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Conway's Game of Life: Mathematics and Construction

Nathaniel Johnston and Dave Greene
Self Published
Publication Date: 
Number of Pages: 
[Reviewed by
Bill Wood
, on
The setup of Conway’s Game of Life is very simple. We cite the community hub (managed by authors of the book under review) for the rules of this “zero-player game”:
Conway's Game of Life is a cellular automaton that is played on a 2D square grid. Each square (or "cell") on the grid can be either alive or dead, and they evolve according to the following rules:
  • Any live cell with fewer than two live neighbours dies (referred to as underpopulation).
  • Any live cell with more than three live neighbours dies (referred to as overpopulation).
  • Any live cell with two or three live neighbours lives, unchanged, to the next generation.
  • Any dead cell with exactly three live neighbours comes to life.

From these simple rules one simply chooses an initial configuration of live and dead cells and sees what happens.  The curious dynamics of the game raise immediate questions about exactly what kinds of configurations can be created.  No formal tools are required to jump in and experiment.

Conway's Game of Life: Mathematics and Construction by Nathaniel Johnson and Dave Greene provides a linear exposition of the questions, results, and techniques behind the game.  It functions as a companion to the website where one can download the ebook.
The material requires no formal background and is appropriate for its target audience of early undergraduate students. A few topics such as counting, number theory, and algorithm analysis appear, but generally at an elementary level and the key concepts are briefly reviewed in the appendices. There are proofs, but they are generally careful deductions using few mathematical tools. It is a perfect topic to hand to a curious undergraduate mathematics or computer science student and let them go.
There are a lot of objects that need names and the vocabulary can be a bit overwhelming. Many of the names are descriptive – gliders, volcanoes, sparks – but there are also Snarks, Sir Robins, and David Hilberts to navigate. This is the reality of the subject and not a complaint about the book. There is no glossary, but the index is good and there are additional resources online.
One needs to be able to zoom in and out in real-time as the configurations evolve to see how small-scale changes impact large-scale behavior. The authors do a fine job of using color diagrams with consistent coloring and iconography – indeed, the book is visually impressive independent of the content – but there is no substitute for going to the website and noodling with it there. The ebook can be downloaded at the website (for free, with optional donation), allowing for the two to be used in parallel easily. There are also a few print-on-demand options.
The book is organized into three parts, each with four chapters. The first part, “Classical Topics,” covers the fundamental structures and their properties. “Circuitry and Logic” examines techniques for putting these structures together into circuits that exhibit more elaborate and precise behaviors. Finally, “Constructions” develops how these circuits can establish some more general properties of the Game of Life itself: universal computation, wherein we can simulate a universal computer, and universal construction, which establishes a sense in which the Game of Life can create and position its own components. Chapters close with notes and plenty of exercises. There are appendices with some mathematical preliminaries, technical details, and selected exercise solutions. Further material is available on the website, which has plenty of tools for simulating the game, finding specific results, and identifying new investigations that a newcomer could engage in almost immediately.
The website is a fantastic resource for the Game of Life community, and the book provides an excellent and structured way to learn the fundamentals. It could also be used as a text for a topics course or for independent study. My plan for the book is to mimic its subject matter – set up an initial configuration by leaving a copy in the math lounge and see what behaviors emerge.


Bill Wood is an associate professor of mathematics at the University of Northern Iowa. He fondly recalls noodling with the Game of Life as a student several decades ago when the literature was considerably thinner.