Kalles Fraktaler 2.9.5

[stardust4ever]

I think I found a similar formula:

f(z)=(absre(z)*absim(z))^2+c

<Quoted Image Removed>

Simon, have you tried doing an order 3 version of your snake brot? The "radioactive" minis with twofold radiation aureola would look gnarly if they were threefold. Like real radiation symbols!!!

Instead of

Code:

				zrabs = abs(zr);
				ziabs = abs(zi);
				tempzi = zi*zrabs+zr*ziabs;
				tempzr = zr*zrabs-zi*ziabs;
				zi=tempzi;
				zr=tempzr;

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

				zisqr = zi * zi;
				zrsqr = zr * zr;

try this

Code:

				zrabs = abs(zr);
				ziabs = abs(zi);
				tempzi = zi*zrabs+zr*ziabs;
				tempzr = zr*zrabs-zi*ziabs;
				zi=tempzi;
				zr=tempzr;

				zrsqr = zr * zr;
				zisqr = zi * zi;
				zi= ((zrsqr * 3) -zisqr) * zi + JuliaI;
				zr= (zrsqr - (zisqr * 3)) * zr + JuliaR;


				zisqr = zi * zi;
				zrsqr = zr * zr;

BTW, I have no idea what the output might look like, but I imagined radioactive logos everywhere...

[TheRedshiftRider]

Simon, have you tried doing an order 3 version of your snake brot? The "radioactive" minis with twofold radiation aureola would look gnarly if they were threefold. Like real radiation symbols!!!

Instead of

Code:

				zrabs = abs(zr);
				ziabs = abs(zi);
				tempzi = zi*zrabs+zr*ziabs;
				tempzr = zr*zrabs-zi*ziabs;
				zi=tempzi;
				zr=tempzr;

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

				zisqr = zi * zi;
				zrsqr = zr * zr;

try this

Code:

				zrabs = abs(zr);
				ziabs = abs(zi);
				tempzi = zi*zrabs+zr*ziabs;
				tempzr = zr*zrabs-zi*ziabs;
				zi=tempzi;
				zr=tempzr;

				zrsqr = zr * zr;
				zisqr = zi * zi;
				zi= ((zrsqr * 3) -zisqr) * zi + JuliaI;
				zr= (zrsqr - (zisqr * 3)) * zr + JuliaR;


				zisqr = zi * zi;
				zrsqr = zr * zr;

BTW, I have no idea what the output might look like.

Hey stardust, I guess you've mistaken me for Simon.

I've had a look at this formula using some other software which simple lets you imput functions, like I did in the previous post.

There does not seem to be a way to do that. f(z)=(absre(z)*absim(z))^3+c will simply change the main fractal but the stretched lines stay the same. I guess we would have to play around with abs values to change that.

[stardust4ever]

Sorry I'm up way past my bedtime. It's already daylight here...

You are probably right about the circular bars. The initial symmetries may be twofold, as are many third order abs() functions. But then again with many 3rd order abs() fractals you get beautiful hexagonal kaleidoscope patterns after the pattern triplicates itself... The third order abs() fractals are beautiful, but the Snake brot is technically fourth order despite having an appearance like a second order set, and the 3-fold variant would actually be sixth order.

[Kalles Fraktaler]

I think I found a similar formula:

f(z)=(absre(z)*absim(z))^2+c

<Quoted Image Removed>

Sure interesting, however I am sorry I cannot extract real and imaginary parts from such format.

Simon, have you tried doing an order 3 version of your snake brot? The "radioactive" minis with twofold radiation aureola would look gnarly if they were threefold. Like real radiation symbols!!!

The radioactive symbol is definitely very interesting!

[TheRedshiftRider]

Sure interesting, however I am sorry I cannot extract real and imaginary parts from such format.

Ok, I'll look whether I can simplify it.



Edit: stardust, shouldn't you go to bed?


Edit2: I tried but I can't do it. (was the code from stardust the code of the original formula or the one I found?)

