This paper derives explicit eigensolutions of the Laplace operator, whose eigenvalue problem is also called the Helmholtz equation. Specifically, the paper showcases all geometries through which the solutions to the Helmholtz equation can be represented in a finite sinusoidal form. These geometries are the rectangle, the square, the isosceles right triangle, the equilateral triangle, and the hemi-equilateral triangle. As a counterexample, the paper also proves that the parallelogram cannot yield a product form of a solution through the method of separation of variables. The solutions for the isosceles triangle and the hemi-equilateral triangle are derived using symmetric properties of the square and the equilateral triangle. 

The paper concludes that symmetry is crucial to solving the Laplacian for these geometries and that this symmetry is also reflected in their respective spectra. However, importantly, the spectrum is unique for the examined geometries.