Notes on approximate golden spirals with whirling squares

TL;DR

This paper extends known properties of approximate golden spirals to generalized m-spirals constructed with whirling squares, revealing that their poles lie on a circumscribed circle, thus broadening the understanding of spiral geometries.

Contribution

It introduces a generalized framework for m-spirals with whirling squares and proves new properties about their poles and circumscribed circles, expanding classical golden spiral theory.

Findings

Poles of m-spirals lie on a circumscribed circle.

Circumscribed circles around squares intercept the pole.

Properties are not exclusive to golden rectangles.

Abstract

We have extended some known results of the approximate golden spirals to generalized m-spirals built with whirling squares for any $m$ ratio ($m>1$). In particular, we have proved that circumscribed circles around squares intercept the pole. This implies that poles of any m-spirals lie on a circumscribed circle about the first square. We show here that these and other fascinating properties are not exclusive of golden rectangles. The classical golden constructors may not be alone.