Fractals and Scaling: Exploring the Sierpiński triangle (and Properties Quiz 1 solutions)


In this video we’ll think about the area in the perimeter of the Sierpinski triangle. And this will be similar to the calculations or considerations we used when figuring out the length of the Koch curve. And in the course of this video I’ll provide solutions to the previous quiz. So here are the steps and the construction of the Sierpinski triangle. And decided that’s step zero initially, I am gonna call this area A. So then what is the area here? Well, you can see what happens is this triangle gets divided into four equal triangles, one of which is missing. So at step 1 the area is three quarters of what we had originally. Right, because this where we go, one quarter, two quarter, three quarter, four quarter, but this quarter is missing. All right, what about the step 2? Well, the same thing happens again. This triangle we’ll remove a quarter of it. This triangle we’ll remove the quarter. This triangle has a quarter removed. So what we left with is 3 quarters of 3 quarters. That’s 3 quarters squared 3 fourths times 3 fourths. At step n, we have three quarter to the n times the original area. And then as n approaches infinity as we go farther and farther in this construction, the area gets closer and closer to 0. So we start with a triangle and by the time we’ll keep the step many many times even more than this. We left with the triangles are getting smaller and smaller and smaller all the time, so the total area of this shape is going to 0. So now let’s think about the perimeter of the Sierpinski triangle. So initially at step 0, let’s say that each side is length 1, so the perimeter is 1,2,3. So we have step 0, perimeter is 3. So here we had 3 sides, each of length 1. Now we have 9… 1,2,3, 4,5,6, 7,8,9 So the number of sides goes up by a factor of 3, but each side itself is half as long. Right, because this side is 1, each of these sides is just the half. So there are 3 times as many things that are half as large, so the perimeter, this is step 1, the perimeter we started at 3, but that gets multiplied by 3 halves. Because there are 3 times as many as before but each one is halves of large. Right at step 2, the story is similar. Each triangle which has 3 sides turns into one of these which has 9… 1,2,3,4,5,6,7,8,9 So again the number of sides goes up by a factor 3, there are 3 times as many sides as before, but each side itself is half as long. This side is half as long as that so that means it’s 3 halves of what we had before. So it’s 3 halves of 3 halves of 3 or 9 fourths of 3 or probably we should write it in this way 3 halves squared of 3. And at step n, it’s 3 halves to the n times 3 our original perimeter here. So the thing to note is that these numbers are getting larger. 3 halves that’s 1.5 You multiply 3 by 1.5 by 1.5 by 1.5 Those numbers are getting larger and larger. And so as n gets larger, as n goes infinity, as we do this more and more, the perimeter is also going to go to infinity. So this shape the Sierpinski Triangle, after we carry out this process many many times, the area is getting closer and closer to 0 but the perimeter of the shape gets larger and larger and larger. And in the mathematical limit that n goes to infinity, this has 0 area but infinite perimeter. So lastly, we think about the dimension of the Sierpinski triangle. Recall from previous subunit that we calculated the dimension be about 1.585 Again a number between 1 and 2. So as I was saying before for the Koch Curve, this means that has some two-dimensional features and some one-dimensional features. So it’s two-dimensional sort of in the sense So it is two-dimensional sort of in the sense that it’s started off as a two-dimensional shape, triangle, but we’re removing more and more and more of it, And what we’re left with is the smaller and smaller triangles, in fact when you carry this out to infinity, really all you have left a line segments which are dimension 1. Another way to think about, this is we started off with something that’s two-dimensional area, triangle, but what we’re left with here is really all perimeter and no area at all. So it’s sort of like that the two dimension the second dimension shrinks away and gets really close to the first dimension, all perimeter, no area. So all perimeter, all lengths, no area that’s one-dimensional but it did start off as two dimensional. So again this is just may be a little bit intuitional way to think about these objects with this funny dimension between 1 and 2. They’re sort of one-dimensional and they’re sort of two-dimensional.




Comments
  1. Very interesting and progressive approach to test prior knowledge and then release the instruction to keep it fresh. How wonderful learning is! Beyond the institutions is there anything more fundamental to the human condition and life itself? Learning that is, providing ordered form to the entropy around us.

  2. Why dont you count the area of the "empty space" in the triangles? wouldnt this need to be considered if we were using something like this to describe a physical process ? Also why do we not count the loss of area from the previous step ?

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