I was wondering if there is a geometric name for patterns of circles and loops created by a Spryrograph toy where one wheel rolls either over or inside the circumference of another circle?

How about an Epicycloid?

The general term for curves that roll on curves without slipping would be Roulette.

(and that’s something new I learned today from Wikipedia!)

According to this, they are called “Roulettes”.

I know nothing about the mathematics of roulettes, I just Googled “spirograph geometry”, and this was the first result.

Also noted on this Wikipedia page (among others):

The classic Spirograph toy traces out hypotrochoid and epitrochoid curves.

It looks to me like an epicycloid is a special case of an epitrochoid, where the tracing point is exactly on the perimeter of the moving circle.

Similarly, the hypotrochoid covers the general case where the moving circle rolls along the *inside* of the fixed circle. Then, a hypocycloid is the special case where the tracing point is on the perimeter.

All four of these curves have Wikipedia pages, incidentally.