Mathematics: Why the brain sees maths as beauty

Brain scans show a complex string of numbers
and letters in mathematical formulae can evoke
the same sense of beauty as artistic
masterpieces and music from the greatest
composers.
Mathematicians were shown "ugly" and
"beautiful" equations while in a brain scanner at
University College London.
The same emotional brain centres used to
appreciate art were being activated by "beautiful"
maths.
The researchers suggest there may be a
neurobiological basis to beauty.
The likes of Euler's identity or the Pythagorean
identity are rarely mentioned in the same breath
as the best of Mozart, Shakespeare and Van
Gogh.
The study in the journal Frontiers in Human
Neuroscience gave 15 mathematicians 60 formula
to rate.
One of the researchers, Prof Semir Zeki, told the
BBC: "A large number of areas of the brain are
involved when viewing equations, but when one
looks at a formula rated as beautiful it activates
the emotional brain - the medial orbito-frontal
cortex - like looking at a great painting or
listening to a piece of music."
The more beautiful they rated the formula, the
greater the surge in activity detected during the
fMRI (functional magnetic resonance imaging)
scans.
"Neuroscience can't tell you what beauty is, but if
you find it beautiful the medial orbito-frontal
cortex is likely to be involved, you can find beauty
in anything," he said.
A thing of great beauty
Euler's identity: Does it get better than this?
To the untrained eye there may not be much
beauty in Euler's identity, but in the study it was
the formula of choice for mathematicians.
It is a personal favourite of Prof David Percy from
the Institute of Mathematics and its
Applications .
He told the BBC: "It is a real classic and you can
do no better than that.
"It is simple to look at and yet incredibly
profound, it comprises the five most important
mathematical constants - zero (additive identity),
one (multiplicative identity), e and pi (the two
most common transcendental numbers) and i
(fundamental imaginary number).
"It also comprises the three most basic
arithmetic operations - addition, multiplication
and exponentiation.
"Given that e, pi and i are incredibly complicated
and seemingly unrelated numbers, it is amazing
that they are linked by this concise formula.
"At first you don't realise the implications it's a
gradual impact, perhaps as you would with a
piece of music and then suddenly it becomes
amazing as you realise its full potential."
He said beauty was a source of "inspiration and
gives you the enthusiasm to find out about
things".
The hugely influential theoretical physicist Paul
Dirac said: "What makes the theory of relativity
so acceptable to physicists in spite of its going
against the principle of simplicity is its great
mathematical beauty. This is a quality which
cannot be defined, any more than beauty in art
can be defined, but which people who study
mathematics usually have no difficulty in
appreciating."
Mathematician and professor for the public
understanding of science, Marcus du Sautoy, said
he "absolutely" found beauty in maths and it
"motivates every mathematician".
He said he loved a "small thing [mathematician
Pierre de] Fermat did". He showed that any prime
number that could be divided by four with a
remainder of one was also the sum of two square
numbers.
So 41 is a prime, can be divided by four with one
left over and is 25 (five squared) plus 16 (four
squared).
"So if it has remainder one it can always be
written as two square numbers - there's
something beautiful about that.
"It's unexpected why should the two things
[primes and squares] have anything to do with
each other, but as the proof develops you start to
see the two ideas become interwoven like in a
piece of music and you start to see they come
together.
He said it was the journey not the final proof that
was exciting "like in a piece of music it's not
enough to play the final chord".
He said this beauty of maths was missing from
schools and yet amazing things could be shown
with even primary school mathematical ability.
In the study, mathematicians rated Srinivasa
Ramanujan's infinite series and Riemann's
functional equation as the ugliest of the formulae.