  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.
