Anyone ever implement the period 3 check as described by Stephenson?

[saijanai]

Stephenson's original papers on higher-period checking are unavailable to me, but I've found various hints of how to implement it. However, my math skills aren't such that I can figure out the final step of actually implementing the test.

Resources:

http://mrob.com/pub/muency/brownmethod.html

http://www.ams.org/journals/proc/1995-123-12/S0002-9939-1995-1301497-3/S0002-9939-1995-1301497-3.pdf (A Parameterization of the Period 3 Hyperbolic Components of the Mandelbrot Set)

This second paper seems like it would be easier to implement but I haven't figured it out. The test might be very useful when exploring the cusp between the main cardioid and the period 3 buibs, ala this very deep image http://www.superliminal.com/fractals/mbrot/wish.jpg

at: -0.13856524454488  -0.64935990748190 size: .00000000045 iterations: 5,000,000,000

at 5 billion(!) iterations per pixel, the overhead of evoking sinh and arcsinh or whatever seems trivial in comparison.


I'm using Smalltalk as my development environment, which allows me to mix external C libraries, inline assembler and dynamic programming with a Morphic GUI, so I'm not bound to a statically compiled algorithm.

Anyway, if anyone has ever implemented the period 3 region check, either using the method described in the muency reference, or as described in the second reference, I'd be grateful for any help.

Thanks.

[Adam Majewski]

Please see :
http://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set#Period_detection
http://commons.wikimedia.org/wiki/File:Mandelbrot_set_Components.jpg
http://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set#Boundary_of_Mandelbrot_set

HTH

Adam

[saijanai]

Please see :
http://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set#Period_detection
http://commons.wikimedia.org/wiki/File:Mandelbrot_set_Components.jpg
http://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set#Boundary_of_Mandelbrot_set

HTH

Adam

Thanks. Yeah, I was aware of those. The cardioid test and the period-2 bulb test are the simple ones. The period-3 test is usually considered too complicated to implement (and I can see why), but I was curious to see if there was any savings possible by using it while zooming in right next to it, but I can't figure out how to turn the mathematical description (there appears to be two different ways to describe it) into an algorithm for programming.

The circle tests are approximate for everything but the period-2 bulb, so when you get *really* close to the edge, they won't be of any use.

[A Noniem]

Usually you don't zoom in directly on the cardoid itself. It won't help you anything in the sample picture, I don't see any cardoid in there. The parts of the mandelbrot which are (partly) black are not very interesting which means that all the period checks are nothing but extra overhead in deeper zooms.

[Adam Majewski]

Maybe perioid checking will be better ?
http://commons.wikimedia.org/wiki/File:Mandelbrot_DEM_Sobel.png
Let me know if you will implement period 3 checking .

Adam

[saijanai]

Usually you don't zoom in directly on the cardoid itself. It won't help you anything in the sample picture, I don't see any cardoid in there. The parts of the mandelbrot which are (partly) black are not very interesting which means that all the period checks are nothing but extra overhead in deeper zooms.

All, well, the period-3 check is on the primary bulbs at the top and bottom of the main cardioid (and on the largest (I think) of the cardioids off to the west on the x + 0i antenna.

There seem to be two equivalent formulas that can be used to do the check, but I have no idea how to translate them into a programming algorithm and I can't find anyone online who has actually done so.

http://mrob.com/pub/muency/brownmethod.html

and

http://www.ams.org/journals/proc/1995-123-12/S0002-9939-1995-1301497-3/S0002-9939-1995-1301497-3.pdf


That is:



                SQRT(3) %I   1
  (D10) [c = (- ---------- - -)
                    2        2
 
                     2                       2
    (D - SQRT(27 D  - 176 D + 1472)   27 D  - 288 D + 1600 1/3
   (---------------------------------- - --------------------)
               384 SQRT(3)                       3456
 
                         SQRT(3) %I   1
                        (---------- - -) (3 D +
                             2        2                                  2
   + ----------------------------------------------------------------- - -,
                          2                       2                      3
         (D - SQRT(27 D  - 176 D + 1472)   27 D  - 288 D + 1600 1/3
     72 (---------------------------------- - --------------------)
                    384 SQRT(3)                       3456
 
 


vs


-7/4 - 20/9 (sinh(w(z) + (2kπi)/)-1/(4sqrt(5)))^2

where w(z) = 1/3arcsinh((88-27z)/80sqrt(5))

and k = 1, 2 or 3 (for the top and bottom period 3 bulbs and the largest cardioid to the west -I think).

I'm not following the proofs very well, but the second paper appears to say that the sinh etc formula is the equivalent to the first (if k= 0).

I'm just not really good at translating the math into a computer algorithm.


L

[Adam Majewski]

http://en.wikipedia.org/wiki/Hyperbolic_function#Inverse_functions_as_logarithms

HTH