In this video, I’ll say more about self-similarity.

So first just the reminder that

self-similarity means that

an object is made up of smaller copies of itself.

So looks the same at different scales.

And moreover, the self-similarity continues across many scales.

So it’s not just like, oh here is a big thing

there is one little copy in it.

But the idea is that it repeats.

It gonna be a big shape

is made up of little shapes,

that shape is made up of little shapes,

that shape is made up of little shapes and so on.

And mathematically, strictly speaking for fractal,

That behavior continues forever

as you keep zooming in onto a shape.

Mathematically fractal, you see more

and more of the same shape.

The Koch curve is an example as you keep zooming in.

This ends infinite resolution and could focus in,

You will keep seeing that same shape

repeating again and again.

Ok, so fractals objects that are self-similar

or often said to be scale free.

They don’t have a sense of length or

typical scale or size associated with them.

So let me illustrate that a couple of different ways.

So in science and everyday life the scale or

size of something usually

is one of the first things we think.

Think about it, so for example a tomatoes.

It’s is the end of the summer here in the Northern hemisphere

and we have just a couple more weeks

maybe a month of getting tomatoes and tomatoes plants.

So this is the tomato

I picked from my garden this morning.

And tomatoes have particular size.

You might have a tomatoes that’s

a bit smaller than this.

Certainly you have tomatoes that larger than this.

This isn’t a giant tomato.

But it would be pretty rare to have a tomato

that’s larger than 2 pounds or kilogram.

And there are smaller tomatoes,

there is this little cherry tomato

that I picked this morning.

And this is about the smallest tomatoes get.

The point is that there is a typical size

associated with tomatoes.

Tomatoes there is average size,

And then if you larger, if you smaller,

we don’t have all different sizes of tomatoes.

We don’t have microscopic tomatoes,

we don’t have tomatoes as big as me,

as big as houses, as big as whale and so on.

Ok so, tomatoes have a scale.

If I another way think about that is

If I wanted to indicate the size of something else

I could put a picture of a tomato,

I could put a tomato next to it.

And you would know, oh, it’s about tomato size.

Because tomatoes have a size.

Ok, in contrast for fractals,

there is a Sierpinski triangle again.

They don’t have a characteristic size.

What’s the typical size of a triangle and a Sierpinski triangle.

Well, they're triangles of all different sizes.

That’s sort of the point as you’ve seen

as we zoom in more and more

we keep seeing more and more triangles.

And mathematically, that would go on forever.

So they’re triangles of all different sizes.

And that’s a very different situation than with the tomatoes

where tomatoes come in a pretty narrow range of sizes.

All right, here is another way to think about this.

So again something is scale free

that aren’t clues that as to its size

Tomato is a clue as to size

but a triangle and Sierpinski triangle is not.

So there is a way you think about that

Suppose that one morning you woke up

and you had been shrunk.

You were suddenly much smaller

than you were before.

May be you got a size of this guy now.

That would be horrible thing to think about.

Hopefully at least you still have two arms.

I just suppose you get shrunk down like this

Well you would know right away that you were small.

Because your bed would be gigantic.

And a tomato could feed you for a week

You would be as big as your cat.

Your cat could be a danger to you

for all sorts of clues as to size

in most aspects of the world in which we navigate

So you could tell right away that we have been,

that you have been shrunk.

On the other hand, imagine

and takes a little bit imagination,

But imagine you live inside a Sierpinski triangle

not our world,

not a normal world, but fractal world like this.

So this world consists of triangles

and triangles and triangles

and as you were seen every time you use

double size you get three times as many triangles

so triangles within triangles within triangles.

If you live in this world

and you were shrunk one morning all of a sudden,

you wouldn’t be able to tell,

there is no sense of scale here.

There is not typical tomato

or cat or bed or pillow.

It’s just the world of triangles

and triangles and triangles.

May be you can also think about

if you live in a tree

and the tree that went on forever

Branches that got bigger and bigger and bigger

And branches got smaller and smaller and smaller

You wouldn’t be able to tell what size you were

All you know is that their branches that have this relationship

that hold across all of these different scales.

So that’s another way to think about

what it means for something to be scale free.

And one of these consequences of shapes that are self-similar.