Fractals and Scaling: Part 1 Summary


So let’s summarize what we were covered in this unit. Began by introducing the idea of fractals in a very qualitative way. Said fractals are objects that are self-similar, small parts of the object are similar to the whole. And we looked at several examples. One of which is the picture of the sycamore tree. where we can see that their branches off of which are branches off of which are branches. And the branches all resemble each other. So it’s self-similar and for something to be said to be fractal, it needs to be self-similar over several scales. So branches off of which are branches off of which are branches, and may be branches smaller still. So that’s self-similarity is over several scales. We would say that we were working with a fractal. Then we consider some mathematical fractals, like the Sierpinski triangle here. And I introduced the idea of the self-similarity dimension. And here is the key equation. The number of small copies inside an object is equaled to the magnification factor, stretch factor, raised to the D power. So for example here, the magnification factor is 2. to go from there to there magnify by 2, stretch by 2. and there are 1,2,3 smaller shapes. So we plugged that into this formula. We can use logarithms and we get that the dimension is about 1.585 As I mentioned before this business of seeing small copies and figuring out the magnification factor it takes a little bit a practice. It’s again it’s a very visual sort of things and so sometimes describing with words feels really inadequate. So if you’re still little confused on this, don’t worry, keep practicing and will be more practice problems in the homework as well. Ok, so this was the self-similarity dimension. Along the way, we did a quick exponent and log review. This was an optional section But I just wanted to include it here, may be as a useful reference. The main thing is an exponentiation is defined to be successive multiplication. And this is the definition for a log. 10 to the logx equals x and from properties of exponents and definitions of logs, all of these useful properties follow. Ok, so that’s the dimension We are working with objects that are between 1 and 2 dimension. So their self-similarity dimension is between 1 and 2. And that’s weird, a little uncomfortable. What might that mean? Well, one way to think about it is that they have both 1 and 2 dimensional qualities or features or properties. So the Sierpinski, excuse me, the Koch curve has infinite length, but it’s in a finite area. Like a box it in, in a finite area. But the curve between my hands as infinite length. So that it fits a finite area That’s sort of a 2d tech thing. It has infinite length, may be that’s a 1d tech thing. So disturbing in between thing that a shape that has combination of features. Sierpienski triangle has zero area but infinite perimeter. So again these are funny sorts of shapes that have different, combination of different features. In other way to think about dimension, which will be more and more important as this course goes on which is I think little bit more concrete to think in terms of scaling. So if I increase an object size, if I double it, all the lengths this way, this way, this way If I put it in a photocopier, hit 200 percent, What happens to the shape? Well, different things happen for shapes for different dimensions. So we have a sphere which is 3 dimensional and I stretch it by a factor of 2. It is now 8 times larger. Because 2 cubed is 8. We take the scale factor, how much we’re increasing it by and raise it to the dimension. So the dimension tells you how the size of an object changes as it is scaled up. I put size in here because it might mean different things and different contexts, It could mean volume, area, you could think of it in terms of mass may be, amount of ink to draw shape. The main thing how does the size change as you scaled it up. Determined by the dimension and there is no reason why dimension can only be 1,2 or 3. Nothing wrong with the dimension 1.5 or 1.6 or anything else. So that’s dimension and scaling. I also talk to little bit about this notion of something being scale-free. So if an object is a self-similar, we often say that it’s scale free. For example for the Koch curve, there is no typical size of the bumps. It is a line with the bump in it, and than bumps on bumps than bumps on bumps. Bumps are all different sizes. In contrast there is a typical size for tomato. Roughly a pound sometimes less sometimes more. But there is a typical size. So if I were showing taking a picture some objects you hadn’t seen before And I wanna to indicate to you what the size of that object was, I put a tomato next to that unknown object so you would have an idea what its size was. If I show you just a segment of the Koch curve, that wouldn’t help. Because you could be looking at small piece or a large piece, they all look the same. Another way to think about it is that any fractal If you were shrunk, you could not tell, so if you suddenly got scale down by a factor 10 and the rest of the world say the same, if you live in the Sierpinski triangle or fractal, you wouldn’t be able to tell. Because there is no object that is set as scaler size. And just I know about self-similarity, it’s worth remembering the real fractals, trees and river basins and so on. It’s supposed to mathematical fractals. Real fractals are not self-similar forever, they can’t be. There are some lower scale or which the self-similarity stops. and some upper scale too. Nevertheless fractals are very useful for very useful abstraction for describing objects. Just like the pure forms of classical geometry, lines and circles and so on. There are also very useful abstractions for describing real objects. So, let me say a little bit about the definition of a fractal. And probably the first thing I should say as what’s done here. That there is not, I mean, airtight, all-purpose, standard definition of fractal. It’s more of a notion, may be than an absolute strict the category. But here some features that fractals have. There is self-similar cross many scales as we discussed. There are not well described by classical geometry, circles and cubes and spheres and triangles. They have a self-similarity dimension that is larger than the topological dimension. Topological dimension is more or less the intuitive notion of dimension. So the Koch curve, we think, oh, that’s a line. Because it’s line, very bending line but it’s still a line. So its topological dimension is 1. But as you’ve seen, its self-similarity dimension is larger than 1. I forget it’s 1.2 , 1.3 something like that. So I don’t wanna go into the topological dimension in much detail. But the main thing is that they have got weird dimensions, often non-integer or not what they would, what would you expect. There are built out of object set up 1 dimension. But the actual fractal itself has a different dimension. So this brings us to the end of this unit. In the next unit, we’ll look at some additional types of fractals. Fractals add have some randomness in them. And so which are perfectly symmetric. And I’ll also introduce another type of dimension closely related to the self-similarity dimension called the box counting dimension. See you next week.




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