When Barney lent me this book, he asked me if I could explain what topology was for. In honesty? Despite my thesis title being “The Magnetic Topology Of The Solar Corona”, I couldn’t — I only studied topology because my PhD boss told me I should. I hoped that reading Euler’s Gem (from the bookmark I found midway, I presume Barney got bored of it) would help me fix that.
It sort of did — for example, it can tell you a great deal about the number and types of solutions to a set of differential equations, or link the numbers of whorls and loops in your fingerprints, or lead to such marvellous scientific breakthroughs as the Beveridge-Longcope equation. So there’s that.
So, what is topology? In essence, it’s the study of shapes — but in terms of counting rather than measuring. For instance, the eponymous gem is the formula ￼[pmath]V-E+F=2[/pmath], relating the numbers of vertices (corners), edges and faces of 3D shapes. That’s topology: you’re not measuring the (solid) angles at the corners or the areas of the faces or anything, you’re simply comparing numbers.
The book takes you through the history of topology (it’s a relatively new name for the field — it was first used in the 18th century), from Pythagoras and his buddies messing about with the regular solids to Descartes (who may have beaten Euler to the formula), to Euler himself, to Poincaré, to Hopf and Schätzli. It explains the problems they encountered as they puzzled about shapes, and it gives sketches of the solutions. It has pictures of Klein bottles, stellated polyhedra, and famous mathematicians.
But it has very little to explain why topology is important. It seems to take as a given that “of course you want to know about the properties of surfaces!” To a certain kind of mind, one that’s already interested in the more abstract ends of maths, that might be true, but — speaking as a folk-topologist — it doesn’t do much to motivate the study of the subject.
It’s also littered with misprints. As someone who’s been taken to task by several readers for not having checked the answers in my books thoroughly enough, I sympathise to a degree — but there were enough glaring ones that I was groaning about getting a decent editor, or at least reading it through.
And one thing that made me frown was that Richeson pays so much attention to 'who should get the credit for what'. It's Lobachevski syndrome (I publish first!) - in a subject where almost nothing (apart from the Beveridge-Longcope equation) is named after its discoverers, and in which every idea is built on a dozen others, it seems a bit silly to spend pages trying to figure out which of two long-dead boffins had a particular idea first.
Personally, I enjoyed it despite the flaws — it got me interested in my research again for the first time in a while (and playing with the possibilities it hinted at) and it made for pleasant reading; however, I suspect that unless you’re already drinking from the fountain of topology, your mileage may vary considerably.
 at least, shapes that fulfil a few rules.
 one of my proudest moments as a researcher was when a reviewer asked for a reference for something I’d stated as fact -- since it was a basic tenet of graph theory, I was able to cite “(Poincaré, 1893).”
 “Klein bottle” is a terribly clever pun in German -- die Fläche (surface) and die Flasche (bottle) sound quite similar.
 Three dimensions and the truth