[stardust4ever]

Edit: stardust, shouldn't you go to bed?

[TheRedshiftRider]

Now that you are still awake. Was that code you posted from the original formula by simon or was it the one I found?


Edit: well...

[Kalles Fraktaler]

Sorry, instead of folding the wifi-radion 3 times, there two of them becomes lines only.
But I am maybe looking in the wrong section of this fractal...

The result has the 6th power...

[stardust4ever]

Sorry, instead of folding the wifi-radion 3 times, there two of them becomes lines only.
But I am maybe looking in the wrong section of this fractal...

The result has the 6th power...

Nevertheless, still a rather cool fractal. There is always the possibility of radiation patterns deeper into the set. But starting around sixth order and beyond, you need to zoom deep just to get away from minis. In my 9th Power B.S. "Madi Gras Masks" zoom video (use the search function), the minis were still quite large around the point at 43 zoom levels when the float point code ended.

[stardust4ever]

Now that you are still awake.

Yep, I crashed pretty hard, but I need to be up again for class at 9am so good nights and I mean it this time. You European people are on very different time zones! The alarm crows early at 7am to prepare for my MWF.

I seem to suffer from delayed sleep cycle / non-24 at times and generally gravitate towards staying up at nights, but skipping bedtime every other night and falling into a 48-hour cycle is bad for anyone. The one time I travelled abroad (Italy, 2005) I experienced zero jet lag effects because I'm so used to constantly resetting my own body clock.

[stardust4ever]

Kalles, first off thank you for your formulas! I am already doing another zoom video into the Quasi B.S. 5th.

I noticed you incorporated my "false" quasi perpendicular but not the real variant. The real variant which can be rewritten with Zr and the brackets inside the abs() to only use one abs() per line, has the best fractal details IMO of the quasi groups. In my opinion the "real" version contains better fractal detail than the "false" variants. Which is why I denoted it "false" because it absolutes neither the standalone Zr or Zi components.

Again I appologize if my 4th order list was long. I am flattered you incorporated so many!

[Kalles Fraktaler]

Kalles, first off thank you for your formulas! I am already doing another zoom video into the Quasi B.S. 5th.

I noticed you incorporated my "false" quasi perpendicular but not the real variant. The real variant which can be rewritten with Zr and the brackets inside the abs() to only use one abs() per line, has the best fractal details IMO of the quasi groups. In my opinion the "real" version contains better fractal detail than the "false" variants. Which is why I denoted it "false" because it absolutes neither the standalone Zr or Zi components.

Again I appologize if my 4th order list was long. I am flattered you incorporated so many!

Thanks stardust. Can you give the full name of the formula you are missing? They are so many now...
If it is "4th Real Quasi Perpendicular" or "4th Imag Quasi Perpendicular / Heart" - they are included in recently uploaded versoin 2.9.7

Fortunately this page exist
http://quickmath.com/webMathematica3/quickmath/algebra/expand/basic.jsp
This enables me to quickly expand the formulas and apply perturbation on them.

If there are any more requested, for example from simon.snake's big file, I can add them as well, however I am having a hard time converting these to real/imag pair formulas.
When we are done adding formulas, I will make a version 2.10 that also implement the formulas in long double and floatexp, so that they can be unlimited zoomed (ok, e100000 is max for this and the next decade )

[stardust4ever]

Thanks stardust. Can you give the full name of the formula you are missing? They are so many now...
If it is "4th Real Quasi Perpendicular" or "4th Imag Quasi Perpendicular / Heart" - they are included in recently uploaded versoin 2.9.7

Fortunately this page exist
http://quickmath.com/webMathematica3/quickmath/algebra/expand/basic.jsp
This enables me to quickly expand the formulas and apply perturbation on them.

If there are any more requested, for example from simon.snake's big file, I can add them as well, however I am having a hard time converting these to real/imag pair formulas.
When we are done adding formulas, I will make a version 2.10 that also implement the formulas in long double and floatexp, so that they can be unlimited zoomed (ok, e100000 is max for this and the next decade )

Ah, never mind. I very briefly looked over the formulas and did not notice an additional grouping of 4th order formula after the 5th order ones. (see attachment) They are all in there it appears. My apologies there were so many 4th order equations but the way the formula factored out gave more combinations to play with.

