Symmetry in Chaos: A Search for Pattern in Mathematics, Art and Nature (Second Edition)
Michael Field and Martin Golubitsky SIAM 2009, 199 PAGES PRICE (HARDBACK) £40.00 ISBN 978-0-898-71672-6
Symmetry and chaos seem unlikely bedfellows; yet Field, from the University of Houston, and Golubitsky, from Ohio State University, have produced a book full of beautiful pictures by combining the two. To quote from the Introduction: “Our pictures are created by merging symmetry and chaos. At first sight this seems paradoxical: a merging of order and disorder or yin and yang”.
The word, ‘symmetry’, carries with it the notion of repetition, while ‘chaos’, although meaning ‘without form’ in everyday life, is not without form or structure in mathematics. The combination of the two is used by the authors to generate pictures that, in the main, exhibit a different character from the more widely known fractal pictures (though the book does contain a few of these fractal pictures). They note that images of symmetric chaos often seem quite familiar. For example, the figure below looks something like a bracelet (it is a low resolution, monochrome version of the high resolution, multi-coloured, Mayan bracelet figure that appears in the book).
The authors use three mathematically different methods to compute images. These result in different ‘flavour’ images, which they call symmetric icons, quilts and symmetric fractals. Luckily for us, they provide details of how each type is generated.
In the first edition of the book they provided computer program listings in the Basic language. However, these are absent from the second edition “… since Basic is no longer a readily available or widely used computer language”. Personally, I think this is a pity, as the Basic programs could have proven useful in clarifying the meaning of occasionally ambiguous written descriptions of the various algorithms. Also, it is often fairly easy to translate from Basic into a more modern programming language. That said, most of the descriptions in the book seem fairly clear – I was able to generate the figure above, using Matlab, without any difficulty, for example.
For several of the images the authors point to similarities with natural and man-made artefacts in the real world; from diatoms to cathedral windows, corporate logos to Venetian enamelled dishes, the Giant’s Causeway to Bénard convection cells!
This is not just a book of pretty pictures – it is full of mathematics. I’ll leave you to choose either to be wowed by the former or to chew over the latter (or both!).
Alan Stevens CMath FIMA
Mathematics Today December 2010
Symmetry in Chaos: A Search for Pattern in Mathematics, Art and Nature (Second Edition) can be purchased at Amazon.co.uk