This article answers an important theoretical question: How many different subdivisions of the hexahedron into tetrahedra are there? It is well known that a cube has five subdivisions into 6 tetrahedra and one subdivision into 5 tetrahedra. However, all hexahedra are not cubes and moving  vertex positions increases the number of subdivisions. Recent hexahedral dominant meshing methods try to take these configurations into account for combining tetrahedra into hexahedra, but fail to enumerate them all: they use only a set of 10 subdivisions among the 174 we found in this article. The enumeration of these 174 subdivisions of the hexahedron into tetrahedra is our main result. Each of the 174 triangulations has between 5 and 15 tetrahedra and is actually a class of 2 to 48 equivalent instances which are identical up to vertex relabeling. This result is purely combinatorial. We also demonstrate that a minimum of 114 triangulations do correspond to hexahedron geometrical triangulations.Each hexahedron is valid for finite element computations (strictly positive Jacobian determinant) . We exhibit the tetrahedral meshes for these geometries and show in particular triangulations of the hexahedron with 14 tetrahedra .
This article answers an important theoretical question: How many different subdivisions of the hexahedron into tetrahedra are there? It is well known that the cube has five subdivisions into 6 tetrahedra and one subdivision into 5 tetrahedra. However, all hexahedra are not cubes and moving the vertex positions increases the number of subdivisions. Recent hexahedral dominant meshing methods try to take these configurations into account for combining tetrahedra into hexahedra, but fail to enumerate them all: they use only a set of 10 subdivisions among the 174 we found in this article. The enumeration of these 174 subdivisions of the hexahedron into tetrahedra is our combinatorial result. Each of the 174 subdivisions has between 5 and 15 tetrahedra and is actually a class of 2 to 48 equivalent instances which are identical up to vertex relabeling. We further show that exactly 171 of these subdivisions have a geometrical realization, i.e. there exist coordinates of the eight hexahedron vertices in a three-dimensional space such that the geometrical tetrahedral mesh is valid . We exhibit the tetrahedral meshes for these configurations and show in particular subdivisions of hexahedra with 15 tetrahedra that have a strictly positive Jacobian .
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This result is purely combinatorial. We also demonstrate that a minimum of 114 triangulations do correspond to hexahedron geometrical triangulations.Each hexahedron is valid for finite element computations (strictly positive Jacobian determinant)
We further show that exactly 171 of these subdivisions have a geometrical realization, i.e. there exist coordinates of the eight hexahedron vertices in a three-dimensional space such that the geometrical tetrahedral mesh is valid
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triangulations of the hexahedron with 14 tetrahedra
subdivisions of hexahedra with 15 tetrahedra that have a strictly positive Jacobian