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arXiv · 2606.25505

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

Arjen Toni Dijksman
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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 ($\theta$) 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 $\theta$ + sin 2 $\theta$ = 1) and an external recursive scaling mechanism (sec $\theta$ and cos $\theta$) 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 ($\Phi$) and the Plastic Ratio ($\psi$) defined through purely planar intersections.

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Arjen Toni Dijksman. 2026-06-24. The Angular Seed Power Map: A Constructive Approach to Recursive Scaling Spirals. https://arxiv.org/abs/2606.25505

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