Fractals and Scaling: Dimension and scaling


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




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