Isospectral Configurations in Euclidean and Hyperbolic Geometry

Abstract

A number of questions related to the length spectrum of surfaces are discussed and in particular the existence of pairs of surfaces which though not isometric are isospectral. Here by isospectral we mean that a pair of bodies have the same distribution of chord lengths. In the Euclidean setting, we study isospectral convex dodecagons found by Mallows and Clark in the 1970's. Starting from their idea, we give constructions for isospectral pairs of hyperbolic surfaces that have no common cover. Since the work of Mallows and Clark is probably unfamiliar to readers with a background in topology/hyperbolic geometry we include expository material on other related topics about the distribution of chord lengths.