Geometry

A New Home for Bivectors Geometric Algebra on a Sphere

Authors: Norm Cimon
Comments: 20 Pages.

The impetus for the work is this quote:"...as shown by Gel’fand’s approach, we can only abstract a unique manifold if our algebra is commutative."[1] Geometric algebra is non-commutative. Components of different grades can be staged on different manifolds. As operations on those elements proceed, they can effect the promotion and/or demotion of components to higher and/or lower grades, and thus to different manifolds. This paper includes imagery that visually displays bivector addition and rotation on a sphere. David Hestenes interpreted the vector product or rotor in two-dimensions: "as a directed arc of fixed length that can be rotated at will on the unit circle, just as we interpret a vectoras a directed line segment that can be translated at will without changing its length or directionu2026"[2]Rotors can be used to develop addition and multiplication of bivectors on a sphere. For those rotational dynamics, rotors of lengthare the basis elements. The geometric algebra of bivectors — Hamilton’s "pure quaternions" — is thus shown to transparently operate on a spherical manifold.This paper also explores the possible generalizations that emerge from the placement of the graded elements which make up a geometric algebra onto separate manifolds.
Category: Geometry