Mathematical Conversations

The Penrose tiling (Roger Penrose(1974)) and the "quasi-crystal" made by Ron Schactman (1985) are beginning landmarks here. Our objects today are tilings $T$, of $\mathbb R^d$, [$d = 1, 2$ mostly] which like Penrose's is aperiodic and can be a subdivision tiling. We will introduce the subdivision function $\Phi$ and its action on the Hull, $X(T)$ consisting of all tilings $T'$ which have the same local patterns as $T$. The action of $\mathbb R^d$ upon $X(T)$, forms our dynamical system. Substitution tilings come with an algebraic integer, $\lambda$. Conjecture: a subdivision tiling is a quasi-crystal iff $\lambda$ is a Pisot-Vijayaraghavan number. Full disclosure: no new news about this conjecture.