A Logarithmic Spiral Formed by a Sequence of Regular Polygons
Juno Park
TL;DR
The paper addresses whether the centers of a left-bending, edge-to-edge sequence of regular polygons with unit side length trace a logarithmic spiral. By placing the construction in the complex plane and deriving an explicit complex center sequence $P_n$, the author performs a detailed asymptotic analysis using Euler–Maclaurin expansions of harmonic sums, obtaining a tractable equivalent for $P_n$ that reveals parity-dependent leading terms. After an appropriate orientation-preserving isometry, the centers are shown to lie asymptotically near the spiral $r = e^{\frac{4}{\pi}\theta}$ with explicit inward offsets: $d(T(P_{2n}),C) \to \tfrac{5}{6}$ and $d(T(P_{2n+1}),C) \to \tfrac{7}{12}$ (plus $\mathcal{O}(1/n)$). The odd-only subsequence yields a further constant offset of $\tfrac{7}{24}$, illustrating a robust, parity-driven structure. These results provide a rigorous link between polygonal center sequences and a canonical logarithmic spiral, with precise geometric offsets and a clear asymptotic description of the convergence behavior.
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(4θ/π). 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.