Powers of Elliptic Scator Numbers

Abstract

Elliptic scator algebra is possible in 1+n dimensions, n∈N. It is isomorphic to complex algebra in 1 + 1 dimensions, when the real part and any one hypercomplex component are considered. It is endowed with two representations: an additive one, where the scator components are represented as a sum; and a polar representation, where the scator components are represented as products of exponentials. Within the scator framework, De Moivre’s formula is generalized to 1+n dimensions in the so called Victoria equation. This novel formula is then used to obtain compact expressions for the integer powers of scator elements. A scator in S1+n can be factored into a product of n scators that are geometrically represented as its projections onto n two dimensional planes. A geometric interpretation of scator multiplication in terms of rotations with respect to the scalar axis is expounded. The powers of scators, when the ratio of their director components is a rational number, lie on closed curves. For 1 + 2 dimensional scators, twisted curves in a three dimensional space are obtained. Collecting previous results, it is possible to evaluate the exponential of a scator element in 1 + 2 dimensions.

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

“…In contrast, components with different director units ěl and ěm (l ̸ = m) do not, i.e., exp (α l ěl ) exp (β m ěm ) ̸ = exp (α l ěl + β m ěm ). An expression for the exponential of a scator with 1+2 components has been derived in [7]. The conjugate of the scator…”

“…A constant magnitude generates the cusphere isometric surface. Other relevant properties of elliptic scator algebra are summarized in [7].…”

“…From Theorem 4 in [7], that generalizes De Moivre formula to S 1+n scator space, provided that the product of the factors and its components are associative,…”