Kalles Fraktaler 2.7

[stardust4ever]

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...?

Honestly, the interpretation that Panzerboy used (and I subsequently included in my list of fractal definitions) is a much better fractal since like the Mandelbrot it is full of minis at all depths. HPDZ's Buffalo is not a "pure" 2nd order fractal since it subtracts Z from Z^2. I'm not really a huge fan of fractals that deteriorate into dust but he did get an impressive zoom video out of it.
www.fractalforums.com/movies-showcase-(rate-my-movie)/buffalo-deep-zoom-to-e227/

Of my 12 fractal formulas, all have minis, some of which resemble the shapes of other fractals. For instance, Burning Ship can have minis shaped like Celtic and Mandelbar Celtic, and visa versa. Mandelbar/Tricorn fractal contains both Mandelbrot and Tricorn minis with a distribution of 50/50. Perpendicular Mandelbrot can have minis shaped like Hearts and Perpendicular Burning Ships.

The "Heart" fractals are the least interesting of the bunch, since minis are only present along the needle and nowhere else. Heart is the submerged side of the Burning Ship flipped across the X-axis, and Celtic Heart is the Buffalo underside flipped across the X-Axis. Likewise, the 3rd order variety "Quasi Heart" has the underside of the Quasi Burning Ship 3rd flipped about the X-axis.

Of the 3rd order fractals I've cataloged so far, Buffalo, Burning Ship, and Quasi Burning Ship (Buffalo/Burning Ship hybrid) seem the most promising for deep zooming. The Quasi Burning Ship (which I believe I am the first to discover) takes the Imaginary side of the Cubic Buffalo and the Real side of the Cubic Burning Ship. Swapping the real and imaginary components generates an identical fractal rotated 90 degrees. For mine, i inverted the Imaginary side so that the action stays above the "surface". The Quasi Burning Ship 3rd has a horizontal mast that resembles the Cubic Burning Ship but with more ornate dendrites, and a vertical mast that resembles the Cubic Buffalo but with slightly less ornate dendrites. Also the fluffy areas in between appear more accessible.

I use the term "quasi" because the suffix means "similar to but not the same", in reference to the fact they look similar to the 2nd order fractals of the same name.

[Kalles Fraktaler]

I've so far tried the following, in addition to what is already available in 2.7:
Cubic Celtic - (very cool!)
Mandelbar - (I don't like it, too skewed, but I will keep it)
Mandelbar Celtic
Perpedicular Mandelbrot
Perpedicular Burning Ship
Perpedicular Celtic
Perpedicular Buffalo
Heart - (I don't like it, too skewed, will not be included)
Heart Celtic - (I don't like it, too skewed, will not be included)
HPDZ's Buffalo - don't work, will not be included

I would like to have cubic equivalents of the ones in my list that I will keep, but I understand there are several per the quadratic fractals.
Because even though I cannot use the power value as a parameter in the formulas like I do for ordinary Mandelbrot, it is used for coloring.
If Cubic "Quasi Burning Ship" is interesting, which of the above should it fall into, "Perpedicular Burning Ship" maybe?

The cubic fractals are really cool, and often they do not get as skewed as the quadratic ones!   

[stardust4ever]

I've so far tried the following, in addition to what is already available in 2.7:
Cubic Celtic - (very cool!)
Mandelbar - (I don't like it, too skewed, but I will keep it)
Mandelbar Celtic
Perpedicular Mandelbrot
Perpedicular Burning Ship
Perpedicular Celtic
Perpedicular Buffalo
Heart - (I don't like it, too skewed, will not be included)
Heart Celtic - (I don't like it, too skewed, will not be included)
HPDZ's Buffalo - don't work, will not be included

I would like to have cubic equivalents of the ones in my list that I will keep, but I understand there are several per the quadratic fractals.
Because even though I cannot use the power value as a parameter in the formulas like I do for ordinary Mandelbrot, it is used for coloring.
If Cubic "Quasi Burning Ship" is interesting, which of the above should it fall into, "Perpedicular Burning Ship" maybe?

