Kalles Fraktaler 2.9.3

[_BlueGuy_]

Suggestion: Accept a file to open as the first command line argument

[claude]

There is only one new function added, the Newton-Raphson method of finding minibrots or zoom to 3/4 of the minibrot, where the pattern of the current view is doubled.

I'm having fun with this!  Even though it does get slow after a few goes around, and I had one instance when KF stopped responding with the find period counter stuck on 880000 or so... had to kill it, didn't get around to retrying yet.

Feature request: extend it to arbitrary-power Mandelbrot (power k needs to iterate z→z^k+c for box-period corners, and Newton method derivative needs to be dz→k dz^(k-1)+1, other than that most of the code should remain the same).  But: I don't know if the size formula will work for higher powers (with iterations of z replaced appropriately), maybe it would work enough to guess the depth?

[stardust4ever]

I'm having fun with this!  Even though it does get slow after a few goes around, and I had one instance when KF stopped responding with the find period counter stuck on 880000 or so... had to kill it, didn't get around to retrying yet.

Feature request: extend it to arbitrary-power Mandelbrot (power k needs to iterate z→z^k+c for box-period corners, and Newton method derivative needs to be dz→k dz^(k-1)+1, other than that most of the code should remain the same).  But: I don't know if the size formula will work for higher powers (with iterations of z replaced appropriately), maybe it would work enough to guess the depth?

I have found the higher order mandelbrots tend to hit the minibrot at a much shallower depth. At larger orders (6 and higher) it becomes blobby and you have to zoom quite deep just to get away from visible minis.

For instance, from past observations:

Second Order:
Pattern (forked path) = 1x depth, 1x iterations
Pattern doubled = ~1.5x (3/2) depth, 2x iterations
Pattern quadrupled = ~1.75x (7/4) depth, 3x iterations
Pattern x8  = ~1.875x (15/8) depth, 4x iterations
Pattern x16 = ~1.9375 (31/16) depth, 5x iterations
Minibrot = ~2x depth, infinite iterations

Third order:
Pattern (forked path) = 1x depth
Pattern tripled = ~1.333x (4/3) depth
Pattern x9 = ~1.444x (13/9) depth
Pattern x27 = ~1.481x (40/27) depth
Minibrot = ~1.5x depth

Forth order:
Pattern (forked path) = 1x depth
Pattern quadrupled = ~1.25x (5/4) depth
Pattern x16 = ~1.3125x (21/16) depth
Minibrot = ~1.333x (4/3) depth

Fifth order:
Pattern (forked path) = 1x depth
Pattern x5 = ~1.2x (6/5) depth
Pattern x25 = ~1.24x (31/25) depth
Minibrot = ~1.25x (5/4) depth

Based on my observational experience, there is a clear pattern to the depth of the final minibort. For order Z=Z^N+C, the minibrot is typically at a depth of ((N+1)/N) relative to the depth of the fork point. Sometimes the final minibrot depth can be slightly less if the fork point zooms into a highly complex Julia side formation. Iteration depth is generally a more accurate metric compared to zoom depth, with each periodic double an exact multiple of the iteration depth at the fork point for order two, although I don't have hard numbers regarding iteration count for orders above two.

As a result, the higher the order, the less far you have to zoom and the less manual zooming required to locate the minibrot. Additionally, only with the order two Mandelbrot is it possible to create long rows of similar objects where the true centroid is difficult to locate visually. For an eXtreme eXample of this effect:
http://stardust4ever.deviantart.com/art/1024X-Long-595637805

Regardless, the new minibrot location method is infinitely useful for finding minitbrots in such locations rather than spending hours manually clicking or scroll-wheeling through thousands of zoom levels.

[Pauldelbrot]

How does KF estimate the minibrot's size?

[Kalles Fraktaler]

How does KF estimate the minibrot's size?

I pasted in the magic code from claude.... (i.e. I have no idea )

[claude]

http://ibiblio.org/e-notes/MSet/windows.htm is where I got the minibrot size estimation algorithm from.