommend? I'm looking for something fairly complete or at least introductory, not just an article about a certain facet. Books are fine, visual or interactive resources are great. Would love descriptions of practical implications and relationships with other fields.

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Quasicrystals are a physical instance of aperiodic tiling. Here is a video of the discoverer describing this finding. –  anon Sep 22 '10 at 17:10
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Chaim Goodman-Strauss has quite a few papers on his web page about this topic, including this specific one:

http://comp.uark.edu/~strauss/papers/newsmall.pdf

Goodman-Strass's publications page:

http://comp.uark.edu/~strauss/papers/index.html

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Try Miles of Tiles by Charles Radin.

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