Hello everyone!

We hope that you had as much fun learning about this little dynamical system as we had making this video! It was amazing to explore how this simple idea connects to so many other mathematical gemstones-- planetary orbits, modular multiplication tables, and, of course, the Mandelbrot set (and more to the point-- Julia sets).

The poem read at the beginning of the film and mentioned is Passage by Victoria Chang https://poetrysociety.org/poetry-in-m.... Jeff and I really did discover this poem on a New York subway.

Thank you to Dr. Chris Brownell who's excellent teaching session of decimal expansions (you thought you knew them!) was the initial spark for this project.

Thanks to Thomas Folderauer for designing the sound effects.

Thank you to ‪@GrantSanderson‬ for hosting the SOME3 competition and giving us a deadline to finish this video! Who knows if it would happen otherwise?

Thanks so much to Will Bricca for being the best director and cinematographer for all our live action footage. You can watch his film projects at @EchidnaCone .

And of course, thanks to Jeff Wack for his infallible enthusiasm for the Mandelbrot set that is the reason this video exists.

If you have read this far, then you're in luck! Here is a list of reading material if you want to see the extension of these ideas into complex dynamics.
- A Brief Introduction to Complex Dynamics by Danny Stoll https://math.uchicago.edu/~may/REUDOC...
- The Spider Algorithm by Hubbard and Schleicher https://pi.math.cornell.edu/~hubbard/...

And here are some questions to think about, if you haven't had enough of the circle map yet!
- Which rational angles are terminating, periodic, or pre-periodic?
- What dynamical system can we use for general base n systems?
- Which rational angles are in the above categories for a general *n*?

Thank you so much for watching!
And, as always, keep exploring!
- Fran

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#some3
Live action filming and directing: Will Bricca
Stop motion filming: Jeff Wack and Fran Herr
Sound effects: Thomas Folderauer
Script: Jeff Wack and Fran Herr
Hand drawn animations: Fran Herr
Programmatic animations: Jeff Wack
Editing: Jeff Wack and Fran Herr