I'm currently doing a zoom into the 5th Quasi B.S. Not very deep (2e65) but has large bailout or 500000 at the final mini so is taking some time. The expanded formulas can get quite unwieldy with the polynomials. I hope you are taking advantage of distributive property and defining variables for repeated terms like Zrsqr and Zisqr as this will reduce the number of multiply operations. The higher order perturbation formulas yield some pretty complex strings of polynomials with dr and di in addition to Zr and Zi, though if my understanding is correct, some of the smaller terms with delta exponents are so infinitesimal they could be safely eliminated in some circumstances.

For instance at arbitrary depth, the delta between two pixels within the image are tiny numbers not far off the end of useful precision. Such a pixel delta value squared, cubed, or raised to higher exponent would result in a smaller number than even arbitrary precision allows. Say the entire image has a width of 2^-96 and you are using 128-bit maths to calculate the orbits. Take any two pixels in the image, and the delta value will be equal to or less than 2^-96.

So with 128-bit calculations, and |dx| <= 2^-96

|dx^2| <= 2^-192
|dx^3| <= 2^-288
|dx^4| <= 2^-384

and so on. If the reference orbit is only calculated to a precision of 128 bits, then terms with 2 or greater exponent for dx^n could be safely discarded from the equation. This effect will continue to hold true at deeper and deeper iteration depths that

dx^n where n => 2

will be smaller than the least significant bit of the reference orbit. That is if my understanding of perturbation theory is correct. I could be way off base however as the delta values will slowly increase each iteration until the orbit finally escapes.

[Kalles Fraktaler]

Ah, never mind. I very briefly looked over the formulas and did not notice an additional grouping of 4th order formula after the 5th order ones. (see attachment) They are all in there it appears. My apologies there were so many 4th order equations but the way the formula factored out gave more combinations to play with.

I'm currently doing a zoom into the 5th Quasi B.S. Not very deep (2e65) but has large bailout or 500000 at the final mini so is taking some time. The expanded formulas can get quite unwieldy with the polynomials. I hope you are taking advantage of distributive property and defining variables for repeated terms like Zrsqr and Zisqr as this will reduce the number of multiply operations. The higher order perturbation formulas yield some pretty complex strings of polynomials with dr and di in addition to Zr and Zi, though if my understanding is correct, some of the smaller terms with delta exponents are so infinitesimal they could be safely eliminated in some circumstances.

For instance at arbitrary depth, the delta between two pixels within the image are tiny numbers not far off the end of useful precision. Such a pixel delta value squared, cubed, or raised to higher exponent would result in a smaller number than even arbitrary precision allows. Say the entire image has a width of 2^-96 and you are using 128-bit maths to calculate the orbits. Take any two pixels in the image, and the delta value will be equal to or less than 2^-96.

So with 128-bit calculations, and |dx| <= 2^-96

|dx^2| <= 2^-192
|dx^3| <= 2^-288
|dx^4| <= 2^-384

and so on. If the reference orbit is only calculated to a precision of 128 bits, then terms with 2 or greater exponent for dx^n could be safely discarded from the equation. This effect will continue to hold true at deeper and deeper iteration depths that

dx^n where n => 2

will be smaller than the least significant bit of the reference orbit. That is if my understanding of perturbation theory is correct. I could be way off base however as the delta values will slowly increase each iteration until the orbit finally escapes.

yes, at least the squares are stored in temprary variables.
One can probably examine each formula and replace much more, example:

Code:

