Constructing new geometries: a generalized approach to halving for hypertopes

TL;DR

This paper generalizes the halving operation from polytopes to a broader class of geometries called non-degenerate leaf hypertopes, enabling the construction of new regular hypertopes, including examples related to cubic toroids.

Contribution

It introduces a new construction method for geometries that extends the classical halving operation to hypertopes, broadening the scope of geometric structures studied.

Findings

Established a relation between $H(Gamma)$ and classical halving

Generalized halving operation to non-degenerate leaf hypertopes

Generated new examples of regular hypertopes from cubic toroids

Abstract

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.