Incidences, tilings, and fields

TL;DR

This paper explores the capabilities and limitations of the master theorem, which uses surface tilings to prove incidence theorems in projective geometry, by formalizing tiling proofs and analyzing different theorem classes.

Contribution

It formalizes the notion of tiling proofs, introduces a hierarchy of theorem classes, and examines the theorem's applicability over real, complex, and finite fields.

Findings

The master theorem can prove certain incidence theorems but not all.

A hierarchy of theorem classes based on topological spaces is proposed.

Finite field cases provide insights into theorem provability.

Abstract

The master theorem, introduced by Richter-Gebert and generalized by Fomin and the first author, provides a method for proving incidence theorems of projective geometry using triangular tilings of surfaces. We investigate which incidence theorems over C and R can or cannot be proved via the master theorem. For this, we formalize the notion of a tiling proof. We introduce a hierarchy of classes of theorems based on the underlying topological spaces. A key tool is considering the same theorems over finite fields.