A Smooth Curve as a Fractal Under the Third Definition

TL;DR

This paper redefines fractals to include smooth curves like circles and spirals, demonstrating that under a new definition, such curves exhibit fractal-like scaling properties through recurring hierarchies of small and large features.

Contribution

It introduces a third, relaxed definition of fractals that encompasses smooth curves, supported by examples like spirals and coastlines, expanding the traditional scope of fractal geometry.

Findings

Smooth curves can be fractal under the new definition.

Scaling of small to large features recurs multiple times in smooth curves.

Examples include logarithmic spirals and coastlines.

Abstract

It is commonly believed in the literature that smooth curves, such as circles, are not fractal, and only non-smooth curves, such as coastlines, are fractal. However, this paper demonstrates that a smooth curve can be fractal, under the new, relaxed, third definition of fractal - a set or pattern is fractal if the scaling of far more small things than large ones recurs at least twice. The scaling can be rephrased as a hierarchy, consisting of numerous smallest, a very few largest, and some in between the smallest and the largest. The logarithmic spiral, as a smooth curve, is apparently fractal because it bears the self-similar property, or the scaling of far more small squares than large ones recurs multiple times, or the scaling of far more small bends than large ones recurs multiple times. A half-circle or half-ellipse and the UK coastline (before or after smooth processing) are fractal, if the scaling of far more small bends than large ones recurs at least twice. Keywords: Third definition of fractal, head/tail breaks, bends, ht-index, scaling hierarchy