To get a started in this video
I’ll review some of the basic ideas of scaling
that we talked about in the very beginning of the course.
And I’ll introduce a little bit about terminology as well.
So remember that the idea behind scaling.
And really the idea behind this entire course.
is to look at how properties of object change,
when the size of the object changes.
So for an example
Suppose we have some shape
And this is, you know some 3 dimensional blob thing
And then if I were to scale this up
So the idea is that picture on the right
is the same as on the left.
It’s just 2 times larger
This is not a perfect drawing
But that’s just sort of the idea.
And we wanna know how might properties
of this shape change when this happens.
So by 2x here, I mean this is 2 times larger
2 times larger in this direction,
in this direction and in that direction.
Alright, so suppose in particular that
we’re interested in the volume.
How the volume of this shape changes.
Well, volume scales with length cubed.
So I would write that volume scales as length cubed.
So what this means is that if I were to double L
V goes up by not 2, not 4 but 2 cubes equals 8 times.
So this shape has a volume 8 times larger than this shape.
And so we’ve seen that sort of relationship
when we’ll be thinking about dimension
the dimension the exponent
that's the sort of an interesting thing
that’s the dimension when we are looking
at self similarity and so on.
And this relationship here
I might write out by saying that
V is proportional to L cubed.
So V is proportional to L cubed.
So what that means is
is that they rise and fall in proportion to each other.
If L cubed doubles, V will doubles
If V goes down by a third, L cubed must go down by a third
So that these two quantities are in proportion to each other
And that will be written mathematically
I’ll just write this again
V equals k L cubed.
And so k here is some constant.
It will be different for different shapes.
If it was a sphere, it would have one form.
If it was a cube, it would have some other form.
Who knows what that number would be for this shape.
But there would be still some L cubed dependent
or L would be some length
could be this height, could be this radius or something
This is also sometimes written
So that V is proportional to L cubed
I kind of prefer this notation
and this is a certainly common.
And then this is how one would say this.
So V is proportional to L cubed.
Let’s think about some other properties.
Suppose I was interested in the surface area.
Well, so areas, we know that it’s 2 dimensional.
So that’s gonna be proportional to L squared, 2 dimensional.
And then if I maybe I wanted to look at the height, to this
That’s gonna be proportional just to L.
Right, height is 1 dimensional.
And L is same as L to the 1
Notice that I haven’t specified, deliberately
what L is here and L is just some length
That could be this length, this could be this length.
that could be that length.
The main thing is we’re establishing the relationship
between a linear dimension of length
and either surface area or height or volume.
So again volume scales as L cubed,
this proportional to L cubed
Surface area or any areas are proportional to L squared.
And height is proportional to just simply L
So the next quiz
If you are feeling rusty with these ideas you can practice this
and then we’ll start thinking about scaling
in the context of metabolic systems.