Geometric Roots of -1 in Clifford Algebras $\G_{p,q}$ with $p+q \leq 4$

Abstract

It is known that Clifford (geometric) algebra offers a geometric interpretation for square roots of -1 in the form of blades that square to minus 1. This extends to a geometric interpretation of quaternions as the side face bivectors of a unit cube. Research has been done [S. J. Sangwine, Biquaternion (Complexified Quaternion) Roots of -1, Adv. Appl. Cliford Alg. 16(1), pp. 63-68, 2006.] on the biquaternion roots of -1, abandoning the restriction to blades. Biquaternions are isomorphic to the Clifford (geometric) algebra $Cl_{3}$ of $\R^3$. All these roots of -1 find immediate applications in the construction of new types of geometric Clifford Fourier transformations. We now extend this research to general algebras $Cl_{p,q}$. We fully derive the geometric roots of -1 for the Clifford (geometric) algebras with $p+q \leq 4$.