Title: 4th AND 5th Order ABS Fractals Are Here!!! Post by: stardust4ever on April 06, 2016, 04:23:18 AM (http://orig09.deviantart.net/07e4/f/2016/099/4/6/4th_order_abs_mandelbrot_variations__complete_set__by_stardust4ever-d9xzypw.png)
http://stardust4ever.deviantart.com/art/4th-Order-ABS-Mandelbrot-Variations-Complete-Set-601300868 Bump: 4th order collection of 24 fractal definitions is now complete! :dink: Code: 4th Order ABS Mandelbrot Variations (Complete Set) Special thanks to Kalles Fraktaler and Command Line Cowboy (Panzerboy) for adding my 2nd and 3rd order fractals to their software plugins. Feel free to use any of the above formulas in any software program, image, or video render. Credit is appreciated but not required. Thanks for viewing... BUMP: Added 5th Order fractals. I'm not going any further as things tend to get a bit blobby with 6th Order and beyond... (http://orig05.deviantart.net/8052/f/2016/097/c/f/5th_order_abs_mandelbrot_variations_by_stardust4ever-d9y3ws9.png) http://stardust4ever.deviantart.com/art/5th-Order-ABS-Mandelbrot-Variations-601484985 Code: Mandelbrot 5th: See Also: 3nd Order ABS Formula: http://stardust4ever.deviantart.com/art/Cubic-Mandelbrot-ABS-Variations-Incomplete-487039945 2rd Order ABS Formula: http://stardust4ever.deviantart.com/art/Mandelbrot-ABS-Variations-Complete-Set-of-Formulas-487039852 Title: Re: 4th order ABS Fractals are here!!! Post by: TheRedshiftRider on April 06, 2016, 06:22:41 AM Nice, I am curious how these look from closeby. :)
Title: Re: 4th order ABS Fractals are here!!! Post by: stardust4ever on April 06, 2016, 06:50:06 AM Nice, I am curious how these look from closeby. :) I would like to know the same. Much has been done with my lists of 2nd and 3rd order fractals, first with Fractal Extreme and later Kalles Fraktaler. Many of the Youtube Zoom movies out there are incredible.Truthfully, I am not a programmer but have a pretty strong grasp of complex math and can edit code. All of the screenshots were generated by editing the floating point code in Fractal Extreme's 32-bit sample Plugin. You can zoom up to 45 zooms before it reverts to plain Mandelbrot. Also I have to overwrite the sampleplugin.dll every time I generate a new definition. Here's a sample of the float point code I wrote up night before last for a 5th order Mandelbrot (untested): Code: Mandelbrot, 5th Order: 5th order polynomials each have 4th power terms in the Real and Imaginary components when you factor out the zi and zr. Because i^2 = -1, all of the even terms are in the real domain and the odd terms in the Imaginary domain. This places the terms (zr)^4, (zi)^4, and zi^2 * zr^2 into the equation. For optimal coding efficiency, it is wise to create new variables to handle the fourth order polynomial terms which each appear twice in the equation. Otherwise, the CPU is performing the same multiplications multiple times. I'll propably stop after 5th order though. The higher the order, the more places there are to insert an abs() or sign change to alter the fractal. From experience I have found Mandelbrots higher than around 6th order tend to get blobby. Title: Re: 4th order ABS Fractals are here!!! Post by: TheRedshiftRider on April 06, 2016, 08:14:48 AM To be fair, I've been trying to understand perturbation rendering for a long time but I still don't completely get it (I did inderstand the explanation in the prevous post). Same for programming. I guess Karl will figure out how to implement them in KF. I do know how to use juliamorphing :) abs-fractals included.
I do hope I get time to clean my poweredge case, I will probably need it if I want to make videos with these within reasonable time. Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: stardust4ever on April 07, 2016, 03:48:56 AM Bump. 5th order variations added. I'm probably going to stop here as higher order fractals tend to get "blobby."
:mandel: Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: Kalles Fraktaler on April 07, 2016, 12:49:06 PM Thanks stardust, many of these looks really awesome!