The cubic fractals are really cool, and often they do not get as skewed as the quadratic ones!    

This is awesome, thank you! Maybe "Hybrid Cubic Burning Ship" or "Cubic Burning Ship / Buffalo Hybrid" is a better name, but the latter is a mouthful. It combines the Real portion of Cubic Burning Ship with the Imaginary portion of the Cubic Buffalo so it really is a hybrid of the two fractals. I have made some preliminary test renders of the Burning Ship Hybrid in high resolution which I haven't uploaded yet.

I initially applied the "Quasi" moniker to several of the 3rd order fractals because they resembled the shape of one of the 2nd order fractals. Perpendicular moniker really doesn't apply to the Burning Ship Hybrid.

With regards to the 2nd order fractals, I applied the term "Perpendicular" to any fractal formulas where the ABS function was applied to one of the Zr * Zi terms in the Imaginary component (Zi = Zr * Zi * 2 + Ci), but not both. "Burning Ship" signifies an ABS operation to the Imaginary component wheras "Celtic" signifies an ABS to the Real component, and Buffalo applies the ABS to both Real and Imaginary. Mandelbar implies multiplying by -2 instead of +2 on the Imaginary component. So the remaining fractal definitions that didn't already have names, I assigned names based on the logistical combinations of the formula properties.

And yes, the Heart brots are boring. Minis exist only on the needle. I included them in the fractal lists only because they were part of a larger superset of fractal definitions, not because they were particularly fascinating.

A lot more permutations exist for 3rd order fractals, but many of them I've encountered so far are "junk" or flips/rotations.  Not every third order fractal necessarily has a 2nd order equivalent and visa versa. And I haven't even scraped the surface of higher orders yet. Burning Ship by definition applies the ABS function to both Zr and Zi prior to squaring. For 2nd order this is equivalent to applying the ABS only to the entire Imaginary side, because for real numbers, (abs(x))^2 = x^2, and the real side of the equation contains only squared figures. For third order, applying the ABS function only to the Imaginary side yields a different result. See also Misunderstood C.B.S. aka "Flying Squirrel".

Basically with each formula I am using MS Visual C++ 2010 to hack the floating point code for the Fractal Extreme SamplePlugin and rerendering the output, as my coding skills aren't quite good enough to develop a finished plugin. Kind of kludgy and it reverts to Mandelbrot after 43 zoom levels, but it works. I'll post some high resolution renders of my 3rd Order Burning Ship / Buffalo Hybrid later. It's way past my bedtime, LOL!

[Chillheimer]

everyone who participated in this rocks majorly!

[stardust4ever]

everyone who participated in this rocks majorly!

[TheRedshiftRider]

Well done, I think this will make exploring much more interesting.

[Kalles Fraktaler]

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?

Thank you

Do you mean to search for minibrot by zooming out instead of in?

[stardust4ever]

Some High Resolution captures of my Burning Ship / Buffalo 3rd Order Hybrid. It is crammed with detail.
[Warning: Images are big @6400x4800 pixels antialiased!]

Overall:
http://sta.sh/01hrnckb4dvc

Miniship:
http://sta.sh/0aiensqbdgu

[laser blaster]

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.

As I mentioned before, the reason the minis disappear is because HPDZ doesn't begin iterating the formula at the critical point. Yes, I know we're dealing with non-analytic formulas here, the critical point stuff still seems to apply here (I guess the abs() formulas are close enough to the Mandelbrot that similar dynamics still apply). To find the critical point, you just strip out the abs()'s from the formula, and take the critical point of that function. If you begin iteration at that point, the fractal will have plenty of minis. The critical point of the HPDZ Buffalo is (0.5 + 0i).

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.

I think the reason that Pauldelbrot's glitch detection fails is because the critical point is not zero. I'm not entirely sure how to fix it. However, Pauldelbrot's glitch-detection involves checking how close a point is to zero... maybe if you checked how close it was to the critical point (0.5 + 0i) instead, it might work? (Obviously you would want to begin iteration at the critical point as well).

