Indra's Pearls

"I put this on the list because, to me, it is the most beautifully visual mathematics book I have ever seen. It is about what is called Fuchsian groups and Kleinian groups. I had heard of these things before I got the book but I had never really been all that interested in them, because I didn’t understand what they were about. Then I got this book, and saw that it is about what happens if you take two or more circular mirrors and create a hall of mirrors effect. You reflect one mirror in the other and vice versa, and you keep on going ad infinitum. With two mirrors you get something fairly straightforward, but with four mirrors you start getting some remarkable designs. The book is full of astonishing pictures of what you get when you reflect images over and over again in these circular mirrors. It turns out all sorts of different figures emerge. In the book they colour code the number of reflections, and as they get more complicated the patterns seem to glow on the page. In the simplest example you have glowing circles. Then they change the mirrors a little bit and you get a sprinkling of glowing dust, or something that looks like a dragon. Fractal books have been around since the 1970s, and they all copy each other in a way. One of the wonderful things about this book is that it has new fractals which I had never seen before. I also like the way that the authors tell you about the wonder they themselves experienced when they started seeing these designs. For example: “The authors, and later the participants in the 1980 Thurston Theory Conference at Bowdoin College, could not suppress their awe at the eerie glowing image of the limit curve snaking its way across an old Tektronix terminal.” Or: “Figure 9.1 shows another level of complexity, an array of interlocking spirals which literally took our breath away when we first drew it.” The point I want to make about this book is that when you see these extraordinary pictures, you want to understand what is going on to make them. Since the book is written by mathematicians, they give you all of the mathematical background behind the shapes. But they write the book in a way that I wouldn’t if I was doing it. As a maths journalist, I think it is more enticing to show the beautiful outcome of the maths first – the shapes – rather than having all the workings in place before the end result. They start by doing the spade work, then they plant the seeds and watch them grow. I would start with the flowers, which are the pictures in the book. To me, that is what you need to excite people about mathematics and get them to see the beauty of it. So if you are a student planning to read this book, or just curious about maths, I would start around page 99 or 100 by looking at the pictures. Let them enthrall you, let them suck you in. Then, if you can’t bear not knowing what their secret is, you’ll be ready to go back and read from the beginning, and find out what Fuchsian and Kleinian groups are."