Recursive formulas for root calculation inspired by geometrical constructions

Rik Verhulst

This article describes a method for calculating arithmetic, geometric and harmonic means of two numbers and how they can be represented geometrically. We extend these mean values to arithmetic, geometric and harmonic thirds, fourths, etc. For this we will only use the tools of the affine planar geometry. Also, we will make allusion to the more general interpretation in the projective plane. From the relations between these means we can deduce a multitude of recursive formulas for $n$-th root calculation and represent them by geometric constructions. {These formulas give a solution for reducing the power of the root}. Surprisingly, one of these algorithms turns out to be the same as the one using Newton's tangent method for calculating zero values of functions of the form $f(x)=x^n-c$, but obtained without use of analysis. Moreover, regarding speed of convergence these algorithms are faster than Newton's tangent method. This geometric interpretation of mean values and root calculation fits into the larger context of affine geometry, where we use multi-projections as generating transformations for building up all the affine transformations. Our focus will primarily be on mean values and roots.