[ellarien]

Thank you

Do you mean to search for minibrot by zooming out instead of in?

No.

When I set the minibrot finder for a certain zoom level it doesn't stop zooming in when it gets close to a minibrot, but carries on and then bombs out with 'cannot find center' when the image is nearly full of black. I guess it is stopping, but it would be nice if it stopped a zoom or two earlier and reported success rather than failure. (I just took it for a spin on the basic Mandelbrot to test my experience, and it's impressively fast on the zooming in.  )

[stardust4ever]

As I mentioned before, the reason the minis disappear is because HPDZ doesn't begin iterating the formula at the critical point. Yes, I know we're dealing with non-analytic formulas here, the critical point stuff still seems to apply here (I guess the abs() formulas are close enough to the Mandelbrot that similar dynamics still apply). To find the critical point, you just strip out the abs()'s from the formula, and take the critical point of that function. If you begin iteration at that point, the fractal will have plenty of minis. The critical point of the HPDZ Buffalo is (0.5 + 0i).
I think the reason that Pauldelbrot's glitch detection fails is because the critical point is not zero. I'm not entirely sure how to fix it. However, Pauldelbrot's glitch-detection involves checking how close a point is to zero... maybe if you checked how close it was to the critical point (0.5 + 0i) instead, it might work? (Obviously you would want to begin iteration at the critical point as well).

I think you're missing the point here. The fact that this fractal disintegrates to dust means it just isn't good for deep zooming. All of my list of abs() fractals have plentiful minis in most areas of the sets, with the exception of the underside of the BS and Buffalo fractals, as well as the Heart variants. Hunting for minis around the so-called critical point is moot because you will eventually have to explore away from this point, at shich point the fractal would eventually become dust. Maybe such a fractal would benefit from perturbation of the seed value (ie starting point or Z0 is some value other than Z or C).

When bailout is set to a higher value (4 or more), even the trailing edge of the needle on HPDZ's Buffalo fractal can be shown to disintegrate into dust.

UltraFractal 5 is loaded with presets similar to HPDZ's Buffalo, that look interesting on the surface (and some that look like mistakes) which don't yield fruitful zoom attempts. The fact that I can take a set of parameters to manipulate the fractal definition in such a way to produce a variety of interesting fractals, shows that simply inserting abs() values in various locations can produce a wide array of related patterns. The presence of minis really only serves to bring closure to the zoom movie, ie the animation has an obvious beginning and an end point. Still I feel that a proper fractal should contain minis at all depths, not just the main fractal.

[laser blaster]

I think you're missing the point here. The fact that this fractal disintegrates to dust means it just isn't good for deep zooming. All of my list of abs() fractals have plentiful minis in most areas of the sets, with the exception of the underside of the BS and Buffalo fractals, as well as the Heart variants. Hunting for minis around the so-called critical point is moot because you will eventually have to explore away from this point, at shich point the fractal would eventually become dust. Maybe such a fractal would benefit from perturbation of the seed value (ie starting point or Z0 is some value other than Z or C).

When bailout is set to a higher value (4 or more), even the trailing edge of the needle on HPDZ's Buffalo fractal can be shown to disintegrate into dust.

You misunderstood me. I'm not advocating for HPDZ's fractal, I'm saying it's a flawed implementation of a perfectly good function. The critical point isn't an area you explore around, it's the starting value you use for z, and it actually has a very specific significance. If you don't start at the critical point, not only will the result completely lack minis, but the resultant fractal will completely fail to serve as an accurate connectedness map of the corresponding Julia sets. Technically, it has not yet been proven that the parameter-plane (mandelbrot-style) fractal for an abs() formula serves as an accurate atlas of the Julia sets, but it seems true enough in practice, as long as you begin at the critical point.

