Snowflakes

[Tglad]

I found a way to make my previous binary rules fractals (https://sites.google.com/site/tomloweprojects/scale-symmetry/automataFinder) have hexagonal symmetry:





You can apply the system to a random start:




I also tried using an even simpler formula, which also has some good results:




and applying this with a random seed, give a nice web-like effect:


The trick to getting hexagonal (rather than square) symmetry is to actually make a 3d cubic fractal and the 2d image is a slice perpendicular to the long diagonal of the cube.

[matsoljare]

Fascinating, now can you animate them?

[Tglad]

Maybe.. but no time to try unfortunately, it wouldn't be straightforward.

[KRAFTWERK]

Lovely results! That third image with the "simpler" formula has parts which looks like a power 8 mandelbulb  (with minibulbs)

Is it possible to go 3D?

[Tglad]

Yes, you construct a 4D grid, then run rule sets on the coarse set of 4d voxels to generate progressively half size voxels. Then a cross section of the long diagonal (x + y + z + w = 0) gives a 3d fractal.

I'm pretty sure it would give octahedral symmetry... which isn't a lot more than cubic that I have already (8 vs 6 sides), but it is different and might look cool.

There is a very interesting 'grid' in 4d called D4 which may give more symmetry in its 3d cross section... but who knows.