A Guide to Penrose Tilings

This book provides an elementary introduction, complete with detailed proofs, to the celebrated tilings of the plane discovered by Sir Roger Penrose in the '70s. Quasi-periodic tilings of the plane, of which Penrose tilings are the most famous example, started as recreational mathematics and soon attracted the interest of scientists for their possible application in the description of quasi-crystals. The purpose of this survey, illustrated with more than 200 figures, is to introduce the curious reader to this beautiful topic and be a reference for some proofs that are not easy to find in the literature. The volume covers many aspects of Penrose tilings, including the study, from the point of view of Connes' Noncommutative Geometry, of the space parameterizing these tilings.

Introduction.- Tilings and puzzles.- Robinson triangles.- Penrose tilings.- De Bruijn’s pentagrids.- The noncommutative space of Penrose tilings.-Some useful formulas.

Francesco D'Andrea has a master in Theoretical Physics (Univ. Sapienza of Rome) and a Ph.D. in Mathematics (SISSA, Trieste). He is currently an associate professor in Geometry at the University of Naples Federico II. In the past, he has been a junior research fellow at the Erwin Schroedinger Institute of Vienna, a postdoctoral researcher at the Catholic University of Louvain-La-Neuve, Belgium, a visiting professor at IMPAN, Warsaw (Simons Professorship), and at Penn State University, USA (Shapiro Visitor Program).

His main interests are in Connes' noncommutative geometry, C*-algebras, and differential geometry.