In this video I’ll give a slightly different way
of thinking about the dimension.
When I introduced the self-similarity dimension
the picture was we have an object
and we’re trying to figure out the dimension
by looking at how many small copies fit in big copy.
In this video, I’ll take slightly different perspective.
We’ll imagine that we already know
an object’s dimension and then we can use
that to predict what happens to that object
when we scale it up or down.
I’ll start by looking at some simple geometric
objects with integer dimension.
So let’s think about what happens to shapes
if we scale them up.
I’ll explain what I mean that in just a second.
So here are 3 shapes that I drew,
not very well but I did myself.
So here’s a line, that’s one-dimensional.
Here’s a square that’s two-dimensional.
And here is a cube that’s three-dimensional.
So what would happen if I took these shapes
and double them in size.
And I did that by going
downstairs in my office building
and making a photocopy of this drawing,
and asking the machine to make it twice as large.
So this gets doubled.
This is double.
This length is doubled and so on.
So the question is what would happened
to the total, saying, master size of these shapes.
Ok, so we have a line, and I double it.
Well, this line is now twice as big as before.
If this was, this would use twice as much ink,
It would if this was a piece of metal
this would be weight twice as much.
If this, say, a piece of carpet
and I doubled it.
This carpet is now four times as big.
If this has side, side 1,
This has side 2, 2 times 2 is 4.
So an area 1 goes to an area of 4
And this is basically
same sort of calculation we did,
same sort of thinking we did,
when we calculated the self-similarity dimension.
Number of small copies equals magnification factor to the D.
So for the line,
we have when we magnify it.
So we double it 2 to the D, sorry ,
the line 2 to the 1 is 2
So we double the magnification factor now is 2,
we have twice as much line,
twice as many small lines as we did before.
For the square, it is 4.
So now I am thinking, not so much
for not only number of small copies,
May be something like the overall mass
demount of material,
This is a square carpet or something
demount material in the carpet.
An then for the cube, cube is three-dimensional.
If we have this cube and picture of a solid cube here
and then we have a magic machine
that doubles the size of the solid cube
We would have 8 times as much
If this is length 1, this cube has a volume of 1.
This is length 2.
2 times 2 times 2 will give me 8.
You can also picture that 1,2,3,4,5,6,7,8
These cubes fit in here.
So for cube,
we would have 2 cubed equals eight.
So if you put a line, a square, a cube
in the copy machine and ask
how much bigger they are,
how much more massive they are,
that copy machine does different things: 2, 4 and 8
You can an imagine a copy machine that works on volume as well.
So may be this is some sort of
crazy 3d printer or photocopier. Alright.
So now, let’s think about the different shapes.
So this is the Koch curve.
So here is a small Koch curve.
And then I did the same thing.
I put in the photocopy machine
and I said alright make it twice as big.
So now the question is if I go from this to this,
how much bigger is this?
Bigger measure in terms of,
may be the total amount of ink
I would need to make this.
or If I was building a sculpture,
how much heavier would this be than that one.
And you might think, oh well,
it’s a line so just to be twice as big.
But it doesn’t have the dimension of one.
It’s actually not a line, it’s a fractal.
It’s in between dimension one and two.
So the dimension of the Koch curve was about 1.262
So how much bigger is the twice as large Koch curve.
Let’s see. I’m gonna need to take 2
and raise it to the 1.262
and I get 2.399, let’s call around 2.4
So dimension between one and two,
a doubling behavior between lines and squares,
So this is may be a less obviously
geometric way to think about dimension.
It’s really the same as a self-similarity dimension before.
It’s just the different perspective.
But it’s another way to think about
what these numbers mean,
if something that’s 1.26 dimensional
seems too far out of strange,
may be more down to earth way is
just to think about what happens
when you double the size of a shape.
Some shapes when you double the size,
they double.
Some you double it, they go up by 2 to the 2 is 4.
Sometimes they go by 8.
And here is a shape that
the behavior is 2 to the 1.262
So that’s another way of thinking about dimension,
It tells how the overall size or mass
or quantity or shape changes
If we scale it up or down