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Author Topic:  Matchmaker^3  (Read 202 times)

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Offline pauldelbrot

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« on: January 11, 2021, 10:38:19 PM »
Ladies and gentlemen, I'd like to unveil for you ...

Matchmaker Cubed!

Many of you are no doubt familiar with the matchmaker, matchmaker 2, and triple matchmaker rational mappings, and the varied and interesting Julia sets (including spacefilling ones) that can be produced.

Well, I've found another one of interest of the same general sort:

\[ \frac{((z-1)^3 + a)((z+1)^3 + b))}{((z-1)^3 + c)((z+1)^3 + d))} \]

Two of the critical points are -1 and 1. I've only had time to take a quick look at some Julia sets and parameter space slices, but it appears to have all of the following:

  • Quadratic minibrots
  • Cubic minibrots
  • Quartic minibrots
  • Herman rings and their associated phenomena
  • Spacefilling parameter regions and Julia sets

Furthermore, it's relatively easy to get enclosed "lakes" on both sides of the rings, unlike with triple matchmaker. Triple matchmaker was also apparently incapable to
produce cubic minibrots, or at least could not do so easily (perhaps on a parameter set of measure zero?) ...

Actual renders, including some spacefillers, will be done eventually, but I have so many miscellaneous things on the go right now that I'm going to have to back-burner it for the time being. However, this family of rational mappings looks so promising as a potential source of beautiful fractals that I felt the need to immediately share the math, so perhaps other people could start kicking it around and exploring it.

Linkback: https://fractalforums.org/index.php?topic=3979.0

Offline kosalos

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  • Posts: 142
Re: Matchmaker^3
« Reply #1 on: January 12, 2021, 10:24:00 AM »
I don't know what I'm doing.
Here's my attempt to understand your idea.

sample image params
cpu -------------------------------
power = 3.1233356
aa = float2( -1.7369139 -0.018962642)
bb = float2( 1.1940293 -1.9503945)
cc = float2( 2.0 -2.0)
dd = float2( -0.67151946 -0.23513868)

shader ----------------------------
( c is the original value from Mandelbrot z^2 + c)
float2 n1 = complexPower((z-1),control.power) + (c - control.aa);
float2 n2 = complexPower((z+1),control.power) + (c - control.bb);
float2 d1 = complexPower((z-1),control.power) + (c - control.cc);
float2 d2 = complexPower((z+1),control.power) + (c - control.dd);
float2 nn = complexMul(n1,n2);
float2 dd = complexMul(d1,d2);
z = complexDiv(nn,dd);

look forward to seeing a correct implementation.