Henry Segerman, Saul Schleimer, Will Segerman

Statement

Saul is a geometric topologist, working at the University of Warwick. His other interests include combinatorial group theory and computation. He is especially interested in the interplay between these fields and additionally in visualisation of ideas from these fields. Henry is an associate professor in the department of mathematics at Oklahoma State University. His research interests are in three-dimensional geometry and topology, and in mathematical art and visualization. He wrote the book "Visualizing Mathematics with 3D Printing". Will is a self-employed sculptor in analogue and digital media. He works from his home/office/workshop in Manchester, UK. His current main source of income is as an escape room puzzle designer.

Artworks

The white and blue regions of this carving have a common boundary. This boundary is our approximation of the Cannon-Thurston map: a space-filling curve. Using veering triangulations we give approximations that fill space more evenly than Thurston's algorithm, which suffers from the problem of "thin necks". The space-filling curves of Peano and Hilbert (and many others) are artificially constructed. Those of Cannon and Thurston arise naturally in the study of hyperbolic three-manifolds. Each curve is canonically associated to a surface contained in a three-manifold. The curve and the space that it fills come from the boundaries at infinity of the universal covers of the surface and manifold.

In this example of a carved (approximation of a) Cannon-Thurston map we begin with a veering triangulation of the SnapPy census manifold s227. This is the smallest example in the veering census where the taut triangulation carries a surface, but is not in fact layered. Thus this Cannon-Thurston map does not come from a fibring. In particular this example goes beyond those considered by Cannon and Thurston.