I was inspired by some posts in reddit.com/r/spirograph by u/HomegrownTomato.
These three posts inspired me to try and understand this technique:
This technique uses:
In this specific setup gear D makes a square in hoop C. Once it has been iterated once the next step is to move hoop C by one step.
In the picture above I was returning gear D back to being near gear B so it is easy to lift up hoop C where it touches hoop A and rotate it by one tooth. Lets call this Outer Stepping
Another way to do this would be to put gear D at the bottom of hoop C near the A label so that the hoop C can be lifted up where it touches gear B and rotated by one tooth. Lets call this Inner Stepping.
With no other changes Outer Stepping and Inner Stepping produce different designs.
The Outer and Inner stepping techniques produce designs that are differently wobbly and folded based on their the relationship between different gear ratios of the 4 parts in the system.
The pieces used in this setup were:
How Do You Fit a Stationary Gear In The Middle?
This seems like it would be really daunting to find a combination of ring, gear, and hoop that fits together correctly for this technique but there is a simple math answer to the problem.
Add up two times the hoop (C) tooth count and the tooth count of the center gear (B). This number needs to equal the ring (A) tooth count. 2C+B=A.
So you've found your gear combo. How to center it?
Place hoop C in ring A and do your best to put gear B in the middle of the ring A. This won't be perfectly centered but if you hold down gear B and move hoop C about 90 degrees around the ring and reposition gear B to be snugged up to the hoop again. By repeating this step at least 4 times it should iteratively center the stationary gear. Also, while working on the design it may be beneficial to snug the center stationary gear B up to stationary hoop C to keep it really solid and snug. This is a very small adjustment that is optional but may make things work better.
I found that using magnets and a metal board made this setup work really good. Using putty, tape, or weights would probably work well too to keep the stationary parts stationary.