Hamiltonian cycles on the edge graphs of the regular polytopes in three and four dimensions are investigated with the primary goal of finding complete multi-colored coverages of all the edges in the graph. The concept of a Hamiltonian path is then extended to the notion of Hamiltonian two manifolds that visit all the given edges exactly once. For instance, the 4D simplex can be covered by a strip of 5 triangular facets that form a Moebius band! The use of Hamiltonian cycles to create physical dissection puzzles as well as geometrical sculptures is also investigated. The concepts are illustrated with computer graphics imagery and with small maquettes made with rapid prototyping techniques.