A Logarithmic Spiral Formed by a Sequence of Regular 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.