Pentaflake Chaos

Dan Anderson commented on my Pentaflake post to observe that the pentaflake can also be created by a random process, sometimes called the Chaos Game. In this game you start with an arbitrary point and dilate it toward a target point that’s randomly chosen from some set of points that you’ve established. You then dilate the new point, by the same scale factor, toward another randomly-chosen target from your set of points.

I find it quite amazing the way in which such a random process can end up creating a beautiful pattern. In fact, a wide variety of patterns can be generated this way, depending on the scale factor and the set of target points you’re using. The pattern Dan described involves a set of five target points, which are (no surprise!) the vertices of a regular pentagon.

Dan used the programming language called Processing to play his Chaos Game, but there’s no reason we can’t play the same game in Sketchpad. So here’s a sketch that describes how to play the Sketchpad Chaos Game in order to make a pentaflake. (Though you can use the sketch below to view the completed construction, you have to use regular Sketchpad to do the construction yourself.)

Page 1 describes the initial construction, up to the iteration. Page 2 describes how to do the iteration, and shows the result. Because this iteration only adds one point with each step of the iteration, it requires a much greater depth to see the pattern.

Page 3 asks another question: What if you dilate toward all five vertices simultaneously?

Finally, all three methods (the one from my last post and the two in this one) generate lots of questions. What happens if you move the initial seed (point Q) outside the circle? For a particular method, how many objects does the iteration produce at each depth? What happens if you change the scale factor, and why does this scale factor (approximately 0.38) produce the result it does? What if you tried the same construction with a square, or a triangle? The Chaos Game pentaflakes are missing the middle pentagon that is part of the construction from my earlier post. Why is that, and is there any way to make it appear in the Chaos Game patterns?

I hope you’ll be inspired to experiment with your own pentaflake, but if you like you can download mine. (And even if you don’t have Sketchpad, you can get the free trial version and use it to do your own construction or to play with mine.)