... at least one more amateur saved your paper to disk

LOL Thanks

By the way, there is also a measure called sinuosity, which is the length divided by the displacement.

Also, once the isosurface of a quaternion Julia set has been calculated by using Marching Cubes (like in this compute shader example that I wrote), one can test the surface for its dimension, somewhere between 2D and 3D. As far as I know, it's one of the simplest ways to calculate the fractal dimension of a surface. As you'd expect, the equation Z = Z, which forms a 2-sphere, results in a surface of dimension 2.0 because the local curvature vanishes. For Z = Z^2 + C, the result is somewhere between 2.0 and 3.0 because the local curvature does not vanish -- that's the definition of a fractal surface.

I do just this in http://vixra.org/abs/1812.0423

The code is at https://github.com/sjhalayka/meshdim

As you can see, the main function is super short, trivial.

Note that the code only runs properly when every triangle is guaranteed to have 3 neighbouring triangles -- the mesh must be closed, with no holes or cracks.

17 hours ago, taby said:

Thanks. I'm really happy with the result that I found, but I do not have credentials, and so I could not get my paper published.

What you write is interesting, and I hope it all works out for you if you decide to get back into fractals.

Yeah, the most fruitful research in fractals seems to be done by amateurs. Can't get no respect. LOL

You do not need credentials to get your paper published. Most high quality academic journals and conferences use a double blind review process, so if you "credentials" are the issue it is probably not a good venue in the first place  If you try to publish at a conference, you have to be able to afford the registration fee and the trip to the conference, though.

This is a bit different than what I experienced -- the journals required that I provide the names of possible reviewers. So, they were not blind at all. Do you have a journal to suggest?

15 hours ago, taby said:

As you can see, the main function is super short, trivial.

I think it's almost too trivial, because it ignores triangle area and so the final result depends a lot on the tessellation. (Marching Cubes generates very bad / nonuniform tessellation with many sliver triangles.)

I'm also curious how the results would be for other (non fractal) meshes. Maybe the resulting range 2-3 is just a property of manifolds in general?

Not sure if i remember this correctly, but interesting in case you don't know: One can calculate the euler characteristic from curvature like this:

eulerCharacteristicA = numVertices - numEdges + numTriangles

sum = 0

for each vertex sum += 2PI - sum of all triangle angles adjavent to vertex

eulerCharacteristicB = -sum / 2PI

So if you would calc curvature this way using angles, you get the genus of the object by (2 - eulerCharacteristic) / 2.

Which means you always get the exact number 2 for a fractal (or any other manifold mesh) with no holes, and this is independent of triangle areas or meshing resolution.

There are many ways to calculate curvature, maybe the one used here is called 'Gaussen Curvature', which seems most frequently used.

3 hours ago, JoeJ said:

I think it's almost too trivial, because it ignores triangle area and so the final result depends a lot on the tessellation. (Marching Cubes generates very bad / nonuniform tessellation with many sliver triangles.)

I'm also curious how the results would be for other (non fractal) meshes. Maybe the resulting range 2-3 is just a property of manifolds in general?

Thank you for the idea -- I'll find the triangle with the largest area and then use that area to normalize the area from the remaining triangles.

Yes, generally there will be some curvature, but if you continue to have a larger and larger sampling resolution, for parametric shapes say, then the curvature disappears and the dimension drops to 2.

2 hours ago, JoeJ said:

Not sure if i remember this correctly, but interesting in case you don't know: One can calculate the euler characteristic from curvature like this...

I'll look at this further. Thank you.

P.S. the one un-named journal that sent my papers for review also required that the reviewer had credentials. So, none of the amateur fractal researchers counted. The one reviewer who did have credentials must have thought it was a blind review because he was not just critical, but a complete maniac on a rampage. LOL