Classifying Strictly Tessellating Polytopes

Abstract

Abstract This thesis consists of a paper and additional results. The paper shows a connection between the geometry of polytopal domains in Euclidean space and the eigenfunctions of the Dirichlet Laplacian. The necessary and sufficient geometric properties of a polytopal domain are shown for the first eigenfunction to extend to a real analytic function on the whole space. Furthermore, alcoves are essential for the proof of the main theorem. Additionally, the paper discusses how the results relate to crystallographic restrictions and lattices. Strictly tessellating polytopes are defined and used in connection to the main theorem. The paper concludes by formulating a conjecture akin to Fuglede’s, replacing tessellation by translaton with strict tessellation. In addition to the paper, results on the geometric properties of strictly tessellating polytopes are presented, and bounds on the number of strictly tessellating polytopes up to equivalence are shown.