Mathematics of a Tetrahedron Chain and the Hamiltonian Cycle Problem

In this article we consider the problem of finding Hamiltonian cycles on a tetrahedral mesh. A Hamiltonian cycle is a closed loop through a tetrahedral mesh that visits each tetrahedron exactly once. Using techniques of a novel discrete differential geometry of n-simplices, we could immediately obtain Hamiltonian cycles on a rhombic dodecahedronshaped tetrahedral mesh consisting of 24 tetrahedrons. For easier visualization, we firstly consider Hamiltonian cycles on triangular meshes and provide some foundations for the study of the tetrahedral mesh.