Hyperbolic paper

Date started: September 2019
Leads: Aaron Abrams, Stepan Paul
Contributors: Alba Málaga-Sabogal, Rotem Tamir, Kelly Delp, Steve Trettel, John Edmark, Samuel Lelièvre


We are aiming to make paper with constant negative curvature so that we and other participants in the program will have a new medium for illustrating properties of the hyperbolic plane. While a theorem of Hilbert prevents one from isometrically embedding the entire hyperbolic plane into \mathbb{R}^3, it is also true that for any large $R$, there are surfaces in \mathbb{R}^3 with constant curvature -1 with injectivity radius R. In theory one could multiply cover the pseudosphere to obtain patches with large injectivity radius. Like the sphere or the plane, any patch of the pseudosphere fits perfectly anywhere on the pseudosphere, and in any orientation. Here is a proof-of-concept demo video: https://youtu.be/JRd928WpY9w.

We hope to use the paper to illustrate hyperbolic polygons, tilings, straight-edge and compass constructions, and origami, and we foresee the paper as being useful for educational purposes, e.g. in a college geometry course. For instance, how do you construct a regular right-angled pentagon? What space curves can be realized as geodesics on a piece of the hyperbolic plane embedded in \mathbb{R}^3? Also, just as there is an origami “flat torus’’ (see https://www.youtube.com/watch?v=M-m-hKtCQVY), is there a similar origami construction of a higher-genus surface embedded in \mathbb{R}^3 by folding a piece of the hyperbolic plane?

There are significant challenges to making paper with intrinsic Gaussian curvature, and so the quality of our models so far have been limited. A few ideas we plan to try in the future include using higher quality paper fibers, using kombucha fiber, thermoforming acetate film, and using paint-on silicone.

This builds on earlier work of Alperin, Hayes, and Lang.


A piece of hyperbolic paper laid on a pseudosphere mold with the same constant negative curvature.
Pieces of hyperbolic paper laid against a pseudosphere and Dini’s surface (all with the same constant negative curvature).

Video demo of hyperbolic paper: