Three years ago, Dan Asimov wrote (as part of a math-fun thread with the subject line "big sunflowers"): I was also about to mention the same book.
In fact, the author, Isaac Amidror, has written a second volume, "The theory of the Moiré phenomenon. Vol. II" as well as a 2nd edition of the first volume, now called "The theory of the Moiré phenomenon. Vol. I".
According to Math Reviews, these are considered the definitive reference on the Moiré phenomenon.
Also, the same guy has coauthored a recent paper:
Isaac Amidror & Roger D. Hersch (2010): Mathematical moiré models and their limitations, Journal of Modern Optics, 57:1, 23-36,
which I have downloaded and which looks quite readable.
I still haven't gotten around to looking at Amidror's book, but I did look at the online table of contents at http://diwww.epfl.ch/w3lsp/books/moire /prefaceKluwer.html (thanks, Thane!), and I also leafed through Amidror's article with Hersch, and one thing that frustrated me was the apparent absence of a framework for limit-objects. I was expecting that, just as there are graphons in graph theory and varifolds in geometric measure theory, there'd be some sort of limit-objects that ordinary Moire pattern pictures converge toward (though perhaps one would want various notions of convergence here, depending on what features of the Moire patterns one wishes to understand in the limit). As far as I can tell, Amidror has no interest in devising such an artsy 20th/21st century infrastructure; he just gets his hands dirty in the details of specific Moire patterns, using Fourier theory in a very 19th century way. (But I've only skimmed; maybe I'm missing key features of Amidror's approach.) I'd be interested in more abstract approaches to the question "In what sense do Moire patterns converge?", since they'd be relevant to things like the Abelian Sandpile Model and the Rotor-Router Model that I think about from time to time; for such models one sees a variety of effects at a variety of scales, with mesoscopic effects that are quite different from both the macroscopic effects and the microscopic effects, and it's a challenge to devise the right language in which one might describe these effects and formulate conjectures, let alone prove things. Jim Propp