Fractals and Scaling: Dimension of fractal “snowflake”


In this video I’ll introduce a mathematical fractal known sometimes as the snowflake fractal and then we calculate its dimension. So the snowflake fractal like many fractals as we’ll see is made by an iterative process. So we start with a single dot. in the every step, I am gonna take the shape that I have make four copies of it and place it on the corners. So here is my initial phase and then the next phase I am gonna take four copies of original dot and put them here 1, 2, 3, 4 Ok, that is the first step. Now I’m gonna iterate that step. The next step I am gonna take this shape all five dots make four copies and place them here 1, 2, 3, 4. So I’ll do that. There is copy 1. There is copy 2, copy 3 and copy 4. So, and next phase, next step, I would take this shape, there are let’s see… 25 dots here. And I would make four copies and put one up here, one up here, one down here and one down here. And I’ll keep doing that, I’ll keep doing that and we would get a fractal. If we carry out this process for a few more steps, we’ll end up with a shape that looks like this. And we can see that this is a fractal. It’s a self-similar object. It’s this cross of x shape and the x is made up of x’s which are made up of x’s which are made up of x’s and so on. So we see this same x or cross shape appearing again and again at many different scales so it’s a fractal. By the way I should mention that although I refer to this as snowflake fractal, only sort of looks like a snowflake. So snowflakes are six arms and not four. But nevertheless reminds me of snowflakes so I call it the snowflake fractal. I don’t think it has a hundred percent standard name. Ok, so in any event we wanna know what’s the dimension of this. We’ll do like we did for that triangle example and other examples before. Look at small copies and the large copy and so on. So to do that we will again use a dimension equation And number of small copies in an object, it is the magnification factor or the stretch factor raise to the D power, the D is a dimension. In here is another view of the process of creating of this fractal. We started at the zero step with the single square. I drew it as a dot originally. And then we go here and then this is step 2 and this is step 3. So let’s see, number of small copies, let’s look at this picture. So number of small copies, I see 1, 2, 3, 4, 5, So number of small copies is 5 What’s the stretch factor or the magnification factor, well that’s 3 And so to see that, imagine this shape how much we have to stretch it so its this long. Well, you can see that this length is made up 1, 2, 3 of those. So the stretch factor is 3. This one is a third of size of the big one. So the magnification factor is 3. And that gets raise to D power. Now the question is what’s the dimension D. Let’s see, could the dimension be 1? Let’s try, I plug it in, 3 to the 1 is 3. That’s not 5. Could the dimension be 2? Let’s see, I plug it in 3 squared, 3 squared is 3 times 3, that is 9. That’s too big. So this object does not have the dimension of 1 and it does not have the dimension of 2. It turns out that the dimension is between 1 and 2. And so in order to solve this equation for D, remember D is the dimension that’s we’re interested in. We’re gonna have to use logarithms. Yes, logarithm. So that’s probably is a good time for me to remind you that there is an optional section in this unit that reviews exponential and logarithms. There is not a lot of algebra in this course but exponents and logarithms are going to come up a lot. And you can already see why. The dimension, this important way of characterizing fractal as you see is an exponent. So you’re going to need to know about exponents. And logarithms are used to solve the equations when the unknown is upstairs as an exponent. So exponents and logs are kind of move back and forward. So you’ll need to be fairly fast with basic exponent and log properties. It’s not a big deal. And I review them in that optional section. So if you are uncertain about logarithms or what’s about to follow doesn’t make sense, check out that review section and will be fine. Remember you can also, I am always ask for help in the forum, if there is something, you are unsure about. Ok, back to the fractal. So here is the situation, we have this equation 5 equals 3 to the D and I’d like to solve for D. So to do so we’ll take the logarithm of both sides of this equation. So log5 equals log3 to the D. So then by the logarithm property for exponents, log3 to the D, it is the same thing as Dlog3. So log5 equals to Dlog3. In the last step is this all for D. So just divide the both sides of the equation by log3 and I’ll end up with this. log5 over log3 equals D. So that’s my answer. That’s an exact expression for the dimension D. And we can get a number out of this with a calculator. So let’s see, I’ve got a log5 over log3. So here is my trusty calculator and I’ll do 5 logarithm divided by 3 logarithm. And I get approximately 1.465 So that tells me that D is approximately 1.465 So we found that the dimension of the shape is not an integer. It’s a number between 1 and 2. That’s kinda weird. We’re not used to non-integer dimensions and we’ll talk a bunch more later in this unit and throughout the course about what this means. But for now, let’s keep up head down and keep sort of working with this basic equation for the self-similarity dimension. So in the next video I worked through another example calculating the self-similarity dimension D for fractal. And then you’ll get to practice this in a couple of quizzes.




Comments
  1. How do you know what your terminal / end units are? for example in N=3 did we have to use N=2? can we use N=0 when counting the number of small copies ?

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