//4th Celtic Real Quasi Heart
Dnr = lb_abs_db(x2*x2 + y2*y2 - 6*x2*y2,4*y2*y*b-12*y2*a*x+6*y2*b2-6*y2*a2-12*x2*y*b-24*x*y*a*b+4*b2*b*y-12*b*y*a2+4*x2*x*a-6*x2*b2+6*x2*a2-12*b2*x*a+4*a2*a*x+b2*b2-6*b2*a2+a2*a2) +a0;
Dni = 4*y*lb_abs_db(x2*x-x*y2,-a*y2-2*b*x*y-2*a*b*y+3*a*x2-b2*x+3*a2*x-a*b2+a2*a) + 4*b*abs(- x*y2-a*y2-2*b*x*y-2*a*b*y+x2*x+3*a*x2-b2*x+3*a2*x-a*b2+a2*a) + b0;

The compiler is having a hard time optimizing all of this in release mode, it takes some 15-30 minutes to make a release build...

Regarding "small add-ends", as Botond called them...
Yeah, I have been on that track. However I found it hard, and Mandel Machine do fail to render some locations correctly. So I simply skipped it all togheter...

[stardust4ever]

yes, at least the squares are stored in temprary variables.
One can probably examine each formula and replace much more, example:

Code:

//4th Celtic Real Quasi Heart
Dnr = lb_abs_db(x2*x2 + y2*y2 - 6*x2*y2,4*y2*y*b-12*y2*a*x+6*y2*b2-6*y2*a2-12*x2*y*b-24*x*y*a*b+4*b2*b*y-12*b*y*a2+4*x2*x*a-6*x2*b2+6*x2*a2-12*b2*x*a+4*a2*a*x+b2*b2-6*b2*a2+a2*a2) +a0;
Dni = 4*y*lb_abs_db(x2*x-x*y2,-a*y2-2*b*x*y-2*a*b*y+3*a*x2-b2*x+3*a2*x-a*b2+a2*a) + 4*b*abs(- x*y2-a*y2-2*b*x*y-2*a*b*y+x2*x+3*a*x2-b2*x+3*a2*x-a*b2+a2*a) + b0;

The compiler is having a hard time optimizing all of this in release mode, it takes some 15-30 minutes to make a release build...

Regarding "small add-ends", as Botond called them...
Yeah, I have been on that track. However I found it hard, and Mandel Machine do fail to render some locations correctly. So I simply skipped it all togheter...

Kalles, first off, thank you, thank you, thank you for implementing our formulas. What you are doing is an incredible service to all fractal explorers.

While I have not made any qualitative tests with regards to render speed and efficiency, it seems the higher order formula (4th and 5th) are taking significantly longer compared to lower orders (2nd and 3rd). I am currently doing a zoom to ~e65 in the 5th order Quasi (hybrid) B.S. and chose a zoom location with relatively high bailout (500,000 at the final mini but this drops rapidly to under 100k as you zoom out), but my desktop is still chugging along about 44 hours in and has progressed roughly ~e10 outwards from the minibrot. For th lower orders, I would expect this slower rate of initial progress further out, ie beyond e300 or so.

My floating point code samples obviously did not include the expanded polynomials present within the perturbation code, but due to the long-winded polynomials with many terms required by perturbation, laws of diminishing returns take effect when using higher powers. It may even be plausible, if perturbation equations are significantly more complex, that for higher orders, full arbitrary precision might be slightly faster at lighter zoom levels (ie 128-256 bits). It is unfortunate if as you claim that nested powers, ie (Z^N1)^N2 do not "work" as I feel there may be great potential for code optimization using the nested power technique with prime factors for formulas of 6th order and higher.

Deep zooms using higher order formula also tend to be not as deep due to the simple fact that one does not have to zoom as far to reach a minibrot. My 5th Quasi B.S. zoom location currently rendering is highly ornate and I can't wait to share it.

I also cannot wait to explore the "3rd" (technically 6th) order Snake brot. I may explore it on my laptop, which has mobile quad core AMD with roughly one third the effective speed of my large desktop. With summer temps fast approaching in humid Louisiana, the room containing my desktop PC is currently isolated from the house AC and getting uncomfortably warm. The exhaust fan is like a mini hair dryer...