However the calculations tend to get very complex in the 3rd power already. I'll take cubic burning ship as an example. The reference calculation is pretty straight forward: Code: xrn = (sr - (si * 3)) * xr.Abs() + m_rref; But then the perturbation calculation, with the abs method from laster blaster, is much trickier. This abs method is necessary otherwise precision limitations will make it impossible to go deeper than e18. The reference points needs to be subtracted from the delta, and the resulting fully expanded expression needs to be analyzed in order to identify how to apply the abs method abs(c+d) - abs(c): if c>0: if c+d > 0 (e.g. d > -c): result = d else if d == -c: result = d else if d < -c: result = -d -2c else if c==0: result = abs(d) else if c < 0: if c+d>0 (e.g. d > -c) result = d + 2c else if d == -c: result = -d else if d < -c: result = -d So here is the perturbation implementation Code: yr = m_db_dxr[antal] + Dr; I tried to cheat, like doing 2nd power burning ship operations twice in each loop, however that also give precision errors beyond e18. Here is the full topic from laser blaster http://www.fractalforums.com/new-theories-and-research/perturbation-formula-for-burning-ship-(hopefully-correct-p)/ Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: stardust4ever on April 07, 2016, 10:52:33 PM Thanks for the link. I needed to reread that thread. BTW, my intention is not to pressure anyone into including some or all of these fractal types in their software, but to document these formula.
The forth order is larger because with the optimized equation, there are three operands on the imaginary side which to which the abs() command can be applied. The partial Burning Ships imaginary and real components are both located on the imaginary side. These can be optionally combined with the Celtic (Buffalo Partial Real) and every partial real variation has it's own Mbar variant. The Quazi perpendicular and Heart variants apply the abs() to the distributive polynomial. And I have yet to combine those with the partial B.S. (six formula not tested or included on the list). Fifth order is very similar to third, with a Zi factored out on the Imag side and a Zr on the real side. Since the fifth Mandelbrot is four bulbs, the real and imag components of the Partial BS and Partial Buffalo are identical, only rotated 90 degrees, but each has a distinct Mbar formulation. Fpr the 3rd order Partials, the Mbar variations are 90 degree rotations of the partials. And the fifth order equations contain a ton of multiply operations even with additional variables created to replace the polynomial terms. I explored a few of these up to the floating point FX limit of ~45 zooms, but as 5th order equations they are already getting a bit on the "blobby" side. And I'm aware that these abs() commands are difficult to factor out, hence the need for multiple if/then statements. They are a nightmare to derive or integrate, requiring two cases for each abs() command based on positive or negative. Calculus sucks anyhow. But again, I'd like to iterate my primary purpose with these formula posters is to document the fractals. Sometimes less is more, and I'm not sure adding 18 different 4th order equations (24 if you include the formula I skipped over) and ten 5th orders would be that productive. The 2nd and 3rd orders seem to be in a sweet spot for beauty versus complexity. Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: Kalles Fraktaler on April 08, 2016, 09:06:35 AM This is much appreciated and some of them will most probably be implemented eventually ;)
These formulas are at least very easy to implement without perturbation now when you documented them! :thumbsup1: Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: stardust4ever on April 08, 2016, 11:41:03 AM Some other thoughts I have regarding the ABS() variations...
