Fractals and Scaling: Chaos game variations


One can vary the rules of the chaos game a little bit, and come up with some additional interesting shapes and have a little bit of fun. So here I am, back at the program that we used before, and let’s start just going back to what we did previously so 10,000 points with some small dots. There are some things that we can change down below. One is, we can change the movement ratios. Right now, this says red point: 0.5
That means that you move halfway to the red point. This 0.5 means you move halfway to the
green point, halfway to the blue point. I think that if they are set a little bit differently, one gets some fun shapes so let’s try this: suppose you move 54 percent to the red, green, and blue
points instead of 50 percent. then, things shrink up a little bit and we have a Sierpinski Triangle of sorts that has kind of broken apart into smaller Sierpinski Triangles… it’s not fully connected anymore. We could do something else. We could go over here and change this into 0.47… and then we get some overlap it’s not quite as clean
a Sierpinski Triangle as before The triangles overlap themselves a
little bit. Let’s get a few more points… so the shape is a little bit different. Let’s go back to 10,000 Maybe we can exaggerate this a little more Let’s go to 0.44 and see what
that looks like At some point it starts to look like a fractal plaid pattern to me. We can make this colorful if we click on “Color code” … That, then is just going to make the top ones red, the left ones blue, the right ones green. That can be a fun thing to play around with. Let’s go back to 0.5 and let me show you one more thing. We can also change the relative probabilities. In the version of the game that we played, you were equally likely to go in the red direction when I called “a” in this direction, or in this direction. We could change that up, so maybe I’ll make red, now, three times as likely and hit “Enter” so that change
goes into effect. And the tops of the triangles, now are heavier. Because the tops correspond to those “a” moves and so now we see a Sierpinski Triangle… the shape is the same, it’s just not uniformly filled out anymore. The dots spend more time in some parts of the triangle than others. And this is interesting to me. It gives it a 3-D effect, or a faded effect. Let’s try it with 10,000 points… It’s a little less dramatic. So, there’s lots you can play around with. You can change the movement ratios You can make them different. I was keeping them all the same. You can change the relative probabilities as well… one last really fun thing
you can do… You can change the number of points. There’s no reason why we have to have 3 points. We could have 4 or 5 or 6 or 7. And then we would just, with equal
probability move halfway or whatever way to each point and iterate that. Let me show you just one more thing here You can also move these points around so let’s say I want to move the red point. Then I would click this on “Move” and then wherever I click my mouse that’s where the red point’s going to go So that can give you… if you want to change the shape of the triangle. Here, I’ll move the blue point. You can do that as well. So there’s lots that you can experiment with… and I’ll suggest a few explorations that you can do for the homework for this week.




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