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.

Roots of Second Order Polynomials with Real Coefficients in Elliptic Scator Algebra

“…If b, the polynomial linear coefficient vanishes, the hypercomplex roots become square nilpotent so that the only roots of minus one are ±ě 1 and ±ě 2 . Arbitrary integer powers of scators and nilpotent elements are discussed in [15]. The S 1+2 scator roots can be visualized geometrically in a three dimensional space, where the scalar (real) axis and the two hypercomplex axes are drawn in orthogonal directions.…”