Kalles Fraktaler 2.7

[Kalles Fraktaler]

I have uploaded a new version, Kalles Fraktaler 2.7

4 new fractal types added:
 * Cubic Burning Ship
 * Buffalo
 * Cubic Buffalo
 * Celtic

I have unfortunately not have had time to test the limits (i.e. the transition between double and long double at e300, and long double and floatexp at e4900)

Also - flat values can be applied on infinite waves, as suggested by stardust4ever.
If an negative value is entered as period, the meaning will be percent on all iteration values.
Then the rainbow colors should be achievable.

Get your free copy from http://www.chillheimer.de/kallesfraktaler

[TheRedshiftRider]

Thanks. This is great. The cubic burning ship will add a lot. I also really like the buffalo and the celtic.

[TheRedshiftRider]

Btw, about the buffalo fractal. I thought It looked like this:



The formula for this one is ((abs(z)^2)-(abs(z)))+c

I learned that the formula you used (abs(z^2)+c) is a type of kaliset. Am I right?

[Kalles Fraktaler]

No I think that one is called perpendicular burning ship.
I will implement that as well eventually, because I found more formulas, including that one, in an old thread from master stardust4ever
http://www.fractalforums.com/fractal-exteme/fractal-extreme/60/

[TheRedshiftRider]

Oh, I see.

[laser blaster]

Great job getting all those formulas to work!  Wow, 4 new types, this is really exciting!

And about the Buffalo fractal... both of those fractals have been referred to as the Buffalo. For example, this video on HPDZ.net showcases the Buffalo fractal in Toofgib's post: http://www.hpdz.net/Animations/BuffaloDeep1.htm

The perpendicular burning ship is a different fractal altogether, although it does look similar to Toofgib's buffalo.

Also, a little something interesting I discovered about the Buffalo formula that Toofgib mentioned... It's an odd fractal, because it doesn't really have any minisets, just a lot of Julia dust. Which is because iteration is usually started at 0. I recently discovered that if you instead begin iteration at (.5+0i), you get minisets again (which is probably because that's the critical point of the abs-less version of the formula).

By the way, what formula are you using for your Buffalo, Kalle? I can't seem to find the formula for that one anywhere...

[Kalles Fraktaler]

Thanks
It got rather easy when I got easier formulas. But they all needed to be investigated to find where to apply the abs-conditions.
I used the formulas stardust4ever posted in this post
http://www.fractalforums.com/index.php?topic=19165.msg77045#msg77045

I still believe series approximation is not possible since it is not possible to derive abs.
But I hope I will be proved wrong

[stardust4ever]

This is awesome news; thank you so much!!!

Here are the ABS fractal formulas I have cataloged so far. The 2nd order fractal formulas (set of 12 formulas) are complete:

http://sta.sh/0ag7nkkbjri

3rd order has a long way to go (bigger superset but lots of duplicates and "junk") but here are some of the more interesting fractals I've discovered so far.

http://sta.sh/0263mwwvgaxn

Click the download link for full size view (1920x1920). Floating point code for each fractal has been embedded in the image.

I thought I ought to make the formulas public so that other people can play with them! Also I don't quite have the programming skills to incorporate them into functional plugins... :wink:

[cKleinhuis]

Btw, about the buffalo fractal. I thought It looked like this:

<Quoted Image Removed>

The formula for this one is ((abs(z)^2)-(abs(z)))+c

I learned that the formula you used (abs(z^2)+c) is a type of kaliset. Am I right?

[stardust4ever]

The formula for this one is ((abs(z)^2)-(abs(z)))+c

Guys are we sure this is correct? All my formulas (except Mandelbrot) have to be split into their native Zr and Zi components in order to work. Complex math no longer works when you start tinkering with the definition of z^2 or z^3. It does however, produce some lovely fractal patterns.

From HPDZ.net:
http://hpdz.net/Animations/BuffaloDeep1.htm#Formula

Fractal Formula

The formula is easiest to understand if it's written in two steps:

(1) Let z' = |Re(zn)| + i |Im(zn)|
(2) z(n+1) = z'^2 - z' + c

In other words, first take the absolute values of the real and imaginary components of zn, then apply the formula on the second line to get the next iteraton, zn+1.

EDIT: Got it. I cracked the code. Please note that this fractal definition is deviant and does not take the standard form of my ABS Mandelbrot variants.

Code:

		zi = abs(zr * zi) * 2.0 - abs(zi) - JuliaI;
		zr = (zrsqr - zisqr) - abs(zr) + JuliaR;
		zisqr = zi * zi;
		zrsqr = zr * zr;

EDIT2: One caveat about the HPDZ interpretation of the Buffalo fractal is no minis. He has posted one video and the zoom fades into nothingness at the end without finishing with a minibrot. In the Buffalo version that takes the absolute value of both sides, there are plenty of them north of the needle.

[laser blaster]

When he says abs, he's really referring to taking the absolute values of the real and imaginary components seperately. It's probably better to write cabs() instead , for clarity. cabs(z)^2 does not equal z^2.

[stardust4ever]

When he says abs, he's really referring to taking the absolute values of the real and imaginary components seperately. It's probably better to write cabs() instead , for clarity. cabs(z)^2 does not equal z^2.

Nevermind, I already figured out the formula...

[TheRedshiftRider]

I see that I made a small mistake.

I know It was funny.  

I made the mistake that these have to be calculated in a different way.  

[Kalles Fraktaler]

Nevermind, I already figured out the formula...

Thanks for all the formulas. The power-2 formulas are all rather easy to implement with perturbation.

Unfortunately Pauldelbrot's glitch detection doesn't work on HPDZ's formula. It works for all the other power-2 formulas though - and the higher powers I tried so far. From what I can see the difference is that HPDZ's formula has two separate abs on each axis. That might also explain why there are no minibrots.
So unless there is a different way to detect glitches on double abs, it is not possible to render it with perturbation...

Addition: maybe the x and y axis needs to be validated separately for glitches...?

[ellarien]

This is great work and I love the new formulae -- the Celtic in particular is very pretty. It's obviously a close cousin to the Burning Ship -- it even looks a bit like some of the distorted mini-ships I've seen.

I have one small suggestion/plea: would it be possible for the minibrot finder to stop when it finds one before the preset zoom level?

Thanks again!