
A Smooth Curve as a Fractal Under the Third Definition - anacleto
https://arxiv.org/abs/1802.03698
======
gjm11
It seems like this could be uncharitably summarized as follows. "The usual
definitions of fractal are rather narrow. The second author of the present
paper has proposed a broader definition of fractal. In this paper we show that
according to this definition curves such as the semicircle and the logarithmic
spiral are 'fractals', but rather than concluding that the definition was a
bad one we choose to conclude instead that those curves really are fractals."

Which is, of course, absurd. Am I missing something?

~~~
tgb
I agree with you - I can't pick any sense out of this paper. Note that they
have not given a single example of a curve that is _not_ a "fractal" by their
definition. What would such a curve look like? A straight line is the obvious
guess - but since all the "bends" are zero for a straight line it's not at all
clear what their definition would even mean.

Intuitively, a fractal curve is something that has _larger_ bends than
expected yet their definition is something has a disproportionately many
_small_ bends. Why would we want to consider such a thing fractal? Frankly if
I were to try to find a curve that was _not_ fractal (under their definition),
I would search amongst curves that _are_ fractal (for the usual notion): how
else would you get large bends even as you subdivide the problem?

------
mlevental
outside of this there already exist smooth fractals

[https://www.emis.de/journals/em/images/pdf/em_24.pdf](https://www.emis.de/journals/em/images/pdf/em_24.pdf)

~~~
contravariant
That's not exactly 'smooth' in the differential geometry sense.

~~~
mlevental
what is 'smooth' in the diff geom sense?

~~~
contravariant
In that context 'smooth' typically refers to things that are infinitely
differentiable. The manifold described in the paper you linked only has
derivatives of order 1.

