The Extension of the Desargues Theorem, the Converse, Symmetry and Enumeration

TL;DR

This paper generalizes Desargues' theorem to all dimensions, proves its converse, explores its configurations in finite fields, and reveals fractal-like properties and extensions of classical theorems.

Contribution

It extends Desargues' theorem and its converse to all dimensions, analyzes configurations in finite fields, and introduces fractal-like properties in higher-dimensional projective spaces.

Findings

Desargues' theorem extended to all dimensions

Desargues configurations correspond to arcs in projective spaces

Revealed fractal-like properties in Desargues configurations

Abstract

This work is, in part, a generalization of the article by A.A. Bruen ,T.C Bruen and J.M.McQuillan on Desargues Theorem in arXiv:2007.09175[mathCO]July 17,2020. We prove the extension of Desargues theorem in all dimensions, using 4 different arguments.We also show the converse theorem. It is shown that the Desargues configuration in projective n-space, for all n at least 2, corresponds to an arc in a projective space of dimension n+1 containing the n-space. Thus, in principle, one can enumerate the number of Desargues configurations in n dimensions when the underlying field is finite. The Desargues configuration in n-dimensions is studied in detail and is shown to exhibit new self-replication or fractal-like properties. In section 11 we extend a classical theorem on semi-simplexes for all n at least 3.