First the executive summary: read these excellent books, and make sure all your friends and relations and bright pupils (if you are a teacher) or teachers (if a bright pupil) do so too. Mathematical Vistas (MV) is the sequel to the same authors' earlier Mathematical Reflections (MR). Each book is a series of explorations of mathematical topics, informed by a definite idea of what mathematics is, and how it should be taught.
This book is by the same authors as "Why do buses come in threes?", which was reviewed in Issue 10 of Plus. Like its predecessor, it consists of entertaining and thoughtprovoking questions on topics not obviously related to maths, and a discussion of each. The authors say that "give us a topic that we care about, and we all become mathematicians", and set out to prove it.
If "How to solve it" really contained an infallible recipe for doing so, mathematics would not be mathematics and the world would be quite different. Of course it doesn't - it can't - but it can - and does - contain a great deal of food for thought for the budding mathematician. Like many other Central Europeans, Pólya relocated to the US at the beginning of the Second World War. There he worked at Stanford University and wrote this immensely successful book (more than a million copies sold) in 1945.
This book is built on an extended metaphor, which casts equations as the poetry of science. According to the editor Graham Farmelo (head of Science Communication at the Science Museum in London), great equations and great poems are alike in a number of ways. Both suffer if anything is added, changed, or taken away, both are a rich stimulus to the prepared imagination, and both draw much of their power from their conciseness.
As Tony Gardiner says in at the beginning of this book, "the last ten years or so has seen a remarkable blossoming of public interest in mathematics [but] most of the books produced have been for adults, rather than for students. Moreover, most are in prose format - for those who want to 'read about' mathematics, rather than those who want to get their hands dirty solving problems."
If you watch a steam engine, you may not know how it works but you can soon get a fairly good idea of its behaviour, and you can predict its future behaviour accurately. Even though you don't understand its workings, you can see it's a pretty simple machine, so you can trust it to behave in a simple way: you have confidence in your predictions based on a short sample of its behaviour.
Research on the Universe leads to many such startling conclusions and this book attempts to describe some of the surprising phenomena which occupy astronomers and cosmologists. Our Universe, Martin Rees' laboratory, allows its natural laws to be cleverly interpreted at arm's length, by observing the 'extreme' physics which we could never replicate in a laboratory. The biggest questions have an almost philosophical tenor.