Most of the Partial BS fractals seem like they are divided into halves, where the half of the overall shape resembles the Mandelbrot and half of the overall shape resembles the Mandelbar. This is less obvious with the second order fractals, but going by my makeshift definitions, the 2nd order Perpendicular is actually the Mbar variant of the partial burning ship real (only the real component has the absolute value applied, with the imaginary side negated), the 2nd order perpendicular buring ship is actually the partial BS Imaginary, and of course for second order and only second order, the Burning Ship is also the Buffalo Partial Imaginary. With even orders, the Zi and Zr get factored out with both results on the Imaginary side of the equation, and with Odd orders, the Zi gets factored out on the Imaginary Axis while the Zr gets factored out on the real axis. But there is always exactly one instance of Zi and Zr in each order. With Zi and Zr factored out of the equation, you are left with a string of polynomial terms remaining using the distributive property. These terms inside the parentheses are always square thus are all positive by default. However they are often subtracted from one another, with Zrsqr - Zisqr on the real side of the second order, and other more complex polynomial terms on both sides in higher orders. In all orders higher than two, there is an option to apply an absolute value function to the value of the polynomial inside the bracket. This results in the Quasi Heart and Quazi Perpendicular fractal variants due to their visual resemblance to the 2nd order equations. In the 3rd order and presumably all odd orders, applying absolute value only to one or both polynomial brackets results in "junk" however doing the Z component on one side and the polynomial component on the other results in the afformentioned Quazi Heart. The Quazi Heart is not interesting in and of itself, but like the 2nd order variants, the Mbar variation of the Heart leads to the Perpendicular equation. As a side note, the "heart" fractal is actually a duplicate of the boring side of the Burning Ship, and the Celtic Heart likewise duplicates the boring side of the Buffalo. A similar instance occurs in the 3rd order ABS set, with the Quasi Heart duplicating the boring side of the Quazi Burning ship, technically the Burning Ship / Buffalo hybrid. Ironically, the Quazi Heart and Quazi Perpendicular are derived slightly differently with the 4th order fractal set. The 4th order "Heart" appears to have much more fractal detail, and slightly less so for the 4th order Quazi Perpendicular. In fact the front half of the 4th Quazi Perpendicular and the rear half of the Heart sort of resemble the uninteresting side of the Buffalo Partial Real. And I have yet to combine these with the Partial Burning Ship variations and investigate the results. There are six yet undiscovered fractals to complete the 4th order set. Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: stardust4ever on April 08, 2016, 12:46:39 PM This is much appreciated and some of them will most probably be implemented eventually ;) Thanks for your interest. I know I posted a metric ton of 4th order formulas. The complete set should have 24 unique fractals sans rotations and flips. To attempt to replicate all of them would be silly.These formulas are at least very easy to implement without perturbation now when you documented them! :thumbsup1: I feel the following are good candidates: 4th Order: Burning Ship, Burning Ship Partial Imag, and Burning Ship Partial Real Buffalo, Celtic, and Buffalo Partial Imag (the Buffalo Partial Imaginary bears and uncanny resemblance to the hybrid or Quazi Burning ship from the 3rd and 5th orders) The Mbar variants of the BS Real and Celtic might also be worth investigating. The Heart, Perpendicular, and Celtic Burning Ship variants seem more like filler to me. 5th Order: Definitely the Burning Ship and Buffalo are must haves, as they have that same beautiful diagonal symmetry that made the third order versions so unique. Overall most of these are pretty good analogues of the 3rd order collection. I will say this though, starting with around 6th order or so, fractals start to get blobby. The Abs variants are no different. I did a small amount of exploring in the 5th order (limited to 45 zooms) and it's hard to get away from the minis at such low depth. Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: stardust4ever on April 08, 2016, 01:39:52 PM Well I rendered the six missing fractals and it appears I've erred in my naming convention on the 4th order fractal set. Expect some major tweaking later. The Quazi Perpendicular fractals require an absolute value on the the real Z component in addition to the polynomial, just like it's 3rd and 5th order sister fractals. :hurt:
Here is the updated formula: Code: Real Quasi Perpendicular 4th: Now I've got to get ready for class in two hours. Been up all night again. Monster Energy to the rescue... Edit: Thank you spell check. I just realized I've been spelling "Quazi" wrong for the past two years. It should be "Quasi," meaning "similar to but not the same." :banginghead: Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: TheRedshiftRider on April 08, 2016, 02:16:31 PM The real quasi perpendicular 4th looks really impressive.
Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: stardust4ever on April 08, 2016, 11:03:02 PM The real quasi perpendicular 4th looks really impressive. Yeah I'm gonna redo the formula sheet sometime. I haven't slept yet...I'll just call them "Real Quasi Perpendicular, Real Quasi Heart, and Imag Perpendicular to differentiate. Buffalo / Celtic will go on the same row, third row down not counting the 4th Order Mbrot and Mbar. Perpendicular / Heart variants fil the bottom rows. Only I have ten fractals left to make into rows of four. This will create two holes. Place the very strange looking Imag variations at the bottom. And fix the spelling errors "Quazi" -> "Quasi" in all four posters. That will be a royal PITA because I flattened all the layers in GIMP long ago when I saved as PNG... :hurt: Screw typos! 4th order set is now complete! :dink: Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: Kalles Fraktaler on April 15, 2016, 02:01:39 PM I tried to implement 4th power burning ship, but was not successful.
The result get distorted beyond e14 already, due to precision issues, in the same way as if I make burning ship or cubic burning ship without laster blaster's abs method... :sad1: The real part is straight forward, since it doesn't include any abs (first parentesis is the delta, minus the second from the reference) Code: double &r = m_db_dxr[antal]; // reference real part However the imaginary part has the abs, for the delta minus the reference Here is without laser-blaster's abs method: Code: Dni = 4*abs(dr*di)*(sdr-sdi) - (4*abs(r*i)*(sr-si)) + dbD0i; Expanding dr, di, sdr and sdi, and rearranging it to define the abs(c+d)-abs(c) Construction: Code: Dni = 4*(sr-si)*(abs(r*i + r*Di + Dr*i + Dr*Di) - abs(r*i)) + 4*abs(r*i + r*Di + Dr*i + Dr*Di)*(2*Dr*r+Dr*Dr-2*Di*i-Di*Di) + dbD0i; In order to be able to use laser blaster's abs method on the first abs: Code: double c = r*i; Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: stardust4ever on April 16, 2016, 12:57:31 AM I tried to implement 4th power burning ship, but was not successful. I wanted to say thank you for attempting to render these fractals. I do have one more trick up my sleeve that may or may not work with the 4th order abs fractals.The result get distorted beyond e14 already, due to precision issues, in the same way as if I make burning ship or cubic burning ship without laster blaster's abs method... :sad1: The real part is straight forward, since it doesn't include any abs (first parentesis is the delta, minus the second from the reference) Code: double &r = m_db_dxr[antal]; // reference real part However the imaginary part has the abs, for the delta minus the reference Here is without laser-blaster's abs method: Code: Dni = 4*abs(dr*di)*(sdr-sdi) - (4*abs(r*i)*(sr-si)) + dbD0i; Expanding dr, di, sdr and sdi, and rearranging it to define the abs(c+d)-abs(c) Construction: Code: Dni = 4*(sr-si)*(abs(r*i + r*Di + Dr*i + Dr*Di) - abs(r*i)) + 4*abs(r*i + r*Di + Dr*i + Dr*Di)*(2*Dr*r+Dr*Dr-2*Di*i-Di*Di) + dbD0i; In order to be able to use laser blaster's abs method on the first abs: Code: double c = r*i; I won't pretend to understand exactly how the perturbation rendering method works (aside from using deltas to calculate orbits at lower precision), but the burning ship and buffalo fractals (as well as their partial |Zr| and |Zi| variants) can be computed to arbitrary power N. Since complex Z can be defined as: Code: Z = Zr + Zi*i Assume complex absolute value is applied separately to the Zr and Zi components. Code: |Z| = |Zr| + |Zi|*i Burning Ship arbitrary power N: Code: Z = |Z|^N + C Buffalo arbitrary power N: Code: Z = |Z^N| + C Next, take the standard reduced polynomial 4th power Mandelbrot: Code: Mandelbrot 4th Order (Reduced Polynomial): Z = Z^4 + C The above formula uses seven bignum multiply operations (four of which are squares) not counting simple integers. Alternately we can use nested exponents to express higher order exponents N which have smaller prime factors. For instance, Z^4 = (Z^2)^2 Code: Mandelbrot 4th Order (Nested Exponent): Z = (Z^2)^2 + C The above formula uses six bignum multiply operations (four of which are squares), making it slightly more computationally efficient than the reduced polynomial 4th. To convert the 4th order Mandelbrot to the generalized Burning Ship, the absolute value