Tuesday, 4 September 2007

The Equation that Couldn’t be Solved – Mario Livio *****

A book we recently reviewed (Unknown Quantity by John Derbyshire) claimed to provide an engaging history of algebra, but failed to deliver. This book, by contrast, does much more than it claims. Not only does provide a genuinely readable history of algebra, but this is just a precursor to the development of group theory, its link to symmetry, and the importance of symmetry in the natural world. (If you are wondering what this has to do with an equation that couldn’t be solved, along the way it describes how it was eventually proved that you can’t produce a simple formula to predict the solutions to quintic equations – if that sounds painful, don’t worry, it isn’t in this book.)
I can’t remember when I last read a mathematics book that was so much of a page turner. Mario Livio has just the right touch in bringing in the lives and personalities of the mathematicians involved, and though he isn’t condescending in his approach, and occasionally readers may find what’s thrown at them a little hard to get their mind around, provided you are prepared to go with the flow and not worry too much if you understand every nuance, it is superb. Just an example of the throw-away brilliance – I’ve read a good number of books on string theory, but this is the first time I’ve seen it made clear how the mathematical basis of the theory is put together. Just occasionally it’s possible that Livio is skimming over a point in a little too summary a fashion – but that’s rare.
If you have read any other maths histories, you may already have come across some biographical detail of Abel and Galois, two very significant men in this story, who have the added biographical mystique of dying young. However, I really felt that Livio has added something to what has been said before, especially in his exploration of Galois’ mysterious death, and also in the way he sets the scene in France at the time, entirely necessary for those of us who haven’t studied history.
The one disappointment with the book is its final chapter, in which Livio tries to examine what creativity is and why some people are creative mathematicians. It sits uncomfortably, not fitting with the flow of the rest of the contents, and it’s clearly a subject the author knows less about than maths and physics. He makes a classic error (which may be one that mathematicians are particularly prone to) of assuming there is a single right answer to a real world problem. Livio challenges us with this problem: “You are given six matches of equal length, and the objective is to use them to form exactly four triangles, in which all the sides of all the four problems are equal.” He then shows us “the solution” in an appendix. The fact is that almost all real world problems, outside the pristine unreality of maths, have more than one solution. In this case, his solution (to form a 3D tetrahedron) is not the only solution, and arguably is not even the best solution.*
However, despite the aberration of this chapter, the rest of the book is a tonic – absolutely one of the best popular maths books we’ve ever seen. Highly recommended.
* Here’s one other solution. For four equilateral triangles, you need 12 identical length sides. So cut each matchstick in half. You now have 12 identical length pieces and can make the four triangles. This is arguably a better solution because it is freestanding – Livio’s solution has to be held in place – and because it is more mathematically pleasing. If you take one triangle away from this solution (4-1=3) you end up with 3 triangles. if you take one triangle away from Livio’s solution, you end up with 0 triangles. (4-1=0). We can think of at least one other solution, and there are almost certainly more.
Review by Brian Clegg

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