The Apollonian Spider I

This is an introduction to a series of videos that is going to appear on this channel.
All the mathematical content of the series is largely inspired by the book
"Indra's Pearls" by David Mumford, Caroline Series and David Wright.
The English captions for this video are created by: ‪@NonTwinBrothers‬
Here is a rough outline of the content planned for the next videos:
II: The ancient construction - and some nice surprises
III: The mathematical description of curves - elementary tools
IV: Möbius transformations - turning the Riemann sphere
V: The Magic
As always, recommendations and feedback is highly appreciated.

[@nice3294]

Man the different iterations of the circles layered on top of each other looked spectacular

[@Number_Cruncher]

There more animations like this in the pipeline.

[@Zenzicubic]

My God, that's beautiful. Reminds me of the book Indra's Pearls.

[@Number_Cruncher]

Yeah, Indra's Pearl is the source of inspiration.

[@Zenzicubic]

@@Number_Cruncher I'm actually thinking of programming that. What are the positions and radii of the generator circles you used?

[@Number_Cruncher]

@@Zenzicubic The first circle (a) is the upper half plane, the circle (b) is centered at 1-i with a radius of 1, the circle (A) is centered at -i/4 with radius 1/4 and the circle (B) is centered at -1-i with a radius of 1.

[@TheRecklessGamer3169]

Great vid

[@dohduhdah]

Is there a reason the color map of the complex domain is the same for both transformations for the rectangular case, but looks different for the polar case?

[@Number_Cruncher]

The polar grid is centered at the fixed point in either case. This shift changes the colors of the grid.

[@dohduhdah]

@@Number_Cruncher Ah ok... I see. Thanks!