The Secret Life of Chaos Page #5

he really wanted to do. To

Mandelbrot, it seemed perverse that

mathematicians had spent centuries

contemplating idealised shapes

like straight lines

or perfect circles.

And yet had no proper or systematic

way of describing the rough

and imperfect shapes

that dominate the real world.

Take this pebble.

Is it a sphere or a cube?

Or maybe a bit of both?

And what about something much

bigger? Look at the arch behind me.

From a distance,

it looks like a semi-circle.

But up close,

we see that it's bent and crooked.

So what shape is it?

Mandelbrot asked if

there's something unique

that defines all

the varied shapes in nature.

Do the fluffy surfaces of clouds,

the branches in trees and rivers,

the crinkled edges of shorelines,

share a common mathematical feature?

Well, they do.

Underlying nearly all the shapes in

the natural world is a mathematical

principle known as self-similarity.

This describes anything in which the

same shape is repeated over and over

again at smaller and smaller scales.

A great example are

the branches of trees.

They fork and fork again,

repeating that simple process

over and over

at smaller and smaller scales.

The same branching principle applies

in the structure of our lungs

and the way our blood vessels are

distributed throughout our bodies.

It even describes how rivers

split into ever smaller streams.

And nature can repeat

all sorts of shapes in this way.

Look at this Romanesco broccoli.

Its overall structure is made up

of a series of repeating cones

at smaller and smaller scales.

Mandelbrot realised self-similarity

was the basis of an entirely

new kind of geometry.

And he even gave it a name -

fractal.

Now, that's a pretty neat

observation.

But what if you could represent this

property of nature in mathematics?

What if you could capture

its essence to draw a picture?

What would that picture look like?

Could you use a simple set

of mathematical rules

to draw an image

that didn't look man-made?

The answer

would come from Mandelbrot.

Who had taken a job at IBM

in the late 1950s

to gain access to

its incredible computing power

and pursue his obsession

with the mathematics of nature.

Armed with a new

breed of super-computer,

he began investigating

a rather curious

and strangely simple-looking equation

that could be used to draw

a very unusual shape.

What I'm about to show you

is one of the most remarkable

mathematical images ever discovered.

Epic doesn't really do it justice.

This is the Mandelbrot set.

It's been called

the thumbprint of God.

And when we begin to explore

it, you'll understand why.

Just as with the tree

or the broccoli,

the closer you study this picture,

the more detail you see.

Each shape within the set

contains an infinite number

of smaller shapes.

Baby Mandelbrots

that go on for ever.

Yet all this complexity stems from

just one incredibly simple equation.

This equation has

a very important property.

It feeds back on itself.

Like a video loop, each output

becomes the input for the next go.

This feedback means that an

incredibly simple mathematical

equation can produce a picture

of infinite complexity.

The really fascinating thing

is that the Mandelbrot set isn't

just a bizarre mathematical quirk.

Its fractal property

of being similar at all scales

mirrors a fundamental

ordering principle in nature.

Turing's patterns, Belousov's

reaction and Mandelbrot's fractals

are all signposts pointing to a deep

underlying natural principle.

When we look at complexities

in nature, we tend to ask,

"Where did they come from?"

There is something

in our heads that says

complexity does not arise

out of simplicity.

It must arise from something

complicated. We conserve complexity.

But what the mathematics in

this whole area is telling us

is that very simple rules naturally

give rise to very complex objects.

And so if you look at the object, it

looks complex, and you think about

the rule that generates it,

it's simple.

So the same thing is both

complex and simple

from two different points of view.

And that means we have to rethink

completely the relation between

simplicity and complexity.

Complex systems

can be based on simple rules.

That's the big revelation.

And it's an astonishing idea.

It seems to apply

all over our world.

Look at a flock of birds.

Each bird obeys very simple rules.

But the flock as a whole does

incredibly complicated things.

Avoiding obstacles, navigating

the planet with no single leader

or even conscious plan. But amazing

though this flock's behaviour is,

it's impossible

to predict how it will behave.

It never repeats

exactly what it does,

even in seemingly

identical circumstances.

It's just like

the Belousov reaction.

Each time you run it, the patterns

produced are slightly different.

They may look similar,

but they are never identical.

The same is true of

video loops and sand dunes.

We know they'll produce

a certain kind of pattern,

but we can't predict

the exact shapes.

The big question is, can nature's

ability to turn simplicity

into complexity in this mysterious

and unpredictable way

explain why life exists?

Can it explain how a universe

full of simple dust

can turn into human beings?

How inanimate matter

can spawn intelligence?

At first, you might think that

this is beyond the remit of science.

If nature's rules are

really unpredictable,

should we simply give up?

Absolutely not.

In fact, quite the opposite.

Fittingly, the answer to this problem

lies in the natural world.

All around us, there exists

a process that engineers

these unpredictable complex systems

and hones them to perform

almost miraculous tasks.

The process is called evolution.

Evolution has built

on these patterns.

It's taken them as

the raw ingredients.

It's combined them together

in various ways,

experimented to see what works

and what doesn't,

kept the things that do work

and then built on that.

It's a completely

unconscious process,

but basically

that's what's happening.

Everywhere you look,

you can see evolution

using nature's

self-organising patterns.

Our hearts use Belousov-type

reactions to regulate how they beat.

Our blood vessels are

organised like fractals.

Even our brain cells

interact according to simple rules.

The way evolution refines

and enriches complex systems

is one of the most intriguing

ideas in recent science.

My interest in my PhD research

in complex systems was to see

how complex systems

interact with evolution.

So, on the one hand you have systems

that almost organise themselves

as complex systems, so they exhibit

order that you wouldn't expect,

but on the other hand, you still

have to have evolution interact with

that to create something that is

truly adapted to the environment.

Evolution's mindless, yet creative,

power to develop

and shape complex systems

is indeed incredible.

But it operates on

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