How does a spirograph work?
A small toothed wheel rolls around the inside of a bigger ring, and a pen in one of the wheel’s holes traces a looping curve. Mathematicians call it a hypotrochoid. This page draws the same curve, from the sizes you set.
Why do some settings give many loops and others only a few?
The pattern closes when the wheel’s teeth line up with the same teeth of the ring again. With 96 and 52 teeth that happens after 24 loops, because 96 and 52 share the factor 4. Sizes that share few factors make dense patterns with many loops.
What does the pen hole do?
It’s how far the pen sits from the wheel’s center, as a percentage of the wheel’s size. Near 100% you get sharp points, as with the outer holes of a real wheel. Lower values give rounder, softer curves.