  Given a residually connected incidence geometry $\Gamma$ that satisfies two
conditions, denoted $(B_1)$ and $(B_2)$, we construct a new geometry
$H(\Gamma)$ with properties similar to those of $\Gamma$. This new geometry
$H(\Gamma)$ is inspired by a construction of Percsy, Percsy and Leemans [1]. We
show how $H(\Gamma)$ relates to the classical halving operation on polytopes,
allowing us to generalize the halving operation to a broader class of
geometries, that we call non-degenerate leaf hypertopes. Finally, we apply this
generalization to cubic toroids in order to generate new examples of regular
hypertopes.
