Constructing New Geometries: A Generalized Approach to Halving for Hypertopes

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 Lefèvre-Percsy, Percsy and Leemans (Bull Belg Math Soc Simon Stevin 7(4):583–610, 2000). 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.