This is a talk I gave at the Bridges conference on mathematics and the arts (http://bridgesmathart.org/), on 9th August 2016, about my paper with Saul Schleimer with the same title. The slides are available at https://www.math.okstate.edu/~segerman/talks/spherical_transforms_talk_bridges.pdf Videos that appear in the talk: 12:08, A two-fold branched cover of my apartment, https://www.youtube.com/watch?v=UUW_ZU3_TQM 13:14, Three person passing patterns, six person passing patterns, https://www.youtube.com/watch?v=OcDJb6ZzfBo 18:30, Spherical Droste video, https://www.youtube.com/watch?v=qvh-EAipIUk 21:58, Funhouse spiral staircase, https://www.youtube.com/watch?v=BNcoPi1k2pg 28:52, Five castles, https://www.youtube.com/watch?v=n5B3__hjdPA 30:44, Palace of Fine Symmetry, https://www.youtube.com/watch?v=TSaZeNI3ZvQ 32:45, Portal to the mirror world, https://www.youtube.com/watch?v=iZt1mqwW5C4 34:09, Spherical video stabilization, https://www.youtube.com/watch?v=2gXIfNR5YPc 35:56, The Spherical Droste Effect, with added twist and recursion., https://www.youtube.com/watch?v=Yp12c3-IL-I The paper is available at http://archive.bridgesmathart.org/2016/bridges2016-15.html