The best books on The Beauty and Fun of Mathematics

recommended by Dana Mackenzie

From mathematical card tricks to an arrangement of interlocking pentagons previously thought impossible, the maths writer explains why solutions to maths problems can be as beautiful as paintings or sculpture

Dana Mackenzie is an author and journalist who writes for Science, Discover and New Scientist. He has a PhD in mathematics from Princeton, and was a mathematics professor for 13 years before becoming a full-time writer. His latest book is TheUniverseinZeroWords

You have picked five books that describe the fun and beauty of mathematics – two words which many of us may not usually associate with the subject.

I chose this topic partly with the idea in mind that it might be surprising for people. But for me, mathematics has always been something that I have found both fun and beautiful. Three of the books that I will talk about here have strong personal connections. And they are all a lot of fun.

I think that the fun in mathematics comes from interacting and participating with it. You notice something curious – perhaps something very simple – and you want to understand it. For instance, that the decimal for 1/7 repeats every six digits, while the decimal for 1/3 repeats every digit. Why? What determines the length of that cycle? Mathematics has the amazing property of being able to lead you through a maze of interacting concepts. The phenomenon I just mentioned will lead you deep into a subject called number theory, which is still not completely understood. The fun of mathematics is finding out where it is going to take you.

“Mathematics produces some of the most stunning images that you will find anywhere – images that will completely bend your mind.”

And the beauty of mathematics has a couple of different interpretations. Sometimes it can be pure visual beauty. Mathematics produces some of the most stunning images that you will find anywhere – images that will completely bend your mind. Another aspect of beauty in maths is closely related to the idea of truth. In mathematics you are always trying to work out what is the truth, and very often the best proof is the most beautiful, because it strips away all of the things that aren’t necessary to your problem and takes you to its very core.

The first of your book choices is George Polya’s

Induction and Analogy in Mathematics.

This is one of the books that has a deep personal meaning to me. In some ways it is the most challenging of the books I have picked, because there is a lot of real maths in it. George Polya, a professor at Stanford University, was a fantastic maths teacher and a real genius. He has a way of taking you gently through the process, and making you feel as if you can do it.

So why does the book have such a personal connection?

I received this book as a prize when I was in high school, for the top graduating maths student. I said “thank you” and then put the book aside, and didn’t even look at it for about two years, until I was in college. Then I picked it up one day and started reading it, and I was absolutely flabbergasted. The particular thing I was captivated by was a passage that Polya wrote about a discovery by an 18th century mathematician called Leonhard Euler, who is often considered one of the greatest mathematicians of all time. Polya wrote about how Euler solved a sum which no one else had been able to figure out. He realised that the sum was pi squared over six, pi being the famous number that gives the ratio of a circle’s circumference to its diameter.

The first amazing thing is that this problem does not seem to have anything to do with circles. In fact, it has to do with squares and what happens when you add up their reciprocals. How on earth do these squares turn into circles? Polya takes us through this discovery process, in which Euler violates just about every rule your high school maths teacher taught you. He says, in effect: “Let’s pretend that the sine function is a polynomial. What polynomial would it be?” Sines and polynomials are two kinds of functions that are central to mathematics, from high school on. So he takes two things that everyone thought were different, and finds a connection between them. That is exactly what great scientists do – not just mathematicians.

This is something that maths books, especially maths textbooks, consistently get wrong. They do not teach students to discover things. At best they teach students how to prove things, but most of them don’t even do that. Polya teaches you how to discover. It’s easy to think that this is just a book about maths, but it’s not. When you read Polya’s book you will learn about science in general. He does an amazing deconstruction of scientific psychology. He talks about making analogies. And he is fond of guessing, which is something you’re not supposed to do in school mathematics. But Polya realises that to discover things, you have got to make leaps of intuition. You need to guess what the answer is. They won’t tell you that in school, but it’s true.

Next up is Persi Diaconis and Ron Graham’s Magical Mathematics, which looks at the magic of maths as revealed through card tricks.

I picked this book because I wanted to have a very recent title on my list. Some of the books I have chosen, such as Polya’s, are fairly old. This book just came out within the last year. Persi Diaconis and Ron Graham are two of the most fascinating and fun mathematicians you will ever meet. They both exemplify mathematics as entertainment, even though they also do very serious maths. Persi Diaconis is a great statistician and Ron Graham is an expert in combinatorics. But a lot of their maths is inspired by simple problems, and the enjoyment they take in them really comes through in this book.

What kind of card tricks do they do?

Diaconis is both a professor of maths and also a skilled magician. As a teenager he completely devoted himself to magic, and learnt tricks from the best in the business. In this book he presents a lot of self-working magic tricks involving cards. This means that the principles involved are not sleight-of-hand but mathematical. For example, he starts the book with something called the Hummer Shuffle – a special way of shuffling cards that anyone can do. You take a deck of cards and cut them anywhere. Then take the first two cards and turn them over. Cut the deck again and turn the top two cards over. Keep doing this as long as you feel like it.

After a while, everyone in the audience will agree that the cards have been completely randomised. But amazingly, there are subtle ways in which you are not randomising it. The magician can send a deck of cards out into the audience, which they shuffle like crazy, and then gets the deck of cards back. Through simple procedures he can then undo what the audience has done, and get marvellous tricks out of it.

