A Smooth Curve as a Fractal Under the Third Definition(arxiv.org)
arxiv.org
A Smooth Curve as a Fractal Under the Third Definition
https://arxiv.org/abs/1802.03698
10 comments
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?
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?
An expansion of a definition is warranted if it is consistent with the original intent of the definition. I can see the log spiral as being construed as fractal although a semicircle is a bit off.
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
Nice paper!
Abstract. This memoir is concerned with isometric embeddings of a square flat torus in the three dimensional Euclidean space. The existence of such embeddings was proved by John Nash and Nicolaas Kuiper in the mid 50s. However, the geometry of these embeddings could barely be conceived from their original papers. Here we provide an explicit construction based on the convex integration theory introduced by Mikhail Gromov in the 70s. Wethen turn this construction into a computer implementation leading us to the visualisation of an isometrically embedded flat torus. The pictures reveal a geometric object in-between fractals and ordinary surfaces. We call this object a C1 fractal.
Abstract. This memoir is concerned with isometric embeddings of a square flat torus in the three dimensional Euclidean space. The existence of such embeddings was proved by John Nash and Nicolaas Kuiper in the mid 50s. However, the geometry of these embeddings could barely be conceived from their original papers. Here we provide an explicit construction based on the convex integration theory introduced by Mikhail Gromov in the 70s. Wethen turn this construction into a computer implementation leading us to the visualisation of an isometrically embedded flat torus. The pictures reveal a geometric object in-between fractals and ordinary surfaces. We call this object a C1 fractal.
Woah!
Just the other day I've had a discussion with my companion about how one would determine they live on a toroidal[1] planet (assuming the atmosphere is too cloudy, and one can only see a small neighborhood).
She suggested measuring circumferences of circles at various points, as a means of getting curvature; the torus is negatively curved at some places.
"But what if we are on a flat torus?", I asked. "We are talking about a potentially realistic scenario here, not a topological abstraction", she objected. I didn't have anything to say.
But now I do.
On page 66, they even show a 3D-printed model!
I knew of Whitney embedding theorem, but somehow, not of the Nash embedding theorem which proves this is possible - and certainly not of this result, which is delightfully amazing.
[1]...which, according to articles like this one, are possible: https://io9.gizmodo.com/what-would-the-earth-be-like-if-it-w...
Just the other day I've had a discussion with my companion about how one would determine they live on a toroidal[1] planet (assuming the atmosphere is too cloudy, and one can only see a small neighborhood).
She suggested measuring circumferences of circles at various points, as a means of getting curvature; the torus is negatively curved at some places.
"But what if we are on a flat torus?", I asked. "We are talking about a potentially realistic scenario here, not a topological abstraction", she objected. I didn't have anything to say.
But now I do.
On page 66, they even show a 3D-printed model!
I knew of Whitney embedding theorem, but somehow, not of the Nash embedding theorem which proves this is possible - and certainly not of this result, which is delightfully amazing.
[1]...which, according to articles like this one, are possible: https://io9.gizmodo.com/what-would-the-earth-be-like-if-it-w...
That's not exactly 'smooth' in the differential geometry sense.
what is 'smooth' in the diff geom sense?
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.
Nice. An interesting challenge for someone on Shadertoy.
Which is, of course, absurd. Am I missing something?