What is it that makes mathematics beautiful?

A UCL study has shown mathematical formulae can evoke the same sense of beauty as artistic masterpieces or music by famous composers

Share

Why do people make scholarly studies of Shakespeare's plays, Rembrandt's paintings, or Beethoven's symphonies? Quite simply, because we recognise in these works a beauty that can be inspirational. They inspire study to those so inclined, as does mathematics despite the fact that most people do not find it at all beautiful. Yet obviously some do, as a recent UCL study confirmed, so what is it about mathematics that makes it beautiful?

Ask this question of different mathematicians and you will get different answers. Some are inspired by geometry, others by algebra. Some think geometrically, others more algebraically, though I know no grounds for supposing any difference in the ways their brains function. My thinking is geometric, and as a child I could see, in a pictorial way why the product of two negative numbers was a positive number. That was beautiful, though a beauty that one could not easily share with others. This is the trouble with beauty — it is in the mind of the beholder, and particularly in mathematics it can be hard to explain it to others.

Sometimes one can explain the beauty of a result, but the other type of beauty — a beautiful way of proving that result, a way that appeals to our sense of symmetry, proportion and elegance — can be harder to get across. A good example is Pythagoras's Theorem for a right-angled triangle, which is usually stated as: the square on the hypotenuse (the slanting side) is equal to the sum of the squares on the other two sides. The most well-known example of such a triangle is one whose sides have lengths 3, 4 and 5, where of course 3 squared+4 squared =5 squared, though there is an infinite array of other examples with whole numbers and different ratios of side lengths.

This is a beautiful result, stated and proved in Euclid's famous treatise on geometry written in about 300 BC, but Euclid's proof is ugly. Since his time, and possibly before, more elegant proofs have been given, and the mere existence of such proofs speaks about the beauty of the result. If it weren't beautiful why would anyone want to find a better proof? But many people have been fascinated by the result, including US President Garfield in 1876. I have my own proof, which is to me a thing of exquisite beauty, and I once looked on the internet to see if it was there — there are zillions of proofs of Pythagoras's Theorem on the internet — but I didn't find it.

Leaving aside geometry, here is an elegant result that usually astonishes students when they first see it. Consider all prime numbers larger than 2. Which ones can be written as a sum of two squares? Each one is an odd number, so its remainder after dividing by 4 is either 1 or 3, and the result is very simply stated. Those having a remainder of 1 are sums of two squares (in a unique way), and those having a remainder of 3 are not. For example, 5 has a remainder of 1 and is equal to 1 squared + 2 squared but 7 which has a remainder of 3 is not a sum of two squares. Try a few examples and you may agree that this is indeed rather beautiful.

React Now

Latest stories from i100
Have you tried new the Independent Digital Edition apps?
iJobs Job Widget
iJobs General

Recruitment Genius: Customer Service Advisor

£15000 - £16000 per annum: Recruitment Genius: A Customer Service Advisor is r...

Recruitment Genius: Plant Fitter - Construction Industry

Negotiable: Recruitment Genius: This well established construction equipment d...

SThree: Trainee Recruitment Consultant

£18000 - £23000 per annum + Uncapped Commission: SThree: As a Trainee Recruitm...

Recruitment Genius: Factory Operatives

£7 - £8 per hour: Recruitment Genius: This high quality thread manufacturer ba...

Day In a Page

Read Next
Bahrainis on an anti-government protest in May  

Hussain Jawad's detainment and torture highlights Britain's shameless stance on Bahraini rights

Emanuel Stoakes
August 1923: Immigrants in a dining hall on Ellis Island, New York.  

When will the Church speak up for the dispossessed, and those that our political system leaves behind?

Stefano Hatfield
The difference between America and Israel? There isn’t one

The difference between America and Israel? There isn’t one

Netanyahu knows he can get away with anything in America, says Robert Fisk
Families clubbing together to build their own affordable accommodation

Do It Yourself approach to securing a new house

Community land trusts marking a new trend for taking the initiative away from developers
Head of WWF UK: We didn’t send Cameron to the Arctic to see green ideas freeze

David Nussbaum: We didn’t send Cameron to the Arctic to see green ideas freeze

The head of WWF UK remains sanguine despite the Government’s failure to live up to its pledges on the environment
Author Kazuo Ishiguro on being inspired by shoot-outs and samurai

Author Kazuo Ishiguro on being inspired by shoot-outs and samurai

Set in a mythologised 5th-century Britain, ‘The Buried Giant’ is a strange beast
With money, corruption and drugs, this monk fears Buddhism in Thailand is a ‘poisoned fruit’

Money, corruption and drugs

The monk who fears Buddhism in Thailand is a ‘poisoned fruit’
America's first slavery museum established at Django Unchained plantation - 150 years after slavery outlawed

150 years after it was outlawed...

... America's first slavery museum is established in Louisiana
Kelly Clarkson: How I snubbed Simon Cowell and become a Grammy-winning superstar

Kelly Clarkson: How I snubbed Simon Cowell and become a Grammy-winning superstar

The first 'American Idol' winner on how she manages to remain her own woman – Jane Austen fascination and all
Tony Oursler on exploring our uneasy relationship with technology with his new show

You won't believe your eyes

Tony Oursler's new show explores our uneasy relationship with technology. He's one of a growing number of artists with that preoccupation
Ian Herbert: Peter Moores must go. He should never have been brought back to fail again

Moores must go. He should never have been brought back to fail again

The England coach leaves players to find solutions - which makes you wonder where he adds value, says Ian Herbert
War with Isis: Fears that the looming battle for Mosul will unleash 'a million refugees'

The battle for Mosul will unleash 'a million refugees'

Aid agencies prepare for vast exodus following planned Iraqi offensive against the Isis-held city, reports Patrick Cockburn
Yvette Cooper: We can't lose the election. There's too much on the line

Yvette Cooper: We can't lose the election. There's too much on the line

The shadow Home Secretary on fighting radical Islam, protecting children, and why anyone in Labour who's thinking beyond May must 'sort themselves out'
A bad week for the Greens: Leader Natalie Bennett's 'car crash' radio interview is followed by Brighton council's failure to set a budget due to infighting

It's not easy being Green

After a bad week in which its leader had a public meltdown and its only city council couldn't agree on a budget vote, what next for the alternative party? It's over to Caroline Lucas to find out
Gorillas nearly missed: BBC producers didn't want to broadcast Sir David Attenborough's famed Rwandan encounter

Gorillas nearly missed

BBC producers didn't want to broadcast Sir David Attenborough's famed Rwandan encounter
Downton Abbey effect sees impoverished Italian nobles inspired to open their doors to paying guests for up to €650 a night

The Downton Abbey effect

Impoverished Italian nobles are opening their doors to paying guests, inspired by the TV drama
China's wild panda numbers have increased by 17% since 2003, new census reveals

China's wild panda numbers on the up

New census reveals 17% since 2003