Fractal clusters

[TGlad]

Hi everyone,

I have been looking at making fractal clusters in 2D since they seem to be more of a neglected fractal type. I also wanted to move beyond square geometry where these things are easy to make, so using sphere inversions. For the boundary to be everywhere fractal it needs ever smaller disks near to all disk boundaries. Here are several:




But the nicest to me has 8 disks around the central one:


A half rotation each iteration (just like on the sphere tree) stops it looking so spikey:


blog: https://tglad.blogspot.com/2023/05/disk-cluster-2.html

[TGlad]

If you grow the subdisks, they can touch each other, and you get a cluster-sponge:  (classification: https://sites.google.com/site/simplextable/)


and grow it more to get a thicker cluster-sponge:


For the half-rotated one, here it is grown but still a cluster:


and grown more is a cluster-sponge:


and in colour for fun:

[claude]

You can find these in some places in some escape time hybrids.  I'll try to find some...

EDIT not the best example, here's one with large scale factor and other decorations,,, still looking for something better

[TGlad]

Nice cluster Claude. Probably more specifically a cluster of cluster-trees! What sort of escape time? squaring complex numbers or folds?

Here's a close up of the cluster-foam with the half-rotation:


But I'm getting distracted, my current interest is in the cluster type.

[mclarekin]

looking good!

[claude]

What sort of escape time? squaring complex numbers or folds?

hybrid of cubic mandelbrot and cubic burning ship (m-bs-m-m in a loop)

[kh40tika]

Hmm interesting. Recently I've been working on something similar in 3D. Here's an example.

[TGlad]

Very nice kh40tika. It looks like a sort of cluster-tree, similar to: https://tglad.blogspot.com/2018/10/nested-spheres.html but different.

The above cluster can generalise to 3D, here I've just got results, for increasing k from 0.5 up to 1:






It's late so I didn't get very good lighting settings, but here's a zoom for around k=0.8:


For all k<1 it is a cluster. I still need to adjust the distance function, it isn't too robust.

[FractalAlex]

Fascinating.Very good representation of the clusters, not gonna lie.

[TGlad]

There is a second cluster possible using a docedahedral hyperbolic tesselation. Here for k=0.7 up to k=1:

[TGlad]

and some close ups:

k=0.7:


k=0.8:


and k=0.1:

[TGlad]

The equivalent of the half rotation in the 2D case is to toggle each cluster child between the icosahedral and dodecahedral shapes. This makes it less pointy at large k, so you can get higher fractal dimensions (of the cluster surface) before it self-contacts and becomes a cluster-sponge.

Here's k from 0.7 to 1:

[TGlad]

The Fragmentarium file is attached.

[mclarekin]

thanks tglad, this will distract me away from what i should be doing

hopefully i will be able to add more parameters

image attached is one of the example files of your sphere tree  in Mandelbulber

[mclarekin]

super fast to render