The mathematics of cellular automata
Cellular automata are discrete models of organisms: arrays of cells, each evolving according to the same pre-established rule and depending on the cell's internal state as well as on the states of its immediate neighbours. As such, they are a fundamental model of how complexity may emerge from very simple building blocks.\N\NThe book under review introduces some of the main features and properties of cellular automata, from a mathematical viewpoint. It should not be treated as a reference (though it does make pleasant reading), but rather as the support of an introductory course on cellular automata, introducing basic mathematical concepts along the way.\N\NIn fact, I believe the reverse is also true: {it is a course on basic mathematics (calculus), motivated by cellular automata as a running example.} In particular, the reader will learn about metric spaces, (equi)continuity, etc. The text contains proofs of the main results, often richly illustrated via cellular automata.\N\NA few caveats for the professional mathematician: the results presented in this book heavily focus on one-dimensional cellular automata, those for which the cells are arranged in a linear array. There is a single chapter on two-dimensional cellular automata, focussing on Conway's ``game of life'', and the celebrated Moore-Myhill theorems, while valid in arbitrary-dimensional grids, are only proven in dimension one. As such, the beautiful connection between \((d+1)\)-dimensional tiling problems and \(d\)-dimensional cellular automata is missing.\N\NFor a mathematically more thorough treatment, I would recommend \textit{T. Ceccherini-Silberstein} and \textit{M. Coornaert}'s [Cellular automata and groups. 2nd edition. Cham: Springer (2023; Zbl 1531.37003)].
scientific article; zbMATH DE number 7814697