# Embedded Hyperbolic Tilings

Date started: October 2019

### Abstract

I would like to illustrate tilings of the hyperbolic plane on surfaces of constant negative curvature. While the entire hyperbolic plane cannot be embedded in $\mathbb{R}^3$, arbitrarily large pieces of the plane can be, so there is in principle no obstacle to illustrating as much of a tiling as desired, although practical considerations quickly come into play. I am currently using 3D printing for these illustrations, but eventually I would like to use bamboo.

### Media

The design of the 3D prints was done in OpenSCAD. As a first pass, I designed a print of a Euclidean tiling illustrated on a cone or cylinder.

Next, I worked on the hyperbolic tiling. First, I needed to know how to parameterize geodesics on a surface of constant negative curvature. Dini’s surface is an example with infinite area, and can be given any desired injectivity radius. Furthermore, the parameterization of its geodesics is known explicitly.

With the mathematical heavy lifting done, I now want to focus on embedding other hyperbolic tilings into $\mathbb{R}^3$ on Dini’s surface and other surfaces of constant negative curvature.

Acknowledgements: I’d like to thank Laura Taalman for her help with using the “sweeper” code to create the designs above on OpenSCAD. Also, I’d like to give credit to Steve Trettel for letting me adapt the Mathematica code on his public site to generate the edges of some of these hyperbolic tilings. Finally, I’d like to thank Kelly Delp, Aaron Abrams, and Alba Malaga Sabogal for several helpful conversations along the way.