Dragon's Tear

[Pauldelbrot]

This is an a-plane slice of the parameter space of

(az² + bz)/(z² + 2z + 1)

which I call Matchmaker 2. Students of the calculus may have noticed that the thing's got a fixed point at zero and that its derivative at zero is b. (The numerator of the derivative is the numerator's derivative -- at zero, b -- divided by the denominator -- at zero, 1 -- minus the denominator's derivative times the numerator. The numerator, at zero, is zero, so the second part of the derivative's numerator sum is zero. The denominator of the derivative is the squared denominator -- at zero, 1 again. Thus the derivative is simply b)

So, we have a fixed point whose stability is exactly controllable; in fact, the fixed point's "twist" is also exactly controllable.

This means we can match any fixed point basin with, potentially, any other self-squared dragon (by varying a after fixing b).

In practice, it goes further than that. In the b-plane, the M-set necessarily has a period-1 disk-shaped component centered on the origin, for all values of a. Furthermore, the roots of bud attachments on the periphery of this disk must be the same independently of a: wherever b is a unit with a rational angle.

So, for instance, where b is a third root of unity other than 1, the period-3 buds must attach. As a varies, those period-3 buds may wobble a bit or shrink or grow but there tends to be a large amount of overlap.

So, if b's magnitude is slightly greater than 1, and has close to a rational angle, there is a roughly-constant attractor of some period higher than 1 for most values of a. This means any given quadratic Julia basin from inside any cardioid-attached bud of the M-set can be chosen and fixed, then a varied to make the other basin into anything one desires.

Unfortunately, this rarely works for buds that are more steps removed from the period-1 disk. But it does allow a certain subset of "designer" paired-quadratic-basin Julia sets.

With Matchmaker, if you wanted, say, a seahorse-dragon basin (e.g. from a period-29 bud close to the period-2 bud) and a "rabbit" Julia (period 3), you had to search the 4D parameter space, highlighting the two desired periods and mutating the parameters until you got one bud of each period to collide in the flexing and rearranging Mandelbrot view. With Matchmaker 2 you can just set b to something that lands it in a period-29 bud for vast swathes of values of a, then get an a-plane Mandelbrot, find a period 3 bud in that, and grab the Julia set.

This image is, in fact, the a-plane Mandelbrot for a b-value that lies in a period-29 bud as described. It looks similar to an inverted standard Mandelbrot set, except that the boring old superattracting basin of infinity has been replaced by an attracting 29-cycle centered on the repelling fixed point 0. As a result all the Mandelbrot dendrites, etc. infiltrate into the crevices of a particular quadratic Julia basin! This is the second charm of this map: the same fairly large selection of quadratic basins can be married to the Mandelbrot set. The third is that the a-plane Mandelbrots for many choices of b lack any points for which the Julia set is the whole Riemann sphere; such points make for slow Mandelbrot images in the other Matchmaker relatives.

Freely redistributable and usable subject to the Creative Commons Attribution license, version 3.0.

Detailed statistics:
Name: Dragon's Tear
Date: February 28, 2009
Fractal: Matchmaker 2 Mandelbrot slice
Location: a-plane; b = -0.9951321 + 0.10822714i
Depth: Very Shallow
Min Iterations: 230
Max Iterations: 951792
Layers: 2
Anti-aliasing: 3x3, threshold 0.1, depth 1
Preparation time: 10 minutes
Calculation time: 30 minutes (2GHz dual-core Athlon XP)