The code has been cracked.

In March 2022 I was contacted by a person named Glagolj who has mathematically described what I’ve been calling “wheel-within-a-wheel” designs made with Wild Gears.

He prefers to call it “second order roulette” as it involves rolling one curve on top of another curve. Fair enough. For me, a mystery. For Glagolj, a mystery to be figured out and encapsulated in a parametric equation.

He wanted to use the patterns I’d drawn by hand and compare them side by side with the renderings generated by his javascript plotter. That’s the ultimate check on the formula. As you can see here, it works.

The plotter can be found here. It allows you to plug in the four gear numbers of a second order roulette, and generate the pattern they should give.

Glagolj’s article on the formula is here on Github.

He prepared a couple of slideshows with some of the interesting patterns he found. These are all computer-generated, so he can use any gear he wants. Of course, working with physical gears we have to work with the ones we have. But the numbers are in the upper-left if you want to try drawing them yourself.

Here is the second slideshow by Glagolj:

Several people have suggested I try third-order roulette patterns, i.e. a wheel-within-a-wheel-within-a-wheel. Well, Glagolj is ahead of me there. Now I have to try to see if I can duplicate his computer-generated results.