Parabolic Hippocampus

[Pauldelbrot]

Distance estimator rendering of an Antimatter Julia set from very close to a point where two buds touch one of those line-like components.

As usual, Julia sets close to parabolic points are difficult to calculate well and efficiently, and this one was no exception, taking five whole days of CPU time to calculate. And even then, it's not perfect.

The image uses the distance estimator and the following gradient:

• black further than one unit from all basin boundary points;
• Shading logarithmically to white 1/10,000 of a unit from any basin boundary point; and
• white within 1/10,000 of a unit of any basin boundary point.

A "unit" is not exactly a pixel, nor a constant distance, because the Antimatter distance estimator is somewhat less accurate than the quadratic one for some reason. The dynamic plane distance estimation tends to underestimate the distance significantly for points near zero.

Still, to a fair approximation, the image shows the Julia set itself in white on black.

The outside diverges; the two large S-shaped lakes are points convergent to zero. Many smaller such lakes exist; and there are lakes near the lakes that converge to zero, in which points diverge. In the bulk of the interior, though, points go to a near-parabolic 21-cycle.

Postprocessed with some edge enhancement.

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

Detailed statistics:

Name: Parabolic Hippocampus
Date: February 1, 2009
Fractal: Antimatter (Herman Ring) Julia set
Location: angle parameter = 0.51803398, c = -3.194097128 + 3.174739687i
Depth: Very Shallow
Min Iterations: 2
Max Iterations: 1,000,000
Layers: 1
Anti-aliasing: 3x3, threshold 0.10, depth 1
Preparation time: 5 minutes
Calculation time: 5 days (2GHz dual-core Athlon XP)

[Adam Majewski]

Hi. Thx for great image. Could you explain :
* what means : Antimatter Julia set ,
* what function is used to creat this image,
* what means angle parameter

Best regards

Adam Majewski

[Pauldelbrot]

Antimatter is ; a is the angle parameter. Critical points are at 2cz³ - (3 + c²)z² + 2cz = 0, so 0 and . Of these, 0 always superattracts since it is also a fixed point of the map. The inner curlicues in the image above are fragments of the basin of 0. The numerator degree is two higher than the denominator degree, so infinity also superattracts; the outer part of the above image is part of its basin. It also has disconnected basin lakelets in the fractal's interior.

[cKleinhuis]

@paudelbrot:   i love your detailed and precise answers very much ! and that you make extensive use of the LaTex feature! keep on!  

to the image:

great variance in the shapes, i like the sharp edges !  

@adam the name "antimatter" here just stands for the type of fractal formula used here, many fractal formulas use sounding names like "Manowar, Burning Ship, ... countless others"

so, the ( iterative, complex )  formula paudelbrot used was :

[Pauldelbrot]

great variance in the shapes, i like the sharp edges !  

You're welcome.

Distance estimator can have that effect on you.

[cKleinhuis]

great variance in the shapes, i like the sharp edges !  

You're welcome.

Distance estimator can have that effect on you.

i really do love distance estimators, they are the only cool method the achieve somewhat like "smooth" borders to those /&()/()&&/)((/&=) ( bad words ) and nasty fractal borders ....

cheers

[Adam Majewski]

Hi, Thx for answer.
I have made image of parabolic Julia set :
http://sourceforge.net/apps/mediawiki/maxima/index.php?title=Image:JuliaRay_2_5.png
It is made with MIIM/J. It is not good ( black Julia set should connect with  green points ) . I have tried with DEM/J but the result was not better.
Could you give me some tips how to improve it ?
Regards. Adam

[Pauldelbrot]

Antialiasing and I think I may have used interior DEM as well as exterior.

[Adam Majewski]

Hi,
Thx for answer. I'm trying to make procedure for internal DEM. It will take some time. Can you make image of Julia set for fc(z)=z*z+0.21650635094611*%i-1.125
using your program ?

Adam

[Nahee_Enterprises]

    Distance estimator rendering of an Antimatter Julia set from very close
    to a point where two buds touch one of those line-like components.
         ....  Calculation time: 5 days (2GHz dual-core Athlon XP)

I missed this image before.  Glad I came back and started looking over some of the past few postings.

For some reason, I really enjoy black/white images.  And this one is quite exceptional.  Thanks for sharing it with us.