This is a book about mathematics. It is easy to read. It is not a popular survey of old knowledge, but an exposition of a distinctive idea by a leading thinker. What it says is true (give or take some of the more speculative parts). That is a combination that does not come along every day. It is quite enough to make the book well worth reading, however irritating some of its features are.

The distinctive idea is easy to state, though it needs some examples to appreciate. It is that simple rules can generate complex—very complex—outcomes when they are repeatedly applied. Isolated examples of this phenomenon have been known for centuries. For example, the number π has a simple definition—the ratio of a circle’s circumference to its diameter—and there is a quite simple formula to calculate its digits. The result begins:



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