Fractals and Scaling: Review of basic scaling

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.

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