Complex tiling: breaking translational symmetryLeonid LevinBoston University
October 2,

ABSTRACT


A tile is a unit size square with colored edges. A palette P is a finite set of tiles with copies of which one can tile the plane so that the colors of adjacent tiles match. Classical works constructed palettes that can tile an infinite plane, but cannot do this recursively. That is, consider all Psquares, i.e. those appearing in Ptilings of a plane. For any c, only Psquares of bounded size can be generated (from their diameter) by cbit programs. We pushed these results to the limit, with a palette for which all such squares have diameter c or less. This bound is tight since specifying the tiles on the square's frame suffices for tiling the square recursively. Thus, this palette allows only tilings with all frames being "nearlyrandom". My talk, however, will only briefly mention this our result. I will mostly talk about just one old tool we are using: a palette P which precludes any translational symmetry in any Ptiling. The set of all Ptilings is, of course, symmetric under translations. Yet, any specific Ptiling is aperiodic, this symmetry spontaneously broken. Classical proofs of this result are very cumbersome. I will try to present a more transparent construction and proof. Joint work with Bruno Durand and Alexander Shen. 
Combinatorics Seminar, Mathematics Department, MIT, sara(atsign)math.mit.edu 

