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The Secret Life of Chaos Page #5
- Year:
- 2010
- 60 min
- 314 Views
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?
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
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.
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.
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.
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
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
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.
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.
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.
organised like fractals.
Even our brain cells
interact according to simple rules.
The way evolution refines
is one of the most intriguing
ideas in recent science.
My interest in my PhD research
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
is indeed incredible.
But it operates on
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"The Secret Life of Chaos" Scripts.com. STANDS4 LLC, 2024. Web. 29 Aug. 2024. <https://www.scripts.com/script/the_secret_life_of_chaos_17702>.
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