Designing Beauty: The Art of Cellular Automata

Cellular automata (CA) are discrete time dynamical systems that consist of a regular grid of cells. Each cell has a finite number of possible states. The state changes from one time instant to the next depending on the current states of the cell and its neighbors following simple rules, uniform over the grid.

The best known example is the Game of Life (GoL) that was designed by John Conway in the early 1970s. It became extremely popular after Martin Gardner's picked it up in his Scientific American column. The rules are simple. Each cell has two states: alive or dead. A live cell with 2 or 3 live neighbors survives, otherwise it dies, while a dead cell with exactly 3 live neighbors will become alive. The ease of programming and personal computers becoming widespread in those days, made a whole generation play this GoL fanatically. Even though the system is deterministic, the ever changing patterns in successive steps makes it impossible to imagine the configuration several steps ahead, unless the steps are actually computed. During the evolution, some nice patterns can occur like all kinds of oscillators, gliders and spaceships (groups of cells that walk away from the group in a straight line), glider guns (that produce a regular sequence of gliders), and exotic configurations like a garden of eden (a pattern without a predecessor). There are many websites where one can experiment with the GoL today or any of its extensions and generalizations. Some are mentioned in this book. It is even possible to implement a Turing machine with particular configurations of the GoL. The CA also feature prominently in the somewhat controversial book A New Kind of Science (2002) by Stephen Wolfram, the founder and CEO of Wolfram Research.

But CA were not only used for recreational purposes. They became a subject of research and were applied for the simulation of all kinds of chemical, biological, or social dynamical systems. They can be implemented in one or two or in any finite dimension, the rules can be made stochastic, there can be many states per cell, etc. When adding colors to the graphical representation, some really nice and appealing pictures can be the result. Because of this somewhat unexpected aesthetic side product, also artists became interested, just for the creation of the graphical effects.

The latter artistic aspect is the one that is the main focus of this book. It is primarily a collection of nice pictures that were generated by CA. Thirty `artists' (that are also mathematicians, engineers, architects, computer scientists,...) contributed to the volume. They briefly situate their pictures in a few lines up to at most two pages and then their graphics are included with extensive captions giving additional information about the particular result, parameters used, etc. Some of them have websites where animations or software is available for the user to experiment. In all cases the reader is referred to publications for further details.

For one-dimensional CA, the dynamics can be represented in a two-dimensional picture, but for two and three-dimensional CA, one may only give a snapshot at some time instant since obviously the picture should evolve in time (unless it would represent a steady state). Most of them are in color, although sometimes a black-and-white pattern can also be fascinating. It is not a coffee-table book though. The format is like for an ordinary proceedings volume, but with a contents that is more graphical than textual. It is not a mathematics or computer science book either because the emphasis is on the graphics and the underlying theory is only briefly mentioned (the rules are usually rather simple anyway), but there is a list of 175 references to books and papers for further reading.

What the book clearly shows is that even though CA are so simple, yet so general, very diverse exiting pictures can be the result. The graphics have an appeal similar to the Mandelbrot and Julia sets and their images that were also very popular in the exhibits and photo books by Heinz-Otto Peitgen and Peter Richter in the 1980s.

The diversity of CA that are represented in this book is too extensive to be enumerated them exhaustively in this review. Some samples: variations on the GoL (like Larger than Life (LtL), Life without Death, enlightened GoL), continuous reaction-diffusion models, toothpick CA (horizontal or vertical line segments —like toothpicks— are added in every step), CA on grids in hyperbolic geometries or on spheres or on hexagonal grids or Penrose tilings, asynchronous CA, CA with memory, ... And we see applications such as prime generators, ecological examples, piston motion, chemical reactions, seismic simulation, Turing machines,...

Thus the book should inspire the scientist to present his or her work in a pleasing and graphically attractive way, and it should invite artists to explore the possibilities to be creative with CA. Fact is that CA are an attractive and simple concept that, because of the nonlinearity, can lead to unexpected amazing and fascinating results. Of course there are also difficult questions to ask and conjectures to make that are very difficult to prove, but these issues are certainly not the topic of this book.

That CA can not only inspire visual artists, but also the composers of music can be heard on a CD Iolet by David Stutz that contains several mathematically inspired compositions. His piece Cellular Automaton is sung by a choir, where each member acts like a cell in a CA and changes his singing depending on what he hears his neighbors sing. The choir leader regularly injects new patterns to start from. It all sounds like a choir of Buddhist monks or Tuvan throat singing. The idea is based on the SF novel Anathem by Neal Stephenson.

If you are not familiar with CA, this is a survey that can get you started to explore a catchy subject. Surely you will be tempted to search the Internet and check out some of the available applets and graphical user interfaces to try some experiments for yourself. Beware not to get hooked too much playing around and getting hypnotized by the graphics.

Adhemar Bultheel
Book details

This is basically a collection of examples of nice pictures that resulted from research on cellular automata. Some technical background is given, but the reader needs to consult the literature for the details.

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978-3-319-27269-6 (hbk)
52,99 € (hbk)