Cannon-Thurston maps: naturally occurring space-filling curves

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