Disk Cluster 2

 In a previous post I was making cluster fractals because these are quite rarely found in fractal examples. Moreover, I was making clusters that are nowhere differentiable (from the outside), which is the purest sort of fractal boundary because everywhere has the same fractal property, with no smooth regions. 

The only problem with these clusters is that they were a little too heavily constructed, and so don't transfer to 3D in any obvious or simple way. 

This post however constructs the cluster based on a very small number of symmetries. It uses two different sphere inversions, octagonal symmetry and a single scale factor k. As such, it makes a proper Kleinian group limit set. To describe it I borrow from figure 6 of this Kajino paper:

We define the outer green circle as radius 1, and the offset green circle l2 is at a distance L=sqrt(1+R^2) where R is its radius: R=2/sqrt(-4 + cosec^2(pi/n)), and n is the polygon number of sides, here it is n=8. The inner red circle has radius r=L-R.

To test if a point is inside or outside the cluster, first you invert the point around the purple circle then:

1. rotate the point to the first octant (in line with the circle l2)
2. if it is inside l2 you invert it around l2
3. if its magnitude is less than rk then divide by rk and then invert around l4 = sqrt(r), sending the red circle to the outer green unit circle

repeat until its magnitude is > 1 (black) or you run out of repetitions (white).

For k=0.9 we get:

For k=1 we get:

and for k=1.1 we get:

Actually this last picture uses slightly less than 1.1 as we need to tweak the value for circles to match properly. I don't have the exact analytic value for k here, but it is very close to k=1.0985.

In terms of fractal classes these are a cluster, a cluster-sponge and a cluster-foam respectively. The first differs from the previous cluster in that the subclusters closer to the centre don't form a uniform pattern, and because it is a proper Kleinian group it generalises cleanly to the two other structures pictured above.

The value of n doesn't have to be 8 but it needs to be more than 6, so here are clusters for n=7 and n=9, both for k=0.8:

Here are the cluster-sponge versions (k=1):

One thing to note about these cluster-sponges it that they are very spikey. That is because the sub-cluster orientation lines up with its parent, causing a cascade of disks all in a row. 

We can avoid that by rotating each point by pi/n in step 3, here for n=8 for comparison with the first three images:

This latter image is again slightly away from k=1.1, at approximately k=1.096. Here's a high resolution close up:

Notice that these are all less spikey, and the k=1 case is no longer a cluster-sponge, so can have larger sub-clusters before touching. In the n=7 case the touch point happens around k=1.064:

For fun here's the prior n=8 image coloured by iteration depth:

The next step is to bring this formulation across to three dimensions. I cover this in the next post, but as a quick preview:

There are basically two ways to do this, we can use a hyperbolic tiling of an icosahedron or a dodecahedron. Here I'm using an icosahedron, which give 20 subclusters (one for each face).

The above image is a sphere at each icosahedron in a hyperbolic tiling, only the tiling has been sphere inverted, sending all radii r to 1/r. We use a value of k of roughly 0.5 above.

We then need to recurse inside each sphere, like so:

For larger k, around 0.8:

and k=1

Here's close up around k=0.8: