Can You Hear the Shape of a Hyperbolic Surface? Now for Real

TL;DR

This paper introduces a musical analogy called the 'hyperbolic marimba' to explore how melodies can encode the geometry of hyperbolic surfaces and multicurves, with interactive examples available online.

Contribution

It proposes a novel method linking hyperbolic geometry with musical melodies and investigates their ability to characterize geometric structures.

Findings

Melodies generated can encode geometric information about hyperbolic surfaces.

The 'HyperMarimba' website allows interactive listening and visualization of the phenomena.

The study explores the extent to which melodies determine the underlying geometric data.

Abstract

We associate a musical instrument, a "hyperbolic marimba", to every pair $(X,\Gamma)$ where $X$ is a hyperbolic surface and $\Gamma\subset X$ a simple multicurve labeled with musical keys. It works as follows: take a geodesic and every time it hits $\Gamma$, play the corresponding note. In this paper we investigate to which extent the so-produced melodies characterize $(X,\Gamma)$ up to isometry. In the accompanying website "HyperMarimba" (available at https://ludox73.github.io/HyperMarimba/story.html ), the reader can actually listen to the produced melodies. They can also visualize some of the phenomena we investigate.