Fractals and Scaling: Self-similar and scale-free


In this video, I’ll say more about self-similarity. So first just the reminder that self-similarity means that an object is made up of smaller copies of itself. So looks the same at different scales. And moreover, the self-similarity continues across many scales. So it’s not just like, oh here is a big thing there is one little copy in it. But the idea is that it repeats. It gonna be a big shape is made up of little shapes, that shape is made up of little shapes, that shape is made up of little shapes and so on. And mathematically, strictly speaking for fractal, That behavior continues forever as you keep zooming in onto a shape. Mathematically fractal, you see more and more of the same shape. The Koch curve is an example as you keep zooming in. This ends infinite resolution and could focus in, You will keep seeing that same shape repeating again and again. Ok, so fractals objects that are self-similar or often said to be scale free. They don’t have a sense of length or typical scale or size associated with them. So let me illustrate that a couple of different ways. So in science and everyday life the scale or size of something usually is one of the first things we think. Think about it, so for example a tomatoes. It’s is the end of the summer here in the Northern hemisphere and we have just a couple more weeks maybe a month of getting tomatoes and tomatoes plants. So this is the tomato I picked from my garden this morning. And tomatoes have particular size. You might have a tomatoes that’s a bit smaller than this. Certainly you have tomatoes that larger than this. This isn’t a giant tomato. But it would be pretty rare to have a tomato that’s larger than 2 pounds or kilogram. And there are smaller tomatoes, there is this little cherry tomato that I picked this morning. And this is about the smallest tomatoes get. The point is that there is a typical size associated with tomatoes. Tomatoes there is average size, And then if you larger, if you smaller, we don’t have all different sizes of tomatoes. We don’t have microscopic tomatoes, we don’t have tomatoes as big as me, as big as houses, as big as whale and so on. Ok so, tomatoes have a scale. If I another way think about that is If I wanted to indicate the size of something else I could put a picture of a tomato, I could put a tomato next to it. And you would know, oh, it’s about tomato size. Because tomatoes have a size. Ok, in contrast for fractals, there is a Sierpinski triangle again. They don’t have a characteristic size. What’s the typical size of a triangle and a Sierpinski triangle. Well, they’re triangles of all different sizes. That’s sort of the point as you’ve seen as we zoom in more and more we keep seeing more and more triangles. And mathematically, that would go on forever. So they’re triangles of all different sizes. And that’s a very different situation than with the tomatoes where tomatoes come in a pretty narrow range of sizes. All right, here is another way to think about this. So again something is scale free that aren’t clues that as to its size Tomato is a clue as to size but a triangle and Sierpinski triangle is not. So there is a way you think about that Suppose that one morning you woke up and you had been shrunk. You were suddenly much smaller than you were before. May be you got a size of this guy now. That would be horrible thing to think about. Hopefully at least you still have two arms. I just suppose you get shrunk down like this Well you would know right away that you were small. Because your bed would be gigantic. And a tomato could feed you for a week You would be as big as your cat. Your cat could be a danger to you for all sorts of clues as to size in most aspects of the world in which we navigate So you could tell right away that we have been, that you have been shrunk. On the other hand, imagine and takes a little bit imagination, But imagine you live inside a Sierpinski triangle not our world, not a normal world, but fractal world like this. So this world consists of triangles and triangles and triangles and as you were seen every time you use double size you get three times as many triangles so triangles within triangles within triangles. If you live in this world and you were shrunk one morning all of a sudden, you wouldn’t be able to tell, there is no sense of scale here. There is not typical tomato or cat or bed or pillow. It’s just the world of triangles and triangles and triangles. May be you can also think about if you live in a tree and the tree that went on forever Branches that got bigger and bigger and bigger And branches got smaller and smaller and smaller You wouldn’t be able to tell what size you were All you know is that their branches that have this relationship that hold across all of these different scales. So that’s another way to think about what it means for something to be scale free. And one of these consequences of shapes that are self-similar.




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