quasicrystal and penrose tiling, mathematical introduction 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.
 A: 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. 
A: 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. 
A: Probably the best advice is to contact Marjorie Senechal at smith.edu (her email name is just her last name), since she has written and reviewed quite a few papers in this area.   See for example an older expository note here and its references.   Though she has recently retired from teaching, she is well-connected with the subject and the people involved.   
There are many technical papers on quasicrystals, not all useful for a newcomer and not all readily accessible online.  I'm not aware of good introductory sources at the textbook level, but I'm certainly not a specialist in this area.   Marjorie might be helpful in figuring out what you need to look at first.
A: 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 $\frac{1}{3} \log 2$.

