TITLE: quasicrystal and penrose tiling, mathematical introduction
QUESTION [9 upvotes]: Starting to research on quasicrystal from material science, I want to know more about how to   understand quasicrystal from a purely mathematical (especially tiling) perspective (probably start from Penrose tiling). I am more familiar with Wang Tile from my previous experience, do you have any suggestion on where shall I start reading on this topic? Thank you.

REPLY [6 votes]: If you really aim for substantial mathematical facts I also recommend "Aperiodic Order" by Baake and Grimm. (My account is so new that I cannot "comment" or "Vote up" or something.) The first 6 or 7 chapters are easy to understand for anyone with some basic knowledge on calculus and algebra. The next chapters are tougher. Already in the first 6-7 chapters you learn a lot not only on tilings but on all the relevant mathematics.