Clusters of minibrots

[mrob]

A while ago someone posed this question (in another topic):

I'm curious about the distribution of "minibrots" within the Mandelbrot set.  I don't recall ever seeing a dense clustering of them.

I wondered something similar the first time I saw the 4 minibrots of roughly equal size near -0.56 + 0.64 i (see them here: http://mrob.com/pub/muency/r225a.html)

So I decided to write a program to find out.

After some exploring it is clear that the clusterings found around the "main cardioid" are just as dense as those found anywhere else, and those can be enumerated by a rational fraction. For example, the one I just used as an example corresponds to the fraction 2/5.

My program locates each radical in order of increasing denominator, then surveys all the islands in the filaments attached to that radical. The "denseness" of the cluster is defined as total area of all the islands divided by the area of the largest island within the cluster.

As it searched, any new cluster that has a higher ratio than all of those searched so far is considered a "new record setter". The record setters are (using my naming):

  R2F(2/5(*B)) ratio=0.340  coordinates -0.550 +0.625 i @ 0.27
  R2F(2/7(*B)) ratio=0.284  coordinates 0.136 +0.643 i @ 0.12
  R2F(5/12(*B)) ratio=0.280  coordinates -0.5752 +0.4837 i @ 0.0554
  R2F(3/19(*B)) ratio=0.274  coordinates 0.3790 +0.1892 i @ 0.0117
  R2F(3/22(*B)) ratio=0.261  coordinates 0.36632 +0.12944 i @ 0.00777
  R2F(3/25(*B)) ratio=0.246  coordinates 0.35097 +0.09173 i @ 0.00543
  R2F(3/28(*B)) ratio=0.238  coordinates 0.33677 +0.06707 i @ 0.00395
  R2F(3/31(*B)) ratio=0.229  coordinates 0.32460 +0.05038 i @ 0.00296

The coordinates are expressed in terms of center and size, and show the "mu-atom" along with all attached filaments including the islands that comprise the "cluster".

An image of that last record-setter can be seen here: http://mrob.com/images/0-muency/r2.3-31.jpg You can see the four minibrots if you look close (they are all white in my pictures). There are three in roughly a straight line and a couple more slightly smaller ones to the right. But the overall grouping isn't really much better than the 2/5 group, seen here: http://mrob.com/images/0-muency/r2.2-5.jpg

So the answer seems to be that 3 equally-sized minibrots, with a couple slightly smaller ones, is about the limit.

- Robert Munafo