In this video, I wanna say a little bit about
the differences mathematical fractals and real fractals.
So mathematically a fractal is
an object that self-similar across all scales.
You can keep zooming in forever and
you continue to see the same shape
repeated again and again.
So in the Koch curve
you zoom in again and again and again
And you can go infinitely small
and you keep seeing the shape appear.
Real fractals obviously don’t have this property
that you can zoom in forever.
So may be this Koch curve looks a little bit
like a rocky coastline of England or our main.
And we’ve seen a picture of coastline
we will look at this next unit.
If you zoom in, so coastline is bumpy
And you zoom in and you see bumps
And you zoom in and you see more bumps and so on.
And that behavior persists over several scales,
many orders are bounded perhaps
But eventually you would start to be zoomed in
so much that you will be looking at
individual molecules if that’s even possible.
So it’s not look like this forever.
There are some cut off scales
Similarly this fern from a couple days ago,
it’s dried out a bit now.
It still looks pretty nice I think.
As a fern shape,
that fern shape is made up of fern shapes.
This fern shape is made up of fern shapes
And it goes on maybe once more.
But it doesn’t continue forever.
So real fractals are different than mathematical fractals
and that the self-similarity doesn’t go on forever.
Sometimes objects like these are said not to be fractals
but to be fractal-like.
So more generally,
the notion of mathematical fractal is an abstraction, it’s an ideal.
It exists in the world of math,
but it doesn’t really exist exactly in the actual world,
the physical world in which we live.
This is not at all unique to fractals.
The same is true really for almost any geometry.
So think about the circle.
Mathematically, geometrically, a circle is defined perfectly.
But there are no exact perfect circles in the world.
Here is a plate from the dining hall
and it looks pretty circular.
But it’s not exactly circular.
It’s little bit rough. It’s pretty old.
It has got a chip missing here.
It’s approximately very well by a circle.
We would say, hey this is a circular plate.
But it’s not a perfect circle.
It’s not a mathematical circle.
And we use circles all the times to describe things
that are circle or circle-like to various degrees.
The plate, the lid on a jar,
somebody’s face or head,
all can be approximated by circles.
These are approximations make sense ?
Sometimes yes, sometimes no.
It depends on the context.
But the geometric concept of circle
is still incredibly incredibly useful.
Similarly, the geometric idea the mathematical idea
fractals are incredibly incredibly useful.
Even though there are no exact perfect
mathematical fractals in the world.
So again this issue,
between abstraction you get a math and real world,
so comparing two is
something to comes up cross the sciences
It’s not unique to fractals at all.
And in term of this course,
the question of when is something efficiently fractal-like
that it makes sense called a fractal.
Well, that question is gonna come up again
in different ways particularly
when we start thinking about scaling
and power laws towards to end of the course