Conway and Aperiodic Tilings

Aperiodic Tilings and Entropy

In this paper we present a construction of Kari-Culik aperiodic tile set — the smallest known until now. With the help of this construction, we prove that this tileset has positive entropy. We also explain why this result was not expected.

Cyclotomic Aperiodic Substitution Tilings

The class of Cyclotomic Aperiodic Substitution Tilings (CASTs) is introduced. Its vertices are supported on the 2n-th cyclotomic field. It covers a wide range of known aperiodic substitution tilings of the plane with finite rotations. Substitution matrices and minimal inflation multipliers of CASTs are discussed as well as practical use cases to identify specimen with individual dihedral symmet...

Fixed Point and Aperiodic Tilings

An aperiodic tile set was first constructed by R. Berger while proving the undecidability of the domino problem. It turned out that aperiodic tile sets appear in many topics ranging from logic (the Entscheidungsproblem) to physics (quasicrystals). We present a new construction of an aperiodic tile set that is based on Kleene’s fixed-point construction instead of geometric arguments. This constr...

Aperiodic Hierarchical Tilings

A substitution tiling is a certain globally de ned hierarchical structure in a geometric space. In [6] we show that for any substitution tiling in En, n > 1, subject to relatively mild conditions, one can construct local rules that force the desired global structure to emerge. As an immediate corollary, in nite collections of forced aperiodic tilings are constructed. Here we give an expository ...

Aperiodic Tilings , Ergodic Theory And