Here solutions to quiz 2.
So first concern a triangle
and triangles are two-dimensional.
So what would happen to the area,
if I scale this up by a factor of 2, sorry 3, excuse me.
3 squared would be 9.
So If I can draw this,
Scaled this up by a factor of 3.
And this area is 9 times what it was before.
Ok, problem 2,
No we’re dealing with a cube
And a cube has a dimension of 3.
And this time the scaling factor is less then 1.
So we are shrinking it down. Let me it draw that.
So we took a cube and shrank it down
by a factor of 3 this way,
factor of 3 this way,
and a factor of 3 that way.
What happens so we’ve got a third cubed,
this 3 is because the dimension is 3.
This is because we’re shrinking by a factor of 3.
It's a third of the size it was before,
So that’s 1 over 27.
So the volume of the smaller cube
is a 27th the volume of the large cube.
The last question on the quiz concerns the Sierpinski triangle.
Recall that the dimension was log3 over log2
which evaluates to around 1.585
So what happens if we scale this up by 2.
What we’ve been doing is we’ve been taking the scale factor,
in this case 2, and raising it to the dimension D.
And now that dimension happens not being integer.
But that’s no big deal.
We can handle on a calculator at least.
2 to the 1.585
It’s around 3, almost exactly 3.
So let’s think about what that means,
It’s not a coincidence.
So if I start with the Sierpinski triangle of this size,
And I scale it up by a factor of 2,
I am left with this.
This length gets scaled up by 2 and
that length gets scaled up by 2.
How much more ink or mass or area is in this picture?
Compare to that,
well, you can see that big shape has 1,2,3.
The little shape 1.
So that’s where this 3 comes from.
So what the dimension like this is telling us
is that every time we scale it up by a factor of 2
We double all of these lengths.
The mass or volume or size of the shape goes up by 3.
So this gives us another way to think about dimension,
It tells us how the properties of an object change,
as the shape, as the size of the object scale up or down.