If you tried initializing z to (0.5, 0) instead of (0), you'll find that the resulting fractal has lots of minis in many places (and never collapses into Julia dust in the mini-dense areas), while still maintaining the overall shape of HPDZ's buffalo. Essentially, HPDZ's buffalo is a slightly corrupted version of the "correct" formula.

Here's a picture of a mini from the "correct" version:

[stardust4ever]

You misunderstood me. I'm not advocating for HPDZ's fractal, I'm saying it's a flawed implementation of a perfectly good function. The critical point isn't an area you explore around, it's the starting value you use for z, and it actually has a very specific significance. If you don't start at the critical point, not only will the result completely lack minis, but the resultant fractal will completely fail to serve as an accurate connectedness map of the corresponding Julia sets. Technically, it has not yet been proven that the parameter-plane (mandelbrot-style) fractal for an abs() formula serves as an accurate atlas of the Julia sets, but it seems true enough in practice, as long as you begin at the critical point.

If you tried initializing z to (0.5, 0) instead of (0), you'll find that the resulting fractal has lots of minis in many places (and never collapses into Julia dust in the mini-dense areas), while still maintaining the overall shape of HPDZ's buffalo. Essentially, HPDZ's buffalo is a slightly corrupted version of the "correct" formula.

Here's a picture of a mini from the "correct" version:

Interesting. That mini resembles the shape of the Perpendicular Mandelbrot, one of my cataloged abs() variants. Any chance I can see what the fractal looks like zoomed out, with your seed value of -.5?

Perhaps more mixed order / polynomial type fractals (example, aZ^3 + bZ^2 + cZ +C or other such expressions) would benefit from this effect. I seem to recall someone on DA who wrote a plugin for UltraFractal 5 that used polynomial based Mandelbrots, sometimes containing minis with multiple orders.

Unfortunately, I have been using Fractal Extreme for rendering test formulas, and it is not currently possible to specify a seed value for perturbated Mandelbrot, only in Julia mode. I used to play around with seed values in Fractal Forge back in the day (2009-ish), and If I chose a seed value near the set, the Mandelbrot would be connected in the vicinity, but turn into chaotic blobs and dust elsewhere.I used to think it was cool looking to render the Mandelbrot set with a perturbation value of (-.75,0) where the elephant valley of the large mini is located.

[laser blaster]

Sure, here's the entire fractal. Doesn't look much different from HPDZ's at this scale.

Polynomial fractals will definitely benefit from initializing z at the critical point(s). If you saw minis in this person's images, then they must have also been starting at the critical point(s).

An interesting thing with mixed-order polynomials of degree greater than 2 is that they can have multiple critical points, and then if you choose just one as the starting value of z, you will typically get a different fractal set for each critical point. You can overlay these sets on top of each other with some blending function, or you can take the intersection of all of the "critical sets", which gives the Julia connectedness map for that function, or you can take the union of all the sets, which results in the "at least a subset of the Julia set is connected" map.

[Kalles Fraktaler]

Still I feel that a proper fractal should contain minis at all depths, not just the main fractal.

I agree. But for maybe a different reason. The minis represent solutions to the expanded polynomials that can be created instead of iterating. These polynomials grow with 2 degrees per iteration.
A deep mini represent the solution to an incredible high degree polynomial. Since each circle around the main Mandelbrot cardioid is a solution, the visible circles on the un-zoomed Mandelbrot view are solutions to already incredible high degree polynomials. What we see as deep minis are solutions to equations that are truly impossible to set up.
That's really fascinating I think. Sorry for gliding too far off topic

I think the reason that Pauldelbrot's glitch detection fails is because the critical point is not zero. I'm not entirely sure how to fix it. However, Pauldelbrot's glitch-detection involves checking how close a point is to zero... maybe if you checked how close it was to the critical point (0.5 + 0i) instead, it might work? (Obviously you would want to begin iteration at the critical point as well).

I think Pauldelbrot's glitch-detection is checking how close the delta orbit is from the reference orbit, not to zero?