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Can anyone suggest a survey (or surveys) that provides an update to Tilings and patterns by Grunbaum and Shepard? If there's a more recent book, that would be fantastic, but I don't see one.

I am most interested in the combinatorics of Wang tilings and other square tilings, with the motivation of applying those techniques to derive upper/lower bounds to constructions in self-assembly.

Thank you.

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    $\begingroup$ I know this doesn't answer your question, but The Symmetries of Things by Conway, Burgiel, and Goodman-Strass (2008), has quite a bit of detailed information on tilings: crcpress.com/ecommerce_product/… $\endgroup$
    – Joseph O'Rourke
    Apr 1, 2011 at 1:11
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    $\begingroup$ @Joseph O'Rourke: Thanks very much. :-) $\endgroup$
    – Aaron Sterling
    Apr 4, 2011 at 13:36

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How is your German? MR2219468 (2006m:05054) Ardila, Federico; Stanley, Richard P. Pflasterungen. (German) [Tilings] Math. Semesterber. 53 (2006), no. 1, 17–43.

MR2133310 (2006e:52036) Zong, Chuanming What is known about unit cubes. Bull. Amer. Math. Soc. (N.S.) 42 (2005), no. 2, 181–211 (electronic).

MR2087242 (2005e:42071) Kolountzakis, Mihail N. The study of translational tiling with Fourier analysis. Fourier analysis and convexity, 131–187, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2004.

MR2005953 (2004k:28011) Jackson, Steve; Mauldin, R. Daniel Survey of the Steinhaus tiling problem. Bull. Symbolic Logic 9 (2003), no. 3, 335–361.

MR1990769 (2004k:52027) Pak, Igor Tile invariants: new horizons. Tilings of the plane. Theoret. Comput. Sci. 303 (2003), no. 2-3, 303–331.

MR1242999 (94g:52026) Schulte, Egon Tilings. Handbook of convex geometry, Vol. A, B, 899–932, North-Holland, Amsterdam, 1993.

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    $\begingroup$ I think the Ardila-Stanley article is actually a German translation of a survey paper they wrote in English (see claymath.org/library/senior_scholars/stanley_ardila_tilings.pdf). $\endgroup$
    – Henry Cohn
    Apr 1, 2011 at 13:54
  • $\begingroup$ Thanks for the references. I've skimmed the non-German ones, and they look very interesting (though perhaps not immediately helpful on my current project). @Henry Cohn: Thanks for the link, I'll check it out. $\endgroup$
    – Aaron Sterling
    Apr 4, 2011 at 13:37
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I would suggest you to look at papers of Chaim Goodman-strauss, especially the one on aperiodic tiles.
https://en.wikipedia.org/wiki/Chaim_Goodman-Strauss

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