There are 174 Subdivisions of the Hexahedron into Tetrahedra

TL;DR

This paper enumerates all 174 possible subdivisions of a hexahedron into tetrahedra, revealing 171 geometrically realizable configurations and providing explicit meshes for practical applications.

Contribution

It provides the first complete enumeration of hexahedron subdivisions into tetrahedra, including their geometric realizability and explicit mesh examples.

Findings

174 subdivisions identified, ranging from 5 to 15 tetrahedra

171 subdivisions have valid 3D geometric realizations

Meshes with positive Jacobian demonstrated for complex subdivisions

Abstract

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.