Shameless plug--I'm the Kerry Mitchell mentioned on that page.

Michael Frame (who created those class notes) told me about the scaling rule some 20 years ago. While investigating it numerically (in FORTRAN), I discovered those deviation features.

That's a pretty interesting page. I believe the fractal structure of those plots is somewhat similar tick marks on a ruler. I'm thinking this: if m/n has one value of rtc/rad, then (1/3)+m/6n and (1/3)+m/3n have a similar structure. Also, (1/4)+m/8n and (1/4)+m/4n would also produce similar values. This might explain the compression that occurs off to the side. Indeed, the structure reminds me of the dyadic rationals...

Did you ever try plotting the values, as sorted by n? Or by m? Or maybe a plot (multivalued) where the x-coordinate corresponds to m+n? I would love to see!!!

i'm wondering if it worth it to pre-compute the 3rd and 4th order disc.

i know they are not real disc, and it certainly doesn't worth it to compute a very good estimation, but i can estimate the size of a rough interal disc contained in the 3rd and 4th order disc/shape. It's very cheap in cpu time. (much cheaper than computing the main cardioid, but considering its size : it worth it)

But that would only save any time if you happen to be zooming in on one of those shapes. And in general, a good Mandelbrot pic doesn't have a huge period-3 or 4 mass off to one side...

Has anyone investigated what shapes the bulbs are? When looking at skewed Minibrots, it seems they don't possess any other deformation. I would hypothesize that the bulbs are all exact ellipses (skewed circles), and that there is some correlation between m/n and their skew angle. The angle could be defined as relative to the normal. However, if we defined the angle that way, in order to fully describe the shape one would also need a stretch factor. The alternative would be the skew angle, and the angle relative to the normal, relative to which it was skewed. If someone wrote a program that gathered data on this, and then maybe tried plotting using both of those, I'm sure it would result in some interesting results.