Reaching for the Extreme - Ian Stewart ****

Ian Stewart is arguably the UK's best raconteur of mathematics - here he takes on some of the extremes of the mathematical world, and in doing so gives us some real insights into what makes mathematicians tick.

There's a good mix here of the flashy fun aspects of maths - think, for instance of the wonders of infinity or the monster group - and the solid everyday that nonetheless can turn up surprises. The book is littered with little insights. For example, if we think it's easy to work out the area of a rectangle by dividing it up into unit squares, what do you do with one that measures square root of two by pi?

You'll find yourself jumping around from what lies beneath calculus to game theory (rock, paper, scissors anyone? - I hadn't realised a version of this game dates back around 2,000 years). One minute you'll be considering colouring maps and the next finding the shortest distance between two points on a curved surface.

Some of the mathematics here has everyday applications, some can be used in fairly abstruse corners of science and some is pure mathematical puzzle solving with no obvious application ever. You pays your money and you takes your choice.

There were a few points where Stewart lost me. Some of these were simply due to going on too far. I enjoyed the tiling chapter to begin with, but as we got onto more and more obscure aperiodic tiles there was a bit of a feeling of 'we get the point, move on'. In other cases, Stewart had the problem of dealing with mathematics that is well beyond the grasp of a general reader like me. He is clearly aware of this, but rather than avoid it, makes the discussion so vague that it's difficult to feel you are getting anywhere.

This was particularly the case in the 'weirdest symmetry' chapter, which brings in the aforementioned monster group. There's some nice material on what groups are, but when we get onto that monster, the 'largest sporadic simple group', it's way beyond comprehension, when we're dealing with something it's was difficult even to prove whether or not it existed. It doesn't really help that Stewart then ties this into string theory, an aspect of physics that many believe is more mathematical play than anything connected to physical reality, which has been notably described as 'not even wrong'.

There's a lot to like, though, in this generous meander through eighteen chapters of different mathematical pondering. It'll certainly stretch your mind - and if occasional it leaves us mere mortals behind, that's not always a bad thing.

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