  Calabi-Yau manifolds can be obtained as hypersurfaces in toric varieties
built from reflexive polytopes. We generate reflexive polytopes in various
dimensions using a genetic algorithm. As a proof of principle, we demonstrate
that our algorithm reproduces the full set of reflexive polytopes in two and
three dimensions, and in four dimensions with a small number of vertices and
points. Motivated by this result, we construct five-dimensional reflexive
polytopes with the lowest number of vertices and points. By calculating the
normal form of the polytopes, we establish that many of these are not in
existing datasets and therefore give rise to new Calabi-Yau four-folds. In some
instances, the Hodge numbers we compute are new as well.
