Multiplicity of Scator Roots and the Square Roots in $\mathbb{S}^{1+2}$

Abstract

This paper presents the roots of elliptic scator numbers in $\mathbb{S}^{1+n}$, which includes both the fundamental $2\pi$ symmetry and the $\pi$-pair symmetry for $n\geq2$. Here, the scator set $\mathbb{S}^{1+n}$ is a subset of $\mathbb{R}^{1+n}$ with the scator product and the multiplicative representation. These roots are expressed in terms of both additive (rectangular) and multiplicative (polar) variables. Additionally, the paper provides a comprehensive description of square roots in $\mathbb{S}^{1+2}$, which includes a geometrical representation in three-dimensional space that provides a clear visualization of the concept and makes it easier to understand and interpret. Finally, the paper handles whether the aspects should be further investigated.