command must be applied to the equation before to the exponent. This yields: Code: Burning Ship 4th Nested: Z = (|Z|^2)^2 + C Code: Burning Ship 4th Partial Imag Nested: Buffalo fractal applies the absolute values after the exponent. Code: Buffalo 4th Nested: Z = |(Z^2)^2| + C Code: Buffalo 4th Partial Real (Celtic 4th) Nested: Code: Buffalo 4th Partial Imag (Quasi B.S. 4th) Nested: These nested exponent formula might give you something to play with since all the lower order abs formula worked but not 4th and 5th. I haven't tested these yet but the output should be equivalent to the reduced polynomial variants... :dink: Again, I won't pretend to understand perturbation theory, but it may be possible to simplify the code by expressing forth order fractals as two nested second order "half-iterations", and only adding complex C every second iteration. Applying the burning ship formula only to the second half-iteration instead of the first should result in a 4th Order Quasi Burning Ship (labeled as "Buffalo Partial Imag 4th" on my formula sheet). The 4th Order Quasi Perpendicular and Quasi Heart variants likely won't be possible using nested second order equations, but Burning Ship, Buffalo, Celtic, their real and imaginary partials, and Mandelbar variants should be possible with nested second order iterations. A nested 5th order, being prime, is out, but 6th (Z^3)^2, 8th ((Z^2)^2)^2, and 9th (Z^3)^3 nested exponents would work. It may also make rendering higher order Mandelbrots more efficient. I recently did a 10th degree zoom video and progress of fractal frames was slow. My 5th order Mandelbrot optimized, reduced polynomial formula (http://stardust4ever.deviantart.com/art/5th-Order-ABS-Mandelbrot-Variations-601484985) uses 7 bignum multiplies, and a nested tenth order version Z = (Z^5)^2 + C would have used only ten bignum multiply operations per iteration. Remember, Burning Ship fractals apply the absolute value to Zi and Zr before the exponent, and Buffalo/Celtic fractals apply the absolute value after the exponent. Again I'd like to thank you for attempting this! :dink: Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: Kalles Fraktaler on April 16, 2016, 12:58:03 PM Thanks for your extensive work stardust, however I am sorry but splitting up the Z^4 to (Z^2)^2 does unfortunately not work with perturbation, that uses the (reference+delta) - (reference) construction.
I found out that the problem is not the imaginary part, but the real part. The expression I showed actually contains r^4-r^4 and i^4-i^4. With the help of this page http://quickmath.com/webMathematica3/quickmath/algebra/simplify/basic.jsp I reduced the real part, and got it to work! :joy: :banana: :chilli: :joy: :banana: :chilli: Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: stardust4ever on April 17, 2016, 04:45:44 AM Thanks for your extensive work stardust, however I am sorry but splitting up the Z^4 to (Z^2)^2 does unfortunately not work with perturbation, that uses the (reference+delta) - (reference) construction. So you found an "island" somewhere around e44 in the 4th order burning ship? Neato! I was only thinking maybe the equations could be factored into two second order half-iterations (which you claimed won't work) but kudos to you regardless!I found out that the problem is not the imaginary part, but the real part. The expression I showed actually contains r^4-r^4 and i^4-i^4. With the help of this page http://quickmath.com/webMathematica3/quickmath/algebra/simplify/basic.jsp I reduced the real part, and got it to work! :joy: :banana: :chilli: :joy: :banana: :chilli: :joy: :banana: :chilli: :joy: :banana: :chilli: You said it contained r^4-r^4 and i^4-i^4. If any of these expressions resulted in the denominator of a fraction, it would invalidate the equation. There was a fairly famous algebraic "proof" that I remember reading from an old algebra textbook in high school. Or was it eighth grade; I cannot remember exactly. It starts with A=B, then manipulates both sides of the equation extensively before resulting in B=A+A and substituting A for B and factoring A out of the equation with the final erroneous result 1=2. Upon further examination of the equation, you have both sides of the equal sign with (A-B)/(A-B) or similar expression in both the numerator and denominator of a fraction, resulting