Standard 2D Crystalline Patterns and Rational Points in Complex Quadrics
  A certain Diophantine problem and 2D crystallography are linked through the notion of standard realizations which was introduced originally in the study of random walks. In the discussion, a complex projective quadric defined over Q is associated with a finite graph. "Rational points" on this quadric turns out to be related to standard realizations of 2D crystal structures. 
Toshikazu Sunada
math.CO math.AG