The main result of this paper may be seen as purely a result about rank 4 lattice polytopes and their duals. However, the authors motivate the result by showing how it is linked to certain results of importance in theoretical physics. Specifically, the authors have classified so-called “reflexive polyhedra” in four dimensions. Reflexive polyhedra are important in the proofs of various results about so-called “Calabi-Yau manifolds”. Readers interested in lattice polyhedra but not in theoretical physics will find this paper useful, and should not be put off by the introductory section. A reflexive polyhedron is defined as a lattice polyhedron whose polar (or dual) is also a lattice polyhedron. In more detail, if \(P\) is a polyhedron whose vertices lie on a lattice \(M\), and if \(N\) is the dual lattice of \(M\), then the dual \(P^*\) of \(P\) must have vertices which lie on \(N\).
In this article, the authors give an algorithm for discovering lattice polyhedra, and then report the results of an application of that algorithm to discover reflexive polyhedra (polytopes) of rank 4. After more than half a year they discovered a total of 473,800,776 reflexive polyhedra (236,879,533 dual pairs and 41,710 self-dual examples). The authors give some interesting examples they encountered before interpreting their results in terms of the Hodge numbers of the corresponding Calabi-Yau manifolds.