Desargues's and Pappus's hexagon theorems on translation-type surfaces in Thurston geometries

TL;DR

This paper extends classical hexagon theorems by proving Desargues's and Pappus's theorems hold on translation surfaces within Thurston geometries, using unified projective models.

Contribution

It demonstrates that Desargues's and Pappus's theorems are valid in Thurston geometries with non-constant curvature, broadening their applicability.

Findings

Classical hexagon theorems hold in Thurston geometries.

Unified projective models facilitate proofs in non-constant curvature settings.

The results generalize known theorems to a wider geometric context.

Abstract

In \cite{Sz25} we generalized the famous Menelaus' and Ceva's theorems for translation triangles in each non-constant curvature Thurston geometry. In this paper based on the described method and results, we prove that the classical Desargues's and Pappus's hexagon theorems are true not only in classical geometries with constant curvature, but also in Thurston geometries with non-constant curvature on the translation surfaces. In our work we use the unified projective models of Thurston geometries.