Series approximation for M set

[skychurch]

Hi all,

Sorry to bore you but I've a couple of questions regarding the Perturbation method. I've looked through the many threads on this subject but I'm still not clear on the detail.

I'm currently rendering using arbitrary length decimals and equation 1 from the perturbation paper with good results. But I'm struggling to get the Series Approximation working correctly. I'm assuming that the iterated coefficients A, B, C are derived complex numbers and not scalars. Is this correct? Do I initialise the starting condition of A0, B0, C0 to be {0,0},  {0,0},  {0,0}, if not, then what?

Any help greatly appreciated.

[Adam Majewski]

http://mathr.co.uk/blog/2016-03-06_simpler_series_approximation.html

HTH

[quaz0r]

the coefficients are indeed complex.  initialize A₀ to {1,0}, the rest to {0,0}.

[skychurch]

Cheers guys, I'll give it a shot. 

[skychurch]

Any point of going beyond e308? Anything new to find?

[quaz0r]

if you want to go further, then i guess there is a point.  if not, then not.  
eventually though you will get bored with that limitation..

[Chillheimer]

Any point of going beyond e308? Anything new to find?

Any point in zooming at all?
At these deeper zoom depth you will get much more repetition in between new shapes.
but there are new shapes, and you are the one who actively decides what they look like by deciding where to zoom towards to.
This technique is called "shapestacking" (though I hope we find a better name in the future)

If you're not familiar with it, here is a simple explanation:
<a href="https://www.youtube.com/v/Ojhgwq6t28Y&rel=1&fs=1&hd=1" target="_blank">https://www.youtube.com/v/Ojhgwq6t28Y&rel=1&fs=1&hd=1</a>

so if at e308 you decide to take a new focus point (that is NOT a already bifurcated point) the next NEW shape will occur at e454  if I remember correctly.
we had this in a thread here somewhere, I think simon posted a list of how the doubling pattern is tied to e...
can't find it right now.

the essence is, with that technique you are able to actively create just about any shape you want, like these:


 the only problem is: you would need so much more cpu power than we currently have to do really complex shapes. the needed cpu amount grows exponentially, the occurence of new shape doesn't. (don't nail me down on this statement)

[quaz0r]

technically though those animal sorts of shapes that karl makes wont be found by zooming deeper; he perturbs the regular orbits with some value to induce those irregular shapes.  i forget what name they came up with for that..

[skychurch]

Thanks guys, Very instructive. So I will have to go deeper at some point and having taken a look at your amazing work I see I've still got a lot of other stuff to do too!

[Chillheimer]

technically though those animal sorts of shapes that karl makes wont be found by zooming deeper; he perturbs the regular orbits with some value to induce those irregular shapes.  i forget what name they came up with for that..

are you talking about the "show inflection" thing that kalle mentions here? http://www.fractalforums.com/index.php?action=gallery;sa=view;id=17228
I read his description as "using a tool to find a spot that would otherwise be hard to find".
kind of like binoculars for shapes that are hidden at that spot when you zoom deeper.

Kalle would you mind to clear things up a bit? a
because if show inflection is working like fractal binoculars, this would be an awesome tool I'd use over and over!
how do I properly use this? I don't understand what is happening when "show inflection" is toggled.
i found this: http://www.fractalforums.com/kalles-fraktaler/kalles-fraktaler-2-5-9-show-inflection/
but i want to know more how to actually use it.


@ skychurch, I hope you don't mind the threadnapping..

[skychurch]

No probs keep napping. I dig the learning curve. 

[Kalles Fraktaler]

are you talking about the "show inflection" thing that kalle mentions here? http://www.fractalforums.com/index.php?action=gallery;sa=view;id=17228
I read his description as "using a tool to find a spot that would otherwise be hard to find".
kind of like binoculars for shapes that are hidden at that spot when you zoom deeper.

Kalle would you mind to clear things up a bit? a
because if show inflection is working like fractal binoculars, this would be an awesome tool I'd use over and over!
how do I properly use this? I don't understand what is happening when "show inflection" is toggled.
i found this: http://www.fractalforums.com/kalles-fraktaler/kalles-fraktaler-2-5-9-show-inflection/
but i want to know more how to actually use it.


@ skychurch, I hope you don't mind the threadnapping..

Hi Chillheimer

The "show inflection" function (named by Stardust4ever) shows how the view will look like another half way to the minibrot, when the current pattern is doubled, because it wraps the pattern around the selected point in the same way.
Yes, I think you don't realize how awesome this function is regarding "shape-stacking"
And thanks to claude's Newton-Raphson method, this function is even more awesome since you don't need to manually center-zoom all they way down to where the pattern doubles, just make sure "3/4 (curren pattern doubled)" is checked in the dialog

[Chillheimer]

omg!
this sounds too good to be true!!
...
but it works! not always, but I'll figure out what to avoid. I'll be playing around with that A LOT!
wow.
great job you two!


ahh.. accidental click.. reminds me of a feature with I wanted to post but forgot:
Undo, please!

[Kalles Fraktaler]

omg!
this sounds too good to be true!!
...
but it works! not always, but I'll figure out what to avoid. I'll be playing around with that A LOT!
wow.
great job you two!


ahh.. accidental click.. reminds me of a feature with I wanted to post but forgot:
Undo, please!

Yes, undo would probably be nice. Perhaps limited to just a couple of clicks.
Tip 1: Learn the Control character combinations!
I use Control+H to toggle "Show Inflection" and Control+D for opening the Newton-Raphson dialog, so that I don't need to move the mouse-cursor from the interesting spots.
Tip 2: Use a high zoom level, I usually use 32. The Newton-Raphson use the same zoom level to limit the area to search for the minibrot so when the zoom level is high there is less risk of failure.

[skychurch]

Okay, I've updated all my software to enable zooming up to the machine memory limit. But I'm now amazed to find I can zoom past 10e640 and still get away with using machine hardware doubles after the SA calculations. How is this possible when the initial delta is far too small to have an effect on the outcome?