A Logarithmic Spiral Formed by a Sequence of Regular Polygons

TL;DR

This paper demonstrates that a sequence of joined regular polygons with increasing sides forms a specific logarithmic spiral, providing mathematical proofs for its precise shape and the convergence of distances to polygon centers.

Contribution

It analytically derives the exact form of the spiral and the convergence distances for sequences of regular polygons with increasing sides, both even and odd.

Findings

The spiral follows the form r=exp(4θ/π).

Distances to polygon centers converge to specific ratios (5/6, 7/12, 7/24).

The same spiral shape occurs for sequences with odd and even-sided polygons.

Abstract

When the sequence of regular polygons with consecutively increasing numbers of sides is joined edge-to-edge in a single direction while minimizing bending, the resulting structure assumes the shape of a logarithmic spiral. This paper proves that this spiral takes the form r=exp(4{\theta}/{\pi}). Specifically, it is derived that the distances between the curve and the centers of the even-sided and odd-sided regular polygons converge to 5/6 and 7/12, respectively, with the centers extending outward along the inner side of the spiral. A similar analysis applied to the sequence of regular polygons with consecutively increasing odd numbers of sides reveals that it forms the same type of spiral, establishing that the distances to the centers converge to 7/24.