Disk Cluster

 Here is a nice example of a cluster.

A cluster is a recursive set of separated solids. It isn't too hard to make an example of one using squares, but harder with disks. In this case I wanted an example with no smooth (differentiable) surface, just as the Koch curve and other fractals have no smooth parts to them. 

It is a little hard to see in the image above, but you couldn't 'land' on any of these disks as there are increasingly small disks towards the surface. 

My first attempt was quite interesting because it contains a Koch snowflake within it (can you see it?):

But the relative structure between clusters changes as you go inwards towards the centre, it also means that the outer shape almost definitely self-connects so isn't a cluster at all. 

The top image however applies Mobius transformations to each cluster so that the gap between clusters remains similar all the way down. Everywhere is disconnected, and it also gives it nice point corners.

A less spikey cluster can be achieved by rotating the shape each iteration: