Crystallographic groups, strictly tessellating polytopes, and analytic eigenfunctions

TL;DR

This paper generalizes the connection between crystallographic groups, tessellating polytopes, and analytic eigenfunctions from 2D to all dimensions, revealing deep links between geometry, analysis, and number theory.

Contribution

It extends known results to higher dimensions, establishing equivalences between eigenfunction analyticity, space tessellation, and Coxeter group fundamental domains.

Findings

Eigenfunctions are trigonometric under the conditions.

Polytope tessellates space if and only if eigenfunctions are analytic.

Connections to Fuglede and Goldbach conjectures are established.

Abstract

The mathematics of crystalline structures connects analysis, geometry, algebra, and number theory. The planar crystallographic groups were classified in the late 19th century. One hundred years later, B\'erard proved that the fundamental domains of all such groups satisfy a very special analytic property: the Dirichlet eigenfunctions for the Laplace eigenvalue equation are all trigonometric functions. In 2008, McCartin proved that in two dimensions, this special analytic property has both an equivalent algebraic formulation, as well as an equivalent geometric formulation. Here we generalize the results of B\'erard and McCartin to all dimensions. We prove that the following are equivalent: the first Dirichlet eigenfunction for the Laplace eigenvalue equation on a polytope is real analytic, the polytope strictly tessellates space, and the polytope is the fundamental domain of a crystallographic Coxeter group. Moreover, we prove that under any of these equivalent conditions, all of the eigenfunctions are trigonometric functions. In conclusion, we connect these topics to the Fuglede and Goldbach conjectures and give a purely geometric formulation of Goldbach's conjecture.