Exceptional Projective Geometries and Internal Symmetries

TL;DR

This paper explores the connection between exceptional projective geometries, octonionic duality, and internal symmetries, revealing new geometric structures linked to Jordan algebras and non-Desarguesian geometries.

Contribution

It introduces a novel mnemonic device related to O(7) tensors and demonstrates the link between classical theorems and octonionic geometries, advancing understanding of exceptional structures.

Findings

Connection between Desargues' and Pappus' theorems via octonionic geometry

Construction of exceptional Hilbert spaces using Jordan algebras

Discussion of Moufang plane and non-Desarguesian geometries

Abstract

A new mneumonic device is shown to emerge in connection with O(7) numerical tensors exhibiting duality and reflecting the natural 7=(4+3) splitting of 7-dimensional space. Then Desargues' and Pappus' theorems are shown to be connected through a geometry that makes use of octonionic numbers exhibiting this duality. Construction of exceptional Hilbert space based on Jordan algebras and exceptional projective geometries is illustrated. A brief discussion of the Moufang plane and non-Desarguesian geometries is presented.