Indra's Pearls, with Caroline Series and Dave Wright
From the Introduction:
This is a book about serious mathematics, but one, which we have written primarily for non-mathematicians. It is an account of our exploration of a family of symmetrical but infinitely convoluted sets, part of the modern investigation of how chaos evolves from very simple rules, producing intricate complexity on every scale from the very large to the very small. In our case, two rules, each on its own producing a pair of spirals, are allowed to interact. Our scheme is not at all arbitrary; it forms parts of a century old mathematical dream, involving much of the deep mathematics of the 19th century, conceived by the great German geometer Felix Klein. With the aid of modern computers for rendering the results, the answers to our `What if...?' questions turned to be not only intellectually fascinating but also strikingly beautiful. Sometimes the outcome is simple, sometimes it is total disorder and sometimes — and this is the most exciting case — it has layer upon layer of structure teetering on the very brink of chaos. There is no religion in our book but we were amazed at how well our constructions reflected the ancient Buddhist metaphor of Indra's net. Mathematicians often use the word 'beautiful' in talking about their proofs and ideas, but in this case our judgment has been confirmed by a number of unbiased and definitely non-mathematical people.
Most mathematics is accessible, as it were, only by crawling through a long tunnel in which you laboriously build up your vocabulary and skills as you abstract your understanding of the world. But the mathematics behind the figures we drew turned out not to need too much in the way of preliminaries. So long as you got through high school algebra with some confidence, everything we say should be understandable, given a bit of careful reading here and there. And if not, then browsing through the figures alone should give a sense of our journey. Our dream is that this book will reveal to a larger audience that mathematics is not alien, cold and remote but just a very human exploration of the patterns of the world, one which thrives on play and surprise and beauty.
See Indra's official home page maintained by Dave Wright with movies and 'much much more'! The title, incidentally, refers to a Buddhist legend that the God Indra strung a vast necklace of pearls in the heavens so that, when you looked at one, you saw the reflection of the others, and in these reflections, double reflections -- and so on. Amazingly, Anish Kapoor has created a sculpture that captures this. In the background to this page, you see his 'pearls' in a photo taken by my wife on a rainy day in the courtyard of the Royal Academy of Art.