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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.
List of contents
Introduction.- Tilings and puzzles.- Robinson triangles.- Penrose tilings.- De Bruijn's pentagrids.- The noncommutative space of Penrose tilings.-Some useful formulas.
About the author
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.
Report
"In this book, D'Andrea presents a succinct and very visual treatment of the much celebrated Penrose tilings alongside a number of supporting and preliminary concepts. ... The book is an efficient introduction to the fundamental and some advanced topics of Penrose and Robinson tiling schemes. ... the figures in the book are all created for the occasion and are very pedagogical and of high quality." (Martin Cramer Pedersen, Mathematical Reviews, March, 2025)
The monograph is extremely well written, it contains all the necessary prerequisites, it explains the topic step by step and contains beautiful pictures. this is a very interesting monograph and I believe it can be highly recommended to anyone seeking an elementary introduction to plane tilings. (Piotr Pokora, zbMATH 1533.52001, 2024)
