Constructing new geometries: a generalized approach to halving for hypertopes

Given a residually connected incidence geometry that satisfies two conditions, denoted (B₁) and (B₂), we construct a new geometry H () with properties similar to those of. This new geometry H () is inspired by a construction of Percsy, Percsy and Leemans 1. We show how H () 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.