Joint work with Saul Schleimer. In this short video we show how various models of hyperbolic geometry can be obtained from the hemisphere model via stereographic and orthogonal projection.

2D figure credits: 4:09 Cannon, Floyd, Kenyon, Parry. 0:49, 1:20, 1:31, 2:12, Roice Nelson. We thank Gazi University, Ankara, Turkey, for its hospitality during filming.

(2,3,7) triangle tiling: http://shpws.me/Dq6q
Geodesic saddle ({3,7} radius 3, subdivided 4): http://shpws.me/ES7b
Pseudosphere: http://shpws.me/GIgq
(2,3,5) triangle tiling (stereographic projection): http://shpws.me/CKVv

Thanks to M.Y. Zhang for writing CC subtitles for this video.