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

IT Project Manager

Competitive: Clearwater People Solutions Ltd: Our client based in Chelmsford a...

Business Intelligence Specialist - work from home

£40000 per annum: Ashdown Group: An established and growing IT Consultancy fir...

Business Intelligence Specialist - work from home

£40000 per annum: Ashdown Group: An established and growing IT Consultancy fir...

IT Manager

£40000 - £45000 per annum + pension, healthcare,25 days: Ashdown Group: An est...

Day In a Page

Read Next
 

Letter from Asia: The battle for the truth behind five journalists’ deaths in Indonesia

Andrew Buncombe
'Irritatingly Disneyfied': fashion vlogger Zoella  

Sure, teenage girls need role models – but not of the Zoella kind

Chloe Hamilton
Two super-sized ships have cruised into British waters, but how big can these behemoths get?

Super-sized ships: How big can they get?

Two of the largest vessels in the world cruised into UK waters last week
British doctors on brink of 'cure' for paralysis with spinal cord treatment

British doctors on brink of cure for paralysis

Sufferers can now be offered the possibility of cure thanks to a revolutionary implant of regenerative cells
Let's talk about loss

We need to talk about loss

Secrecy and silence surround stillbirth
Will there be an all-female mission to Mars?

Will there be an all-female mission to Mars?

Women may be better suited to space travel than men are
Oscar Pistorius sentencing: The athlete's wealth and notoriety have provoked a long overdue debate on South African prisons

'They poured water on, then electrified me...'

If Oscar Pistorius is sent to jail, his experience will not be that of other inmates
James Wharton: The former Guard now fighting discrimination against gay soldiers

The former Guard now fighting discrimination against gay soldiers

Life after the Army has brought new battles for the LGBT activist James Wharton
Ebola in the US: Panic over the virus threatens to infect President Obama's midterms

Panic over Ebola threatens to infect the midterms

Just one person has died, yet November's elections may be affected by what Republicans call 'Obama's Katrina', says Rupert Cornwell
Premier League coaches join the RSC to swap the tricks of their trades

Darling, you were fabulous! But offside...

Premier League coaches are joining the RSC to learn acting skills, and in turn they will teach its actors to play football. Nick Clark finds out why
How to dress with authority: Kirsty Wark and Camila Batmanghelidjh discuss the changing role of fashion in women's workwear

How to dress with authority

Kirsty Wark and Camila Batmanghelidjh discuss the changing role of fashion in women's workwear
New book on Joy Division's Ian Curtis sheds new light on the life of the late singer

New book on Ian Curtis sheds fresh light on the life of the late singer

'Joy Division were making art... Ian was for real' says author Jon Savage
Sean Harris: A rare interview with British acting's secret weapon

Sean Harris: A rare interview with British acting's secret weapon

The Bafta-winner talks Hollywood, being branded a psycho, and how Barbra Streisand is his true inspiration
Tim Minchin, interview: The musician, comedian and world's favourite ginger is on scorching form

Tim Minchin interview

For a no-holds-barred comedian who is scathing about woolly thinking and oppressive religiosity, he is surprisingly gentle in person
Boris Johnson's boozing won't win the puritan vote

Boris's boozing won't win the puritan vote

Many of us Brits still disapprove of conspicuous consumption – it's the way we were raised, says DJ Taylor
Ash frontman Tim Wheeler reveals how he came to terms with his father's dementia

Tim Wheeler: Alzheimer's, memories and my dad

Wheeler's dad suffered from Alzheimer's for three years. When he died, there was only one way the Ash frontman knew how to respond: with a heartfelt solo album