The Angular Seed Power Map: A Constructive Approach to Recursive Scaling Spirals

Abstract

We present the ''Power Spiral Map'', a continuous angular evolution of the linear coordinate grid established in our previous work. While that previous Power Map utilized a seed value translating along a horizontal axis, this work builds upon a seed angle ($θ$) projected onto a unit diameter circle. This operation controls two coupled geometric behaviors: an internal area-preserving partition of unity within a reference square (cos 2 $θ$ + sin 2 $θ$ = 1) and an external recursive scaling mechanism (sec $θ$ and cos $θ$) that dictates the expansion or contraction of successive generations of squares unfolding as a spiral in the 2D plane. We demonstrate that continuous variation of this angular parameter generates discrete geometric alignments that yield polynomial identities, with examples of the Golden Ratio ($Φ$) and the Plastic Ratio ($ψ$) defined through purely planar intersections.