Saul Schleimer and I attempt to explain what a Cannon-Thurston map is.

Thanks to my brother Will Segerman for making the carvings, and to Daniel Piker for making the figure-eight knot animations. I made the animation of the (super crinkly) surface using our app (with Dave Bachman) for cohomology fractals. You can play with the app (on Chrome or Firefox) at https://henryseg.github.io/cohomology....

Also see:
Cannon and Thurston, Group invariant Peano curves, Geom. Topol., 2007.
Mumford, Series, and Wright, Indra's pearls, Cambridge University Press, 2002.

Some of these curves are available in t-shirt form at https://www.neatoshop.com/artist/Henr....

00:00 Introduction
00:28 The Hilbert curve
01:00 Approximations to Cannon-Thurston map
01:36 What space do they fill?
02:01 Symmetry of the Hilbert curve
02:34 Symmetry of the Cannon-Thurston map
03:10 The Hilbert curve is artificial
03:38 The complement of the figure-eight knot
04:39 The universal cover
05:20 Unwrapping the surface in the knot complement
05:51 The crinkling
06:50 Thurston's pictures
07:24 Comparing algorithms
08:23 s227
09:18 Carvings