New crumpled surface fractal

[Tglad]

I added four different coloured local lights... it looks a bit odd but I think it definitely gives a better impression of the 3D shape. Here's bendangle 0 up to 22.5 degrees:

(does anyone remember how to change the size in the 'img' wiki tag? these are too big)

Even though it is a complicated shape, the folds lines on the flat piece of paper are essentially just millions of repetitions of one curve, which looks like this for bendangle 7.5 degrees:


The surface's fractal dimension is
where φ is the bend angle.

[DarkBeam]

and when phi approaches pi/2? Fractal with infinite dimensions? Ouchies

[hobold]

I added four different coloured local lights... it looks a bit odd but I think it definitely gives a better impression of the 3D shape.

I did not explain well enough what I meant. My suggestion was not so much about local lights, but about incoming light having different colour depending on direction. In other words, I had somewhat distant light sources (surrounding the scenery) in mind, that would influence the whole surface more or less evenly. But every single little triangular facet would - depending on its surface normal (i.e. its orientation) - get a strongly varying mix of those surrounding colors.

Say, a facet facing more north would "see" a northern light source and get more of that light's color, but less of the southern light. Likewise for east and west, or however you would want to distribute the lights. I hope that is a bit clearer.

(In a perfect world, I would have provided images as visual examples.)

[Tglad]

Yes that might work better, it might just be a case of trying out several such ideas. Another way to get depth cues is to add an image to the surface... I tried adding texture coordinates to the .ply but couldn't get it to load. The file format doesn't seem to be clear for this.

DarkBeam: That is normal, the Koch and Levy curves both approach infinite fractal dimension as the bend angle increases, this is possible on a 2D plane because they overlap, so the set is 'immersed in' rather than 'embedded' in 2D. For the Koch curve it only overlaps for bend angles greater than its 'maximum' (where it gives a solid triangle), for the Levy curve it overlaps for all bend angles > 0. This crumpled surface fractal is a bit like an intermediate of those two, with a smallish overlap up until its 'maximum' of 45 degrees. The Volcanic surface fractal http://tglad.blogspot.com.au/2012/11/volcanic-surface-fractal.html on the other hand has no overlap until its dimension goes beyond about 2.3.

[DarkBeam]

Very interesting reply thanks

[Tglad]

Here's the executable for Windows 64 bit: https://drive.google.com/file/d/0B8S7Si-yu3Doa19yNy1MaTRPdkU/view?usp=sharing

It is super easy to use, just run it from the command line by typing Folding -f myFileName -b 15 -y 90 -c 128
for a bend angle 15 degrees, yaw angle 90 degrees and to save out as myFileName.ply. Larger values of -c give more detailed meshes
Or Folding -h for help.

The source code and some example meshes are also at the bottom of my blog post: http://tglad.blogspot.com.au/2016/09/foil-surface-fractals.html