Our goal is to add a viewer to the popular three-manifold software SnapPy . It will show what a hyperbolic three-manifold looks like “from the inside”, in the spirit of Thurston’s article  (see especially Sections 2 and 3). This will be accomplished by hyperbolic raytracing. The technique has already been explored in other projects such as  and ; by integrating it into SnapPy, we are able to see, interactively, how Dehn surgery affects the inside view (see ). Other applications include visualising how cusp neighborhoods become “Margulis tubes” as the geometry is deformed away from the complete structure.
A further, more ambitious, goal is to relate the inside view of a hyperbolic link complement to the “outside view”: that of the link diagram. This will require finding an explicit homeomorphism from the exterior of the link diagram to the hyperbolic triangulation. A user could then, for example, fly through the three-manifold in the inside view and at the same time see her or his trajectory in the exterior of the link diagram. Another application would be to show how the hyperbolic systole (or other parts of the length spectrum) is topologically related to the original link.
-  SnapPy, a computer program for studying the geometry and topology of 3-manifolds, http://snappy.computop.org
-  How to see 3-manifolds, by William P Thurston, in Classical and Quantum Gravity, Volume 15, Number 9
-  Ray-marching simulations of Thurston geometries, https://im.icerm.brown.edu/portfolio/ray-marching-simulations-of-thurston-geometries/
-  Better pictures of Cannon-Thurston maps, https://im.icerm.brown.edu/portfolio/cannon-thurston-maps/
-  Path through Dehn surgery space for the figure 8 knot complement, https://twitter.com/henryseg/status/1211226130167910400