Skip to main content

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 triangles 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.

Paperback 
Using these links earns us commission at no cost to you
Review by Brian Clegg

Comments

Popular posts from this blog

Wonderworks - Angus Fletcher *****

If you are interested in both writing and science this is an unmissable book. Reading can not just impart information, but can influence the way that we feel. Angus Fletcher describes 25 literary techniques (many also applicable to film and TV), ranging from those known to the Ancient Greeks to modern innovations, that can have a particular influence on our feelings and state of mind. Interestingly Fletcher describes this as technology - devices to make something happen. But rather than being a purely philosophical exploration, what lifts the book significantly is that for each of these techniques, Fletcher describes how studies of the impact on the brain show the physical effects occurring in different parts of the brain as a result. The chapters (one for each technique) end with a faintly self-help feeling bit - for instance, how you can use the 'clear your head' technique by reading certain texts, but that really isn't the point. This isn't a self-help book, it's

Life is Simple - Johnjoe McFadden ***

This is a really hard book to review, because it has two quite distinct parts and the chances are that if you are interested in one of these parts, you may well find the other part less engaging. The first section concerns the development of Occam's razor - the idea of keeping your explanation of something as simple as possible while it still works - and the impact this would have on philosophy (and proto-science) in the Middle Ages. The second part treads very familiar ground in taking us through some of the major developments in science from Galileo onwards, occasionally tying back to Occam's razor to show that the impact of the idea continued. As it happens, I love the first bit as I find the medieval development of science and its intertwining with religion and philosophy fascinating. Jonjoe McFadden brought in a lot of material I wasn't familiar with. Of course I was aware of Occam's razor itself, but I knew nothing about William of Occam as a person, or the way hi

Being You - Anil Seth ***

The trouble with experts is they often don't know how to explain their subject well to ordinary readers. Reading Anil Seth's book took me back to my undergraduate physics lectures, where some of the lecturers were pretty much incomprehensible. For all Seth's reader-friendly personal observations and stories, time after time I got bogged down in his inability to clearly explain what he was writing about. It doesn't help that the subject of consciousness is itself inherently difficult to get your head around - but I've read plenty of other books on consciousness without feeling this instant return to undergraduate confusion. There are two underlying problems I had with the book. One was when complex (and, frankly, rather waffly) theories like IIT (Integration Information Theory) were being discussed. As the kind of theory that it's not currently possible to provide evidence to support, this is something that in other fields might be suggested not to be science at