  We characterize the essential spectrum of the plasmonic problem for polyhedra
in $\mathbb{R}^3$. The description is particularly simple for convex polyhedra
and permittivities $\epsilon < - 1$. The plasmonic problem is interpreted as a
spectral problem through a boundary integral operator, the direct value of the
double layer potential, also known as the Neumann--Poincar\'e operator. We
therefore study the spectral structure of the the double layer potential for
polyhedral cones and polyhedra.
