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
Labels:

Comments

Post a Comment

Popular posts from this blog

The Laws of Thought - Tom Griffiths *****

In giving us a history of attempts to explain our thinking abilities, Tom Griffiths demonstrates an excellent ability to pitch information just right for the informed general reader.  We begin with Aristotelian logic and the way Boole and others transformed it into a kind of arithmetic before a first introduction of computing and theories of language. Griffiths covers a surprising amount of ground - we don't just get, for instance, the obvious figures of Turing, von Neumann and Shannon, but the interaction between the computing pioneers and those concerned with trying to understand the way we think - for example in the work of Jerome Bruner, of whom I confess I'd never heard.  This would prove to be the case with a whole host of people who have made interesting contributions to the understanding of human thought processes. Sometimes their theories were contradictory - this isn't an easy field to successfully observe - but always they were interesting. But for me, at least, ...
Post a Comment
Read more

The AI Paradox - Virginia Dignum ****

This is a really important book in the way that Virginia Dignum highlights various ways we can misunderstand AI and its abilities using a series of paradoxes. However, I need to say up front that I'm giving it four stars for the ideas: unfortunately the writing is not great. It reads more like a government report than anything vaguely readable - it really should have co-authored with a professional writer to make it accessible. Even so, I'm recommending it: like some government reports it's significant enough to make it necessary to wade through the bureaucrat speak. Why paradoxes? Dignum identifies two ways we can think about paradoxes (oddly I wrote about paradoxes recently , but with three definitions): a logical paradox such as 'this statement is false', or a paradoxical truth such as 'less is more' - the second of which seems a better to fit to the use here.  We are then presented with eight paradoxes, each of which gives some insights into aspects of t...
Post a Comment
Read more

Einstein's Fridge - Paul Sen ****

In Einstein's Fridge (interesting factoid: this is at least the third popular science book to be named after Einstein's not particularly exciting refrigerator), Paul Sen has taken on a scary challenge. As Jim Al-Khalili made clear in his excellent The World According to Physics , our physical understanding of reality rests on three pillars: relativity, quantum theory and thermodynamics. But there is no doubt that the third of these, the topic of Sen's book, is a hard sell. While it's true that these are the three pillars of physics, from the point of view of making interesting popular science, the first two might be considered pillars of gold and platinum, while the third is a pillar of salt. Relativity and quantum theory are very much of the twentieth century. They are exciting and sometimes downright weird and wonderful. Thermodynamics, by contrast, has a very Victorian feel and, well, is uninspiring. Luckily, though, thermodynamics is important enough, lying behind ...
Post a Comment
Read more