Original subtitles

(ambient music)

Mathematics has been described as the science of patterns.

In mathematics, there are patterns in numbers, in shapes,

in probability, in motion.

And nature has visible regularities that I see

more than I ever use to, because now I'm looking for them.

They're everywhere, really, and if we have eyes to see them,

we can really enjoy them as well as try and understand them.

This is the greatest free show on Earth.

And I think that if I could just get my students

to put down their cell phones and concentrate

on this world around them, they can see wonderful beauty.

And I hope it will excite them as it does me.

And what always excites me is that the stunning variety

of shapes and patterns to be found in the natural world

are all the result of Mother Nature following

biological, chemical, physical principles

and the underlying mathematical structures.

And in the world of nature, one of the most basic rules

is always be efficient.

Nature does appear to seek the most economical,

the most efficient ways of achieving Her end.

She seeks to minimize energy.

It is certainly the case that a lot of mathematical models

of natural processes involved minimization schemes.

And one of those natural minimization schemes

is don't expend energy if you don't have to

just like some of my students.

Some of the most primitive creatures,

such as sponges and corals simply let their food

waft over them.

In technical terms, they are asymmetrical.

For most creatures, some sort of symmetry

is both important and revealing.

Look at the sea anemone, it's another creature

that doesn't move or at least not very much.

And it displays what is known as radial symmetry.

You can cut through a sea anemone at basically any angle,

and the cross-section will always be the same.

A nice variation on radial symmetry is rotational symmetry.

The starfish, for example, if you carefully arrange

the stars on the starfish, then you will have

pentagonal symmetry,

which means that there are five axes of symmetry,

each at an angle of 72 degrees from the next one.

We see it in flowers, Vincas, I believe,

are pentagonally symmetric, you've only got to rotate

through a certain angle to leave the flower

indistinguishable approximately from where it was before.

Closer to home, if you carefully cut through an apple,

you'll find the pits are arranged

in a pentagonally symmetrical pattern.

But the commonest type of symmetry by far

is what's known as bilateral symmetry.

Nearly all creatures on the planet exhibit

bilateral symmetry.

And again, it has to do with mobility.

Any designer would put an organism's senses and its mouth

at the head of the organism's motion.

In bilaterally symmetrical animals

that immediately presumes an up and a down,

and also left and right.

There's so much going on in the natural world,

so many different causes and effects,

that one might look for a unifying principle

to describe that.

And I would say elegance and efficiency is probably the one.

Take a look at a beehive for example,

we're all familiar with the hexagons in a honeycomb,

but why hexagons?

Why not circles, which have a minimum perimeter

for a given area.

Surely, that's a superb example of efficiency.

Unfortunately, the problem with circles

is that you can't pack them together without leaving gaps.

So the nearest regular polygon that comes to a circle

is the regular hexagon.

And so those six sides minimize the perimeter

for a given area.

And that may have something to do with the way

beehives are constructed, because the bees,

whether they know it or not are superb engineers.

And it turns out given those space-filling constraints,

the hexagons use the least wax for the most storage space.

And the icing on the cake, if you're a bee,

is that for every six hexagons you make,

you get another one for free.

Hexagons are a recurrent theme in nature,

turning up in geological formations, insect eyes,

and perhaps most famously, in snowflakes.

When you hear the statement, and it's often heard,

that every snowflake is individual, is unique,

it's true but it depends on the scale at which

you're looking at it.

Because if I just look at them on my sleeve

as they fall on my dark coat, yes some are big,

and some are smaller, but I don't really get a sense

of the uniqueness at the level we're talking about.

And if you go down very, very closely with a microscope,

if you could imagine the molecules,

then they're all the same, no one ice molecule

is different from any other.

That six-fold symmetry is, in an amazing way,

perpetuated through the different scales.

And so we see the six-fold symmetry, the hexagonal symmetry

in snowflakes because of that.

And yet, they are different.

There's little irregularities, perhaps,

so they're not exactly hexagonally symmetric.

So in that sense, every one is different.

But the life history depends on, of course,

like human beings, where we've been, what we've experienced,

what we've encountered.

So each snowflake falls a different path to the Earth,

and therefore it falls through different regions,

regions of different humidity, different temperature,

buffeted by the wind, perhaps, melting a little bit

and then refreezing.

So in that sense, yes, they are utterly unique.

They are an excellent example of one of nature's

favorite mathematical features, deterministic chaos.

One of the features of deterministic chaos

is that there is an in-built sensitivity

to initial conditions.

And that applies to the snowflake, in the sense

that each one has its own unique path

through its brief lifetime.

This is one of the features of chaos that,

a small variation in the initial conditions,

and this is related to what's known as the butterfly effect

can induce downstream, so to speak,

in time, very different circumstances.

The butterfly effect seems really, really silly.

A butterfly flapping its wings somewhere in South America

could give rise to some enormous storm over Japan.

Now, this sounds almost sort of zen,

but it's certainly part of what is known

as deterministic chaos.

So that's why the weather is actually unpredictable

beyond a few days because some slight change

can be replicated en masse downstream, so to speak,

a few days later.

And so the structure of the weather

is inherently unpredictable.

We can a broad idea over a few days,

and even perhaps into 10 days,

but nevertheless, there's an ultimate limitation there.

Often, the manifestation of deterministic chaos in nature

produces complex shapes and forms that have

a fractal-like structure.

A fractal is essentially, crudely speaking,

a picture of chaos.

There are several ways of looking at fractals,

but essentially, they are things,

they are geometric entities which have the same geometric

or statistical properties no matter how small,

how far down you go, how much you magnify them.

Let me give you a classic example of the

Cook or Koch Snowflake Curve

which is a beloved of many students,

especially the ones I teach, because it's so fascinating.

You take an equilateral triangle,

every side is the same length.

You then remove the middle third of each side

and put another little equilateral triangle

or at least two sides of it on there,

so you've got a sort of star.

And you continue that process, this is called iteration.

You continue it ad infinitum, and what you end up with

is a crinkly wonderful shape pattern

which actually has infinite length and finite area,

because it can be enclosed in a circle,

its therefore got finite area, and this is the archetype

or one of them of a fractal.

Another fractal of note is the Sierpinski Triangle

or the Sierpinski Gasket.

If you take an equilateral triangle and mark the midpoint

of each side and join those points together,

you've got another triangle pointed downwards as it were,

paint that black or remove it,

and then do the same thing with the remaining three

white triangles.

If you continue that process, you get a strange shape

that seems to appear on some shells in which

has these triangles, these patterns

to several different scales.

And that appears to be the result of some chemical process

that I don't understand that mimics, in some way,

this Sierpinski Triangle, at least to several levels down.

And it's quite fascinating, they're beautiful.

It's wonderful, I try to get across, I don't know,

the beauty of nature.

When I translate all this into the classroom,

it's very easy to get excited and passionate about it,

and I think it's a wonderful tool

for younger children as well to try and instill in them

the curiosity, well they have the curiosity younger children

it's not been dissipated from them.

But just to focus them and ask questions.

What's going on here, what's that?

It's just wonderful.

Science is all about asking questions,

and in that sense, scientists and mathematicians

are still kids.

(gentle piano music)