  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
