A Smoother Pebble


Mathematics and Beauty

I remember the moment I discovered the astounding beauty of mathematics. I was eight years old and my father and I were cycling in the local park with him riding just behind me. I was amusing myself by mentally juggling with numbers, when suddenly I saw that the sum of the first n odd numbers ( 1 , 3 , 5 , 7 ...) is the nth square number ( 1 x 1 = 1, 2 x 2 = 4, 3 x 3 = 9, etc.). It was like a thunderclap - I was overcome by an intense feeling of transcendent beauty and elegance. It was as close to a mystical experience as I have ever had. It was as if I had penetrated into a hidden, mysterious world without the dirt and imperfection of the everyday one, had reached out and touched the sublime.

In my ecstasy I clamped the brakes, hard. Not having time to react, my father cannoned into me, and we both went flying. When we'd got ourselves and our bikes untangled, I wanted to talk about my 'find', but he wanted to talk about what the hell I'd thought I was doing.

From my coversation and reading, it seems that most people who get bitten by the maths bug can look back to a moment in their childhood when they 'got' it. Here is a passage from Bertrand Russell's memoirs:

At the age of eleven I began Euclid with my brother as tutor. This was one of the great events of my life, as dazzling as a first love. I had not imagined that there was anything so delicious in the world. ... From that moment, until Whitehead and I finished Principia Mathematica at the age of thirty eight, mathematics was my chief interest, and my chief source of happiness.

Or Paul Painleve, a French mathematician:

I remember weeping through excess of aesthetic delight, twice on the same day, some time in my fifteenth year. The first occasion was produced by the description in the Iliad of the parting of Hector and Andromache, the second was produced by the definition of acceleration given by Newton.

Well, he was French. The best known story of this kind is about Gauss, the greatest of all mathematicians, at the age of ten. His schoolteacher wanted to occupy his class of peasant boys for an hour and ordered them to find the sum of the numbers from one to one hundred, i.e. 1 + 2 + 3 + ... + 99 + 100. To the teacher's astonishment Gauss almost immeadiately wrote the answer down on his slate and sat back with a smile. He had noted that 1 + 100 = 101, 2 + 99 = 101, 3 + 98 = 101, ... 49 + 52 = 101, 50 + 51 = 101. There are altogether fifty such pairs, so the answer is 50 x 101 = 5050. Although the problem was a quite trivial one, there is in Gauss' neat insight a microcosm of all that mathematicians mean by the elegance of proof.

The morning after the bike ride, I had an experience common not only to mathematicians but to all research scientists at some time or other in their careers: the chagrin of being beaten to the punch, when my teacher told me my result had been known for several millenia. But this pettiness was minimal and fleeting, and in no way spoiled my pleasure at the beautiful thing I had discovered. A Babylonian had found it first, so what? I didn't want to own the result I'd found, anymore than you want to own a sunset. My joy wasn't even the joy of discovery, or of accomplishment. which are not restricted to the mathematician but known to all scientists or to anyone who solves a crossword puzzle or fits together a jigsaw. It had been, above all, an aesthetic experience.

For me, it truly was the ecstasy and devotion of a first love. From that moment on, I had only one desire - to learn more about mathematics. Love has faded since, as first love will, but we are still good friends.

Most people will find what I am saying bizarre and incomprehensible. Maths is just a dull school subject about arithmetic, cosines, rolling dice and hypoteneuses. Only a tiny number of people ever experience the sheer joy of mathematical beauty, and the great majority of people don't even know that it exists. This is a rather exceptional situation. The tone-deaf generally realise that there must be something in music they are missing, virgins grasp sex might prove enjoyable, the blind hear the sighted praise the beauties of nature; mathematical beauty is overwhelmingly a private preserve of mathematicians and the kids who will one day join the exclusive freemasonry themselves. In fact, forget mathematical beauty - most non-scientists don't realise that mathematics, as an independent branch of learning, even exists. Many who do simply assume that it is about programming enormous computers to do ever more enormous calculations.

When mathematicians say 'mathematics', they don't mean the school mathematics of counting, statistics or everyday geometry, the basic maths about the real world where we count apples, measure lengths and areas and gather analyse scientific data. They call that 'arithmetic'. School mathematics bears about as much resemblance to real mathematics as mythology bears to philosophy; not just richer, more complex, more sophisticated, but with different foundations, methods and nature. Mathematics is an art, one that deals with form and abstraction at their most general. It will always remain the preserve of the few.

Anyone can listen to music and start tapping their toes, but doing maths requires effort. Just doing maths is not enough, either. I'd been messing about with maths for years before stumbling into that enchanted garden I never knew existed. And although I enjoyed doing maths over the next few years and read widely, I didn't get the old feeling again with the same burning intensity until I started (much later) to read group theory and the theory of the infinite.

Russell made one of the best attempts to explain to the outsider the aesthetic of mathematics:


Mathematics, rightly viewed, possesses not only truth, but supreme beauty -- a beauty cold and austere, like that of scupture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.
All right, maybe you're sceptical of a lot of maths geeks claiming to be the custodians of sublime wisdom; what about the poet Novalis, who isn't in the trade union:


The Life of God is mathematics; all divine ambassadors must be mathematicians. Pure mathematicians is religion. Mathematicians are the only blessed people.

Postscript: For those readers who didn't get what I was saying about square numbers and odd numbers, the first square number is one, as one times one equals one. One is also the first odd number. So the first odd number equals the first square number. The second odd number is three. Add three to the one and you get four, which the second square number as it is equal to two times two. So the second square number is the sum of the first two odd numbers. The third odd number is five. Add the first three odd numbers together (i.e. one plus three plus five) and we get nine, which is the third square number as nine equals three times three. This pattern is true for all square numbers. In mathematical notation:


12 = 1

22 = 4 = 1 + 3

32 = 9 = 1 + 3 + 5

42 = 16 = 1 + 3 + 5 + 7

52 = 25 = 1 + 3 + 5 + 7 + 9 ,

62 = 36 = 1 + 3 + 5 + 7 + 9 + 11 .

Or more succinctly:



This image may also help:




(Images courtesy Wikipedia)

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