Other Polynomial and Rational Maps

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Offline pauldelbrot

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« on: April 07, 2018, 02:11:01 AM »
Rational mappings of a single complex variable, other than z2 + c.

Christmas Balls

Parameter space of z -> \( \frac{z}{c(z^2 + 1)} \). Different colors indicate different attractor counts and behaviors: grey = 0 attracts and Julia set is a dust; black = no stable attractors and Julia set is whole Riemann sphere; other colors = 1 cyclic attractor of period 2 or above or 2 attractors that are identical, rotated copies of each other.

Matchmaker is a generalization of this formula which, for most values of its parameters, breaks the symmetry this formula forces.

Offline Adam Majewski

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Offline pauldelbrot

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« Reply #2 on: April 14, 2018, 08:15:58 AM »
Wooden Ring

Formula: Supernova (\( az^2\frac{z - c}{1 - cz} + b \))

Minibrot surrounded by rings and spirals. The dense, noisy Julia shapes from the collapsed basin in the rings necessitated downsampling from 32000x24000 for 25x25(!) antialiasing on this image -- 625 samples per pixel of output.

Offline pauldelbrot

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« Reply #3 on: April 24, 2018, 11:37:14 PM »
Twisted Up

Offline pauldelbrot

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« Reply #4 on: May 05, 2018, 01:33:52 AM »
Twofold Seahorse Valley

Matchmaker parameter plane, both critical points studied. There are two overlapping seahorse valleys here, and each influences the other. Green = no stable attractors, gold = two stable attractors, everything else = one stable attractor that captures both critical points.

Offline pauldelbrot

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« Reply #5 on: May 16, 2018, 07:24:44 AM »
Purple Slime Vortex

Offline pauldelbrot

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« Reply #6 on: June 07, 2018, 02:03:19 AM »
Herman Ring

An actual Herman ring. White = the Julia set, black = either A(0), A(∞), or A(the Herman ring). The former two basins consist of the exterior black area and all simply-connected black areas; the black region that is a union of infinitely many annuli is the third basin, and the lower of the two largest, central annuli is the actual Herman Ring itself. The annuli are actually just tangent to one another, including at the critical point located at the image's center of symmetry. The spirals of white that approach that point have smaller branches and smaller that meet there, though these are difficult to discern. Like with Siegel disks, Herman ring Julia sets have the relevant critical point actually contained in the Julia set itself.

The formula used is Supernova (in fact its subset Antimatter, in which d = 0, the original formula found in a dusty old university library book a few decades ago and shared to the Fractint mailing list, this site's spiritual predecessor of those times) and the rendering method is distance estimation.

Offline gerrit

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« Reply #7 on: June 07, 2018, 04:49:46 AM »
Herman Ring

An actual Herman ring. White = the Julia set, black = either A(0), A(∞), or A(the Herman ring). The former two basins consist of the exterior black area and all simply-connected black areas; the black region that is a union of infinitely many annuli is the third basin, and the lower of the two largest, central annuli is the actual Herman Ring itself. The annuli are actually just tangent to one another, including at the critical point located at the image's center of symmetry. The spirals of white that approach that point have smaller branches and smaller that meet there, though these are difficult to discern. Like with Siegel disks, Herman ring Julia sets have the relevant critical point actually contained in the Julia set itself.

The formula used is Supernova (in fact its subset Antimatter, in which d = 0, the original formula found in a dusty old university library book a few decades ago and shared to the Fractint mailing list, this site's spiritual predecessor of those times) and the rendering method is distance estimation.
Nice. I don't have the prerequisites to understand what \( A(\cdot) \), Antimatter etc. is. If you could elaborate I'd be interested. Do you remember which dusty old library book it was?

Offline pauldelbrot

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« Reply #8 on: June 07, 2018, 05:05:12 AM »
Nice. I don't have the prerequisites to understand what \( A(\cdot) \), Antimatter etc. is. If you could elaborate I'd be interested. Do you remember which dusty old library book it was?

A(x) means the attracting basin of x. Antimatter is the fractal formula given for "supernova" higher up in this thread, without the "+ b" (so, with b fixed at zero). The library book was some old highly technical mathematical research volume about iterated rational functions of the complex numbers. I went into the stacks specifically to find any formula that might produce Herman rings, after hearing in multiple places that it had been proven (the Sullivan classification) that such functions divide the complex plane into only a few types of regions: attracting basins for attractive fixed points and cycles (corresponds to parameters inside Mandelbrot bulbs and cardioids); attracting basins for parabolic fixed points and cycles (corresponds to parameters at cardioid cusps and where bulbs are tangent to cardioids or other bulbs); attracting basins for Siegel disks (points on the boundaries of bulbs and cardioids where no bulb attaches and that are not cusps) or cycles of same; attracting basins for Herman rings or cycles of same; and the Julia set itself, which is the boundary of any one of the above (thus, any point belonging to the Julia set has points belonging to all of the attracting basins of that function arbitrarily close by). If a function produces no attractors at all the Julia set is the full Riemann sphere. Furthermore, every attracting basin must capture at least one critical point or value.

The ordinary Mandelbrot fractal maps the possibilities for quadratic polynomials. Infinity always attracts, and counts as a critical point. The other critical point is zero (derivative of z2 + c is 2z, only zero at zero) and may thus either also go to infinity, or else may go to one finite attractor. The Mandelbrot Set itself contains the points where the latter happens, and is the closure of that set. But it cannot produce Herman rings, only the other possibilities named above. I got curious to see one of these things and dug around in the university library and card catalogue (the internet was barely a thing yet back then!) and eventually found the formula I named "Antimatter" presented as an example that could generate Herman rings, if a was an irrational unit (i.e. cos x + i sin x for x/2pi irrational) and c was either pure real or pure imaginary and of magnitude exceeding about 3 or so.

I took the formula and coded it as a Fractint formula, and later as UF formulae (and derived the Supernova variation the same way Nova derives from the Newton's method Julia set). It turned out that formulas capable of producing Herman rings produce some unique and beautiful patterns, both in parameter and dynamic space images, such as seahorses with lakes in the center and smoke-swirl turbulence patterns that alternate spirals on each side along an arc or around a ring. Spacefilling Julia sets from parameters close to ones that produce Herman rings can be especially spectacular ...

Offline gerrit

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« Reply #9 on: June 07, 2018, 05:16:20 AM »
Thanks!

Offline pauldelbrot

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« Reply #10 on: June 18, 2018, 05:35:52 AM »
Ruby Centerpiece

Matchmaker.  Green and yellow goes to one attractor; the little red jewel goes to the other. The corresponding region of parameter space has a tiny red mini in a seahorse tail in a seahorse valley of a larger red mini, in turn inside of a green mini belonging to the other critical point.

Offline pauldelbrot

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« Reply #11 on: June 30, 2018, 12:59:54 PM »
Trefoil

Combination iters and DE of a moderately deep embedded Julia set of z3 + c.

Offline pauldelbrot

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« Reply #12 on: July 12, 2018, 06:03:09 AM »
Crumblestone

A minibrot of the general cubic family \( z^3 - 3a^2z + c \). Critical points are at +/- a. The parameter space is four-dimensional but the Julia sets are two-dimensional.