Ordine privo di periodicit\`a: il fascino matematico delle tassellazioni

TL;DR

This paper reviews the mathematical beauty of aperiodic tilings, covering foundational concepts, key proofs, and recent discoveries, making complex tiling theories accessible to a general audience.

Contribution

It provides a comprehensive overview of aperiodic tilings, including recent breakthroughs like the discovery of an aperiodic monotile, with explanations suitable for non-specialists.

Findings

Existence of an aperiodic set of Wang prototiles proved by J. Kari

Discussion of Penrose tilings and their unique properties

Recent discovery of an aperiodic monotile by D. Smith and colleagues

Abstract

This is a review (in Italian) on aperiodic tilings of the plane intended for a general audience. First, we recall some basic results about lattices and periodic tilings. Then, we move on to one-dimensional (domino) tilings and Wang tilings. We present a beautiful proof of the existence of an aperiodic set of Wang prototiles due to J. Kari. Next, we discuss Penrose tilings and their properties. Finally, we briefly present the recent discovery by D. Smith and his collaborators of an aperiodic monotile.