Abstract: This talk is based on joint work with my students, Max Blom, Henrik Nordell, Oliver Thim, and Jack Vahnberg. In 2008, Brian McCartin proved that the only polygonal domains in the plane which have a complete set of trigonometric eigenfunctions are: rectangles, equilateral triangles, hemi-equilateral triangles, and isosceles right triangles. Trigonometric eigenfunctions are, as the name suggests, functions which can be expressed as a finite linear combination of sines and cosines. In 1980, Pierre Bérard proved that a certain type of polytopes in n dimensional Euclidean space, known as an alcoves, always have a complete set of trigonometric eigenfunctions. We prove that the following are equivalent for polytopes in n dimensional Euclidean space: (1) the first Dirichlet eigenfunction for the Laplace eigenvalue equation extends real analytically to all space (2) the polytope strictly tessellates space (3) the polytope is an alcove. As a corollary, we obtain that if the first eigenfunction extends real analytically to all space, then it is trigonometric, and moreover, all the eigenfunctions are trigonometric. This talk is intended for a general mathematical audience including students! No experience with any of these mathematical concepts is required, as everything will be explained so that everyone can follow.