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.
From: Arjen Toni Dijksman [view email] [via CCSD proxy]
[v1]
Wed, 24 Jun 2026 07:39:42 UTC (1,969 KB)
此内容由惯性聚合(RSS阅读器)自动聚合整理，仅供阅读参考。 原文来自 — 版权归原作者所有。