quasicrystal and penrose tiling, mathematical introduction
Two recommendations:
Senechal, Marjorie. Quasicrystals and geometry. Cambriged Univ Press, 1996. Review by Charles Radin in the AMS Notices: PDF download.
Baake, Michael. "A guide to mathematical quasicrystals." Quasicrystals. Springer Berlin Heidelberg, 2002. 17-48. (arXiv prepub link.)
Maximum entropy equals $\frac{1}{3} \log 2$.
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.
The newly published "Aperiodic Order Volume 1. A Mathematical Invitation" by Baake and Grimm is also good. More daunting than Senechal's book, but clearly written and comprehensive.