Original subtitles

(mellow music)

Why does mathematics work?

Why is it so useful?

Is mathematics invented or discovered?

I happen to think we discover it rather than invent it

but it depends on one's philosophical presuppositions

what conclusions one will draw.

One of the most celebrated discoverers

of the role of mathematics in nature

was a young Italian merchant's son called Fibonacci.

Fibonacci was a 12th-century mathematician living in Pisa,

and he came up with a sequence which is

prevalent in nature today in one form or another,

known as the Fibonacci sequence,

one, one, two, three, five, eight, and so on.

Each number in the sequence

is the sum of the two previous numbers.

So, the next number after one will be one,

because there was nothing before it.

The number after that will be two, one plus one.

After that will be three, two plus one,

then five, then eight,

then 13, 21, 34, 55, 89, 144, and so on.

The whole sequence appears again, and again, and again

in one form or another, with surprising regularity.

And, it is amazing, nobody really knows why

this particular sequence is so important in nature,

but there it is.

That's what we see.

If you look at the distribution

of seeds on a sunflower head,

you see these wonderful spirals,

which are, in a sense optical illusions,

because they're not representative

of the order in which the seeds were developed.

Nevertheless, the number going clockwise,

and the number going anticlockwise,

are 95% of the time adjacent terms

in the Fibonacci sequence, and it's just amazing.

But there's more to the Fibonacci sequence

than the arrangement of seeds on a sunflower head.

The ratio of successive numbers in the sequence

gets closer and closer to a rather special value,

known as the golden number.

This golden number, one plus the square root of five,

all divided by two, is a natural consequence

of the geometry of a regular pentagon.

If you draw a regular Pentagon and you join up the corners,

these cords, these lines joining the corners,

intersect one another in a ratio

that can be shown to be this golden number.

The ratio of the larger part of that cord

to the smaller part is the same

as the whole length of the cord

divided by the larger part,

approximately 1.618, the golden number.

The golden ratio is a linear measure

in the sense that it's a ratio of two lengths.

If one translates this to the geometry of a circle,

one can get something called the golden angle.

It turns out to be approximately 137.5°

and that, again, figures quite commonly in nature

in the arrangement of leaves.

The most effective arrangement of leaves on a stem

is when the new ones sprout

at precisely the golden angle from the one below.

One consequence appears to be minimal blocking of sunlight

from leaves below when the sun is high.

If you look at a cactus from above,

some types of cactus at least,

I have a very good photograph of one

with a picture of my shoe there as well

which you can determine that this golden angle

is approximately represented as the leaves grew

from one to another to another and so,

the successive leaves fill the space with that angle

from the previous leaf, and it's quite magnificent.

I've measured some of these,

and while I can't measure them accurately,

it's a pretty darn good approximation, to 137.5°.

The more you look at natural phenomena

the more you see the evidence of golden ratios,

golden angles, and the Fibonacci sequence at work,

and often, that will result in the emergence of spirals.

They're everywhere if you care to look.

Seashells, the nautilus shell, the shell of a snail.

You can actually relate this spiral

approximately to the golden ratio.

If we go back to our golden rectangle

which has, long side approximately 1.618,

short side one, and you cut off a square,

what you have left in that rectangle

is another rectangle, a smaller one

which is also a golden rectangle.

The relationship between the areas of the squares

is extremely interesting.

They are in the ratio of successive Fibonacci numbers.

We might start with one, and then one, and then two,

and then area three, and then area five,

and then area eight, and area 13, and so on.

And if you keep doing that and then join corner to corner

with an arc of a circle

you get an approximate equiangular spiral,

and this is certainly very reminiscent of the spirals,

particularly the nautilus shell, that we see in nature.

So, once again, these numbers are almost ubiquitous,

and one of these days I'm going to ask God why.

One of the most enduring puzzles of mathematics in nature

relates to the patterns and markings we see on animals.

A leopard's spots, a zebra's stripes and so on.

It fell to one of the greatest mathematicians

of the 20th century, Alan Turing,

to shed light on the topic.

Now, he was a genius.

He published one paper and one only on the chemical basis

of morphogenesis, that was his only foray into biology,

but it was profound, it was seminal.

Morphogenesis relates in biological terms

to the various chemical changes that may take place

in an embryo that will ultimately lead

to patterns in the adult creature.

The equations are quite complicated

but mathematicians have found that by varying the parameters

in these equations you can get spots,

you can get stripes, you can get uniform colors.

These studies have helped answer some age-old questions

including whether zebras are white horses with black stripes

of black horses with white stripes.

People used to think that they were

black-striped white horses,

but the prevailing view is now the opposite.

Zebra embryos are completely black.

The white stripes appear during the last embryonic stage.

So, zebras it seems, are black horses with white stripes.

But not all of the colors in nature are created

by the interaction of chemical pigments in an animal's skin.

Some colors are created by microscopic structures

that split white sunlight into its component colors.

It's a phenomenon known as known as iridescence.

Many birds and insects display these beautiful colors.

Iridescence, I love that word because it is derived

from a Greek word for rainbow, iris and iridos,

and over the years I've studied rainbows in great detail.

Rainbows are all about some basic,

and yet very subtle geometry, and they will only occur

when certain very specific conditions are met.

The sun has to be shining,

there has to be rain somewhere,

and if the conditions are right,

if the sun is not too high in the sky,

then if you stand with your back to the sun,

the sunlight is scattered by the raindrops ahead of you,

and it's scattered in all directions,

but there's a concentration of the light

that is refracted inside the raindrop,

reflected from the backside of the drop,

and refracted out again.

The colors you see are from different raindrops.

There's myriads of raindrops, so it's cumulative effect.

A good portion of them will scatter right into your eye,

a good proportion scatter green, orange, whatever.

In a sense, a rainbow is a highly exotic image of the sun.

I just love rainbows.

I think most people probably do.

If I can go on to another phenomenon

which is related, the glory.

If you've ever flown on a plane

and been on the shadow side of the plane above cloud

you may well have noticed the shadow of the plane

surrounded by circular colored rings,

and that's back scattering of light by cloud droplets.

The smaller the cloud droplets are

the larger the radius of the glory.

This is an amazing phenomenon,

and the more you look into these

atmospheric optical phenomena

the more fascinating they become.

Interactions between sunlight and water droplets,

sunlight and ice crystals,

and always the all-important geometrical configurations

that link you, the observer, the sun,

and whatever water droplets or ice crystals

are creating the effect.

So, you get sun halos, moon halos,

fog bows, son dogs,

circumzenithal arcs,

circumorizon arcs, and so on.

It's a mathematical feast.

As I often say mathematics in nature

is the greatest show on Earth.

And, what thrills me is when a student,

and this happens quite a lot,

will either come to me at the end of the class

or sometimes after the course is over,

and will show me a picture they've taken,

or sketch out something they've seen,

and they're excited by it and they

don't perhaps understand what was going on.

"What was happening here?", they say, and I say

"Well I wasn't there, I don't know,

"but here's a possibility, here's what I suspect."

They're actually thinking, they're taking this stuff outside

into the greatest free show on Earth,

and they're thinking about it and they're thinking

about the underlying principles

and whether or not they're correct it doesn't matter

to that degree, it's the fact that they're thinking

and they're curiosity has been aroused,

and so I feel like I've made a difference

however small, in that students life.