Human beings are absolute pattern detectors.
We spot patterns and we spot structures.
So you can say, look,
here is this beautiful pattern, and underneath that pattern,
there is this wonderful mathematical idea.
Part of the reason we find it beautiful is because of its inherent symmetry in
proportion.
If you understand how something is put together,
if you can see that behind that beautiful form of poetry, art,
and music and literature, the beauty of architecture,
those shapes that nature produces, that's all mathematics as well.
My main research area has been group theory. Really it's the study of symmetry,
that's how I like to think of it.
A group is just a collection of things,
which could be numbers, it could be functions,
it could be matrices,
all of these things can form groups.
But the one I like the most is groups of symmetries.
Numbers, shapes, symmetry, all of those things are there in the universe,
the nature around us.
And this one here, this beautiful flower.
It's got five petals and that we've got a fivefold symmetry.
You can rotate it one fifth of the way round the circle.
You get lots of numbers like 5, 8, 13,
that form part of the Fibonacci sequence, which is very,
very useful in terms of plant growth because it relates to something called the
golden ratio.
If you look at a sunflower,
there are loads of spirals with Fibonacci numbers of seeds in them.
So the golden number is there for sure,
and you see it as well in some of the spiral shapes of seashells.
We have not invented the golden ratio, right?
We've discovered it and we see it all over nature.
This is the absolute lodestone of symmetry where I live.
These beautiful designs created by William Morris with botanical themes,
plants and animals.
He made fabrics, tapestries, he made wallpapers.
There are 17 different kinds of wallpaper.
Now, this seems counterintuitive because of course there are as many kinds of
wallpaper as you could design.
But what we're talking about there is the ways that the patterns can fit together.
These designs, they're all different,
but what they have in common is that they all have symmetry to them.
The famous artist, M.C. Escher,
designed lots of repeating patterns and tessellations.
He learned about the 17 wallpaper patterns from his brother who was a
crystallographer. And then of course,
Escher produces all these wonderful designs that we all know and love.
This collection of structures actually has applications beyond the design of wallpaper.
For instance, crystal structures.
Now, if you think of the layers of atoms that form inside a crystal,
there are these 17 possibilities.
So what's a symmetry?
Something you can do to that design that leaves it looking the same.
There are lots of vertical mirror lines you could draw, reflect this design,
it's going to look the same either way.
You could also shift it along in steps so you could translate it,
that's the mathematical word.
So it's very, very cool the way the symmetries can combine like this.
And that's one of the properties that they must satisfy to form this algebraic
object called a group.
So in this case, we've got a collection of all the symmetries of this beautiful design,
and anytime you combine any of them, the outcome is still another symmetry.
Just check. 1, 2, 3, 4, 5.
There's not one connection between mathematics and music, there are many.
It's got so much mathematics in the rhythms, the patterns, even the counting,
the way that frequencies match up to produce sounds that we find pleasing together.
The mathematician Leibniz said that music is what happens when we do
mathematics unconsciously.
You can't listen to a piece of Bach without knowing that it's mathematical.
The beautiful patterns and symmetries in that music.
If you're listening to a fugue or playing a fugue on the piano,
you'll get a theme that comes in and it comes back,
but it's upside down, inverted. Or it might be retrograde backwards.
Or it might be translated, up or down.
So these are just the same kinds of symmetries that we saw in the wallpaper, right?
People, somehow, they get put off the idea of mathematics.
They think mathematics is about numbers, and numbers are not creative,
they're rigid.
Whereas art is more free and creative.
Actually, of course, that is not right.
The most creative subject for me is mathematics.
It's just that we are creating with ideas, with patterns.
So yes, music is mathematics, just as art is mathematics, and so is literature.
Everything is mathematics.