Unification of Grassmann’s Progressive and Regressive Products using the Principle of Duality

Abstract

This paper suggests that geometric algebra should be based on points instead of vectors and should be metric-free. Whitehead’s 1898 treatise on Grassmann ’s Ausdehnungslehre described just such a metricfree geometric algebra of points. However, Whitehead’s treatise spoiled the natural simplicity of the theory by a lopsided derivation based purely on points which gave rise to two different products (progressive and regressive). The current paper tidies up the theory by invoking the principle of duality to put points and hyperplanes on an equal footing and then shows that the theory has just a single antisymmetric product which evaluates to give the same results as Whitehead’s progressive and regressive products.