Manifold-based Proving Methods in Projective Geometry

TL;DR

This paper compares various proof methods in projective geometry, showing their equivalences and limitations, especially focusing on quadrilateral tilings and Menelaus configurations, with examples in 2D and 3D.

Contribution

It establishes the equivalence and relative strength of different projective proof techniques, including recent quadrilateral tiling methods.

Findings

Quadrilateral tiling proofs are at most as strong as bi-quadratic polynomial proofs.

Quadrilateral tiling proofs are equivalent to Menelaus configuration proofs.

Examples demonstrate transitions between proof methods in 2D and 3D.

Abstract

This article compares different proving methods for projective incidence theorems. In particular, a technique using quadrilateral tilings recently introduced by Sergey Fomin and Pavlo Pylyavskyy is shown to be at most as strong as proofs using bi-quadratic final polynomials and thus, also proofs using Ceva-Menelaus-tilings. Furthermore, we demonstrate the equivalence between quadrilateral-tiling-proofs and proofs using exclusively Menelaus configurations. We exemplify the transition between the proofs in several examples in 2D and in 3D.