Students Anil Damle and Geoffrey Peterson
Dates of Involvement 2008-2009
Faculty Advisors James Curry and Anne Dougherty

Introduction
Understanding of the eigenstructure of an equilateral triangle was presented in “Eigenstructure of the Equilateral Triangle, Part I: The Dirichlet Problem,” (Siam Review, Vol. 45, No. 2, pp. 267-287) by Brian J. McCartin.  Starting there we sought to expand our knowledge of triangular eigenstructures to isosceles triangles. Here, we focus on the Dirichlet problem and seek to develop a solution to the general eigenvalue problem in an isosceles triangle. The method presented by McCartin relies on Lame’s Fundamental Theorem to construct solutions that are either symmetric or antisymmetric with respect to an altitude of the triangle.

Project Description and Results
We first investigate the coordinate system that is used to easily parameterize the isosceles triangle and the obtained solution (see Figure 1).
Figure 1: Isosceles triangle with u, v, w axes

The Cartesian coordinates of points inside the triangle must be transformed into a triangular coordinate system which measures the distance from the intersection point of the altitudes to the projection of the point onto the altitude. This transformation takes the form:








With this new coordinate system, we also must redefine the Laplace operator in order to properly use separation of variables.

We then construct a form of the solution that is symmetric about the bisecting altitude of the isosceles triangle and zero along the edges using sine and cosine functions, and analyze these solutions to find six equations for three eigenvalues, only to find that the only case in which consistent solutions exist for these equations is the equilateral triangle.

  Figure 2: (1,4) mode of an equilateral triangle

Although we can choose certain eigenvalues that satisfy the Dirichlet conditions, we show graphically that they may not also satisfy Laplace’s Equation by analyzing certain extreme conditions.  

We are currently looking at conditions under which complete solutions could exist, although we may not be able to find them using McCartin’s method. These conditions include the ability to tile the plane with the triangle solely through rectangular reflections, the location of nodal lines in extensions of the solution, and whether the solutions are symmetric or antisymmetric. From this discussion, we present triangles where we believe that solutions should not exist and triangles where certain types of solutions may exist, such as the isosceles right triangle.

About Anil and Geoffrey:

Anil Damle is currently in his third year at CU-Boulder. An Applied Mathematics and Computer Engineering double major, Anil has various interests in both fields. Future plans include participation in the 5 year bachelors/masters program at CU possibly followed by the pursuit of a doctorate degree.

Geoffrey Colin Peterson is currently in his third year at CU-Boulder, and he is planning to complete the 5-year MS/BS degree program in Applied Mathematics. His primary interests in this field include probability and statistical theory and simulations. Colin is also interested in computer programming, specifically graphics and game programming.