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.