Theoretical and numerical considerations on the polar (exponential) form of octonions and elements of higher-order Cayley-Dickson algebras

TL;DR

This paper investigates the polar (exponential) form of octonions and higher-order Cayley-Dickson algebras, identifying errors in previous work and proposing a numerical method to improve understanding of their angular representations.

Contribution

It corrects earlier inaccuracies in the polar form of octonions and introduces a numerical approach that could lead to analytical formulas for their angles.

Findings

Identified errors in previous polar form considerations

Developed a promising numerical method for octonion angles

Potential to derive analytical formulas for polar angles

Abstract

The article is devoted to the issue of the polar form of octonions. This is a~continuation of the works initiated by Hahn and Snopek in their articles from 2011. The results presented in the article show errors made in previous considerations and suggest the possibility of their improvement. The presented numerical method gives promising results, which as a result of further work can give analytical formulas for angles in the polar representation.