Gallucci's axiom revisited

TL;DR

This paper revisits Gallucci's axiom by constructing a synthetic model of projective space, deriving key theorems without relying on analytical geometry concepts like cross-ratio or coordinates.

Contribution

It introduces a new synthetic approach to projective space that derives fundamental theorems solely from space axioms, including Gallucci's axiom, without analytical tools.

Findings

Derived Desargues' theorem from space axioms

Established Pappus' theorem within the synthetic framework

Proved the fundamental theorems of projectivities and collinearities

Abstract

In this paper we propose a well-justified synthetic approach of the projective space. We define the concepts of plane and space of incidence and also the Gallucci's axiom as an axiom to our classical projective space. To this purpose we prove from our space axioms, the theorems of Desargues, Pappus, the fundamental theorem of projectivities, and the fundamental theorem of central-axial collinearities, respectively. Our building up do not use any information on analytical projective geometry, as the concept of cross-ratio and the homogeneous coordinates of points.