Enumerating combinatorial triangulations of the hexahedron

Most indirect hexahedral meshing methods rely on 10 patterns of subdivision of the hexahedron into tetrahedra. A recent observation at least one more pattern exists raise the question of the actual number of subdivisions of the hexahedron into tetrahedra. In this article answers we enumerate these subdivisions by exhausting all possible ways to combine tetrahedra into hexahedra. We introduce a combinatorial algorithm that (1) generates all the combinations of tetrahedra that can be built from eight vertices and (2) tests if they subdivide a hexahedron or not. The challenge is that there are 2 combinations of tetrahedra to consider. We use topological arguments and an efficient pruning strategy to drastically reduce this number. Our main result is that there are 6,966 combinatorial triangulations of the hexahedron which can be classified into 174 equivalence classes. Our results are consistent with theoretical results available on the subdivision of the cube.