One of them is to hide in the deck a royal flush – an ace, king, queen, jack and 10 of one suit – and make it magically appear even after the audience have shuffled the pack. He will do some more shuffles, then he will spread out the whole deck and you will see that only five cards are face down. Then he asks someone to turn the cards over and – lo and behold! – they are the royal flush. It looks like magic but he is using maths. In the book he explains how this trick and others like it actually work. All of them come from mathematical principles, such as keeping certain patterns intact (as in the Hummer Shuffle) or tricking the audience into revealing information without realising it. The skill of the magician is turning this into a magic trick and not just a boring mathematical problem.

Indra’s Pearls concentrates on the beauty of maths and explores the ideas of Felix Klein, who was a geometer from the 19th century.

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.

Among the shapes that are really beautiful in the book are fractals.

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.

Tell me about your next choice, Penrose Tiles to Trapdoor Ciphers by Martin Gardner.

In a way, this book is representative of the many books that Martin Gardner wrote. He wrote a column called “Mathematical Games” for Scientific American magazine for many years. If you talk to 10 mathematicians at random in the US – at least of a certain age – probably half of them got excited about maths by reading Martin Gardner’s column and books.

He wrote about recreational mathematics, and encouraged people to participate in mathematics by posing interesting questions. His readers would write in and suggest ideas. I could have picked any of his books, as he has written dozens of them. But this one is of particular interest because it contains the columns that introduced the world to two of the most fascinating mathematical discoveries of the 1970s.

The first one was Penrose tiles. These are tiles that cover a plane with pentagonal symmetry, which had long been considered impossible. The trick is that they have a long-range order that is different from simple repetition, as in a crystal. In fact, analogues of the Penrose tilings were later discovered in chemistry and called “quasicrystals”. The discoverer of quasicrystals, Daniel Shechtman won the Nobel prize for chemistry last year. But what the newspapers mostly failed to report was that mathematicians discovered them first. In particular, one mathematician, Sir Roger Penrose, who discovered that if you slightly change the rules you can get a pentagonal symmetry. It’s very similar to what Euler did to evaluate that sum I talked about earlier. Great scientists are not afraid of bending the rules.

What about the trapdoor ciphers – what were they?

They were also discovered in the 1970s, and before then no one had any clue that these things existed. Basically they are secret codes that let anyone send a message. Before trapdoor ciphers were discovered, it was assumed that if you wanted to have a secret code the sender and recipient both needed to know the key. But then Ronald Rivest, Adi Shamir and Leonard Adleman discovered a “public key” code where the sender of a message doesn’t need to know the secret for decoding it. Public key cryptography is the more common name for it these days. These codes have enabled technology like ATMs and smart cards. Anyone can make a deposit, which involves sending a coded message to the bank, and yet it can’t be stolen because of a public-key or trapdoor code. Nowadays they have gotten quite a bit more sophisticated, but they are all based on mathematical principles, and the fundamental idea is not that difficult. Gardner explains it very well.

It sounds very clever. Finally you have chosen a children’s book, The Dot and the Line.

The Dot and the Line is another book that had a lot of personal meaning to me. My parents gave it to me when I was about nine years old. It’s a simple story about a romance between a dot and a line, and a way of teaching some mathematical concepts to kids without them even realising it.

The line is trying to impress this dot, and the dot is bored by him because he is too straight. She is in love with a squiggle. The line wants to impress her, and so he works and works and finally learns to bend himself and he makes an angle. In the process you are learning a mathematical term, about angles. Then he starts making fancier and fancier shapes and you learn some more mathematical terms like polygons and parallelograms. Finally he starts making curves and – well, I don’t want to give away too much of the plot.

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What makes the book stunning are the exquisite pictures of what he is doing. Many of them are mathematical, but when you are nine years old you don’t even realise you are learning maths. You are just seeing fascinating pictures, and learning mathematics through beauty. That is what makes it wonderful.

So does the line manage to win over the dot?

Yes, she leaves the squiggle behind. Reading it now, I have mixed feelings about it because I kind of sympathise with the squiggle! But Norton Juster – who is better known for another children’s book, ThePhantomTollbooth – had a keen sense of the mathematical aesthetic, that there is beauty in patterns. And these patterns make you want to find out what is behind them.

The beauty in maths is something that you also focus on in your latest book, The Universe in Zero Words.

Yes, one of the key messages I was trying to get across in that book is that mathematics can be beautiful. I give the history of 24 of what I think are the most beautiful equations. I try to explain what they mean and also why they are beautiful, as well as important.

I think the production of the book is absolutely lovely. I have had so many friends tell me that they didn’t expect it to be so beautiful to look at. I can only claim a tiny amount of credit for that. Most of the credit goes to the co-publishers of the book. But one thing I did contribute to was the idea of putting each of the equations into calligraphy. I hope this sends a subliminal message to the reader that I really treasure these formulas, and I think that they should be part of our cultural heritage, just like paintings or sculptures.

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Dana Mackenzie

Dana Mackenzie is an author and journalist who writes for Science, Discover and New Scientist. He has a PhD in mathematics from Princeton, and was a mathematics professor for 13 years before becoming a full-time writer. His latest book is TheUniverseinZeroWords