2 McCartin ’ s Method : Redefinition of Eigenvalue Problem

We examine the eigenstructure of generalized isosceles triangles and explore the possibilities of analytic solutions to the general eigenvalue problem in other triangles. Starting with work based off of Brian McCartin’s paper on equilateral triangles, we first explore the existence of analytic solutions within the space of all isosceles triangles. We find that this method only leads to consistent solutions in the equilateral case. Next, we develop criteria for the existence of complete solutions in other triangles. We find that complete solutions are guaranteed in the equilateral, right isosceles and 30-60-90 triangles. We then use a method developed by Milan Prager to formulate solutions in the right isosceles triangle through folding transformations of solutions in the square.