A Smooth Curve as a Fractal Under the Third Definition

Ding Ma; Bin Jiang

doi:10.3138/cart.53.3.2017-0032

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arXiv · 1802.03698

A Smooth Curve as a Fractal Under the Third Definition

Ding Ma·Bin Jiang

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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

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Ding Ma, Bin Jiang. 2018-02-11. A Smooth Curve as a Fractal Under the Third Definition. https://doi.org/10.3138/cart.53.3.2017-0032

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