  We study geometries occurring in Feynman integrals that contribute to the
scattering of black holes in the post-Minkowskian expansion. These geometries
become relevant to gravitational-wave production during the inspiralling phase
of binary black hole mergers through the classical conservative potential. At
fourth post-Minkowskian order, a K3 surface is known to occur in a three-loop
integral, leading to elliptic integrals in the result. In this letter, we
identify a Calabi-Yau three-fold in a four-loop integral, contributing at fifth
post-Minkowskian order. The presence of this Calabi-Yau geometry indicates that
completely new functions occur in the full analytical results at this order.
