Previously we explored one version of the Chaos Game in which we varied the fraction of the distance traveled at each iteration. This time we'll look at a variation that focusses on the vertices. We'll still allow there to be any number of vertices, but now we may restrict which ones may be chosen each time we plot a dot. For example, we may not want a vertex to be chosen twice in a row.

To make this a bit clearer, suppose we number the vertices clockwise, with one starting at the top. I'll use the term "step" to describe the difference between the two vertices chosen at two consecutive iterations. For example, if the 2nd vertex is chosen at one point, and the 5th vertex is chosen next, we'll call this a step of 3. Similarly, if the 3rd vertex is chosen followed by the 8th vertex, this would be a step of 5. If a vertex is chosen twice in a row, that's a step of 0. Note that if there are n vertices, we can describe all possible differences between consecutive vertex choices with n-1 steps.

You can try out this variation on the panel below. After selecting the number of vertices, you can choose which steps are allowed. For example, deselecting 0 would prevent any vertex from being chosen twice in a row. Leaving everything selected (the default) will just repeat the same procedure we used in previous experiments. And for now, we'll keep the fractional distance r at 1/2 for each of these.

Did you find some nice combinations above? I think 6 vertices with 0 and 1 steps deselected looks pretty nice. For the larger numbers of vertices, you may find better patterns with only a few allowed steps. For example, 10 vertices with only 0, 3 and 7 steps selected looks very interesting.

At this point though, you might be wondering how things look if we both restrict steps and change the fractional distance r. If that's the case, you can use the panel below to try combining these two approaches. And this time I'll even let you choose your favorite color as well. :)

Well, if you've worked your way through all three of these pages, I'm hoping you might have some suggestions for what other variations we might try in the Chaos Game. Or better still, maybe you will write your own program to test some of those ideas. Either way, I'd love to hear your thoughts. So, if you don't mind, please send them to me at dansmith.philomath(at)gmail.com.

Thanks for reading! ~ Dan