Imagine a thick triangular sheet of metal vibrating so much that it produces
a resonant tone. Are there are two qualitatively different types of vibrational
behavior that lead to the same tone? The answer is yes for both the equilateral
triangle and the right isoceles triangle. In joint work with Luc Hillairet (Nantes),
we show that the answer is `no' for the generic triangle without explicitly
identifying a single triangle for which the answer is no. In this talk, I will describe
the underlying mathematics and the new method---asymptotic separation of variables---
which solves this and similar problems.