Spirographs in After Effects had it's 15-minutes of fame years ago and were lost in the mists of time (around here anyway), except for a few tutorial/project posts. Revisiting the topic reveals a variety of approaches and directions.
In ESPL: Dances of the Planets, Ensign Software used their own day trader software to overlay planetary orbits, after inspiration from A Little Book of Coincidence by John Martineau. Here's 8 years of solar orbits of Earth and Venus, a five-petal flower with the Sun at the center of the cage:
There are many examples of After Effects spirograph tutorials. Earlier ones — still interesting with free projects — includes those from Chris Zwar, Ayato, and Michele Yamazaki (now only in her book). Satya Meka showed similar results (he liked the Radio Waves filter) in Create an Ethereal Morphing Letter Canvas and Create an Elegant Concentric Rings Animation. Here's a preview:
More recently, Christopher Kirkman posted Generate Your Own Splendid Spirographical Designs.
Animating Spirograph by Ashish Rastogiis is a short text tutorial.
old Mark Christiansen tutorial with Math.sin(time), Math.cos(time) and with nulls+parenting + WriteOn. He also posted a spirograph project made by attaching Trapcode Particular to a null and using it as an emitter to do some 3D. See also a spirograph in 3D by Brian Charles.2 AE projects based on an
Motion Boutique released a free After Effects script ported from subBlue’s Guilloché Pattern Generator. Guilloché patterns are Spirograph-like patterns often used similar to create fine details on bank notes. Victoria Nece says “Nab’s script lets you make AE masks out of them. … You can’t animate the masks directly inside the script palette itself (although you can generate them at multiple keyframes along a single mask path), but that isn’t really a problem — I managed a lot of neat effects using a single static shape. These patterns work especially well as inputs for the Radio Waves effect — add a generous amount of blur and glow, and boom, instant animated BGs.” Here's some tests by Victoria:
Spirograph is a “condensed animated typeface inspired by the toy that was invented by Denys Fisher in 1965 (no longer available though Spirograph sets by Kahootz seem good). With a lots of customizing options, you will be able to easily create your own look. Besides the usual A to Z and 0 to 9, it also comes with extensive punctuation.” Also available at AEscripts.com.
If you use Apple Motion, check out Mark Spencer's video on how to create spirographs in Motion:
Exploring spirographs in Adobe Illustrator might exercise creative muscles: see Deke’s Techniques: Creating a Spirograph-style pattern from a single path in Illustrator, How to Create Vector Spirograph Designs by Janel, and Illustrator full spectrum spirograph by Veerle Pieters.