Complex Tilings

Abstract

We study the minimal complexity of tilings of a plane with a given tile set. We note that every tile set admits either no tiling or some tiling with O(n) Kolmogorov complexity of its n-by-n squares. We construct tile sets for which this bound is tight: all n-by-n squares in all tilings have complexity at least n. This adds a quantitative angle to classical results on non-recursivity of tilings -- that we also develop in terms of Turing degrees of unsolvability. Keywords: Tilings, Kolmogorov complexity, recursion theory