Collected Essays (41 page)

Detail of Mandel Cubic Zipper

And here’s another, this one found in what we call an Mc set rather than an Mk set.

 Detail of Mandel Cubic Ogre

By slightly varying the two components of the k parameter, one can look at k-sections near each other, and try to visualize stacking them one atop the other. I would very much like to view 3D sets which are stacks of Mk sets that arise as one varies, for instance, the real part of k from -1 to 1. I have a lingering hope that these objects may look bulbous rather than taffy-like, despite the lack of success of some preliminary investigations. What we want to see is a three-dimensional Mandelbrot shape with buds all over it—this may be related to the rather different three-dimensional beast called the Mandelbulb.

The Mandelbulb, which has been under intense investigation in recent years is a quite different kind of thing from the cubic Mandelbrot sets. The Mandelbulb is defined so as to be an inherently three-dimensional Mandelbrot set. The trick is to use spherical coordinates for three-dimensional space, and to define “multiplication” in terms of adding angles. I was in fact one of the first people to work with the Mandelbulb—back in 1988. I have some background information and some links about the Mandelbulb in a
blog post
.

But, again, the Mandelbrot has no essential connection with the cubic, quartic and other Mandelbrot and Rudy sets that I’m showing pictures of in the current essay.

 The Cubic Rudy Set is the True Cubic Mandelbrot Set

 An apparently new fractal which I’ve enjoyed investigating is this.

 R = {c : Jcc is connected}

 = {c : c is in Mc}

 = {c : ( c-fcc—> FINITE ) AND ( -c—fcc—> FINITE) }.

I immodestly call this the Rudy set, although it may be that pros like Branner, Douady, or Hubbard have their own name for it. As I say, I first starting working with this set some twenty years ago, but computers were pretty slow back then. In the April of 2010, using the commercial
Ultra Fractal
program, I saw much more detail of the Rudy set than ever before. Images that used to take hours to render can pop up in seconds.

 Note that the Cubic Rudy Set has an absolute or non-relative quality, in that it avoids the choice between the Mk and Mc Mandelbrot Cubics, each of which are a certain kind of orientation-dependent cross-sections of the Cubic Connectedness Map. By going down to the Jkk in the definition of the Rudy Set, we reach down to something that’s not relative to any specific orientation. Note also that we could equivalently define the Rudy Set as {c : c is in Mc}. For this is just {c : Jcc is connected}, which is the same as {k : Jkk is connected}.

The Rudy Set

 Compare the definition of R as {c: Jcc is connected}to the definition of the Mandelbrot set M as { c : Jc is connected}. This makes me think that R is a good generalization of M, in some ways better than the Mk or Mc.

R is an object which is extremely rich in unusual fractal structures. One good region is the plume between 2 o’clock and 3 o’clock relative to the whole set. I call this area “Mars”.

Rudy Mars

 An image like a rocking horse is found in the Mars region of the Rudy set. This horse is one of my favorite spots.

 Rudy Horse

 Another good region is the spike at the top, at 12 o’clock. There is an interesting structure there that is a bit like a Mandelbrot set, but considerably gnarlier. I call it Fat Bud. This is a wonderful region for extreme gnarl.

 Rudy Fat Bud

I keep finding more and more great stuff in the Rudy set.

The Rudy Hedgehog

Lots of little Mandelbrot sets turn up inside the Rudy set.

Rudy Sanskrit Bud

I put the Sanskrit Bud onto a T-shirt. Very yogic.

I recently found a really powerful region in the first Mandelbrot bud above the top of the Rudy set. There’s a yottawatt particle beam blasting out.

Rudy Particle Beam

And near the Particle Beam are some globs of paired twirly things like bugs you’d find under a log.

Rudy Isopod.

And down inside the very center of that gap at the core of the Rudy Isopod is a mini-Mandelbrot set, a variation on the Sanskrit Bud.