Hearing the Shape of a Trapezoid Drum

Abstract

In this article we take a small step in answering the famous question “Can one hear the shape of a drum?” Recently, Grieser and Maronna have shown that if the drum is shaped like a triangle, the answer is yes.
In this article we investigate this question for trapezoid drums. We define a subset of trapezia and prove that for drums of this shape the answer to the question is also yes. To do this, we need to investigate periodic orbits in a triangle. In particular we prove a lower limit on the length of families of periodic orbits.