  We construct sphaleron solutions with discrete symmetries in Yang-Mills-Higgs
theory coupled to a dilaton. These platonic sphalerons are related to rational
maps of degree N. We demonstrate that, in the presence of a dilaton, for a
given rational map excited platonic sphalerons exist beside the fundamental
platonic sphalerons. We focus on platonic sphaleron solutions with N=4, which
possess cubic symmetry, and construct the two branches of their first
excitations. The energy density of these excited platonic sphalerons exhibits a
cube within a cube.