in 0/0 which is undefined. Anything times zero equals zero, so working backwards, any value could be applied to an invalid fraction such as 0/0, such a conundrum which allowed the equation to be manipulated to 1=2 as a result. :tongue1: My calculus sucks, but there is a great deal of math involved in determining limits, which a curve or infinite series may approach but never actually reach a specific value with finite terms. Discovering the precise limit to a problem, or even whether a series escapes to infinity or has a defined limit, is not trivial. For fractals, we sometimes work out the equation to millions of iterations to determine if a point falls within a set. If the deltas involved with perturbation use infinitesimal values to compute "limits", a zero term anywhere in the equation, especially under a denominator, could really foul the result. Kudos to you again regardless. :joy: :banana: :chilli: :joy: :banana: :chilli: I look forward to future Kalles Fraktaler updates. I would also love to see 5th order B.S. [Z=|Z|^5+C] and Buffalo [Z=|Z^5|+C] at some point if they are possible to implement, since the odd order versions of both these fractals have that beautiful diagonal symmetry that is so intriguing. Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: Kalles Fraktaler on April 17, 2016, 10:40:54 AM You said it contained r^4-r^4 and i^4-i^4. If any of these expressions resulted in the denominator of a fraction, it would invalidate the equation. This is actually the main magic of perturbation, reducing the larger term to be able to calculate with low precision.We calculate the delta, i.e. the reference (x) plus an offset (d) and then subtract the reference. An easy example, (x+d)^2 - x^2. Expaning the parentesis yiels (x^2 + 2xd + d^2) - x^2. The largest term of the reference, x^2, can be reduced, and left is only 2xd + d^2, which fortunately doesn't require high precision to be calculated accurately. I managed to forget the importance of this since last time I implemented perturbation formulas :D In 4th Power Burning Ship there were also a term r*r*i*i that needed to be reduced in order to get it working. 5th order requires some good analysis I assume. Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: stardust4ever on April 17, 2016, 11:47:48 AM This is actually the main magic of perturbation, reducing the larger term to be able to calculate with low precision. I get it now. Working out the polynomials and reducing them is a pain for higher orders, but it's still just basic algebra. The 2zΔ + Δ^2 would start out infintessimally small, considering the deltas are the equivalent of a few pixel widths onscreen. For a distance in the neighborhood of e-44, the Δ^2 term would be on the order of e-88, about twice the current arbitrary precision depth. As a result, the Δ^2 could be safely discarded. Additionally, if floats or scaled integers are used, the required precision decreases as the deltas drift apart with every iteration.We calculate the delta, i.e. the reference (x) plus an offset (d) and then subtract the reference. An easy example, (x+d)^2 - x^2. Expaning the parentesis yiels (x^2 + 2xd + d^2) - x^2. The largest term of the reference, x^2, can be reduced, and left is only 2xd + d^2, which fortunately doesn't require high precision to be calculated accurately. I managed to forget the importance of this since last time I implemented perturbation formulas :D In 4th Power Burning Ship there were also a term r*r*i*i that needed to be reduced in order to get it working. 5th order requires some good analysis I assume. But I see how the equations get ugly for higher orders. (z+Δ)^5 would be Code: z^5 + 5*z^4*Δ + 10*z^3*Δ^2 + 10*z^2*Δ^3 + 5*z*Δ^4 + Δ^5 Code: ((zr + zi*i) + (Δr + Δi*i))^5 - (zr + zi*i)^5 But essentially we are dealing with powers of sums (a + b + c + d)^N which seems really nasty to expand, even if a bunch of terms from -(a + b)^N ultimately either get cancelled out or are so infintessimally small we can safely ignore them. I've only ever done polynomial powers of sums with two variables each (for which one can use the horizontal rows of Pascal's triangle as a cheat sheet). Four variables (necessary for summation of two complex numbers) would be a PITA. Only the terms containing only Zr and Zi can be algebraically eliminated. Edit: (a+b*i+c+d*i)^5-(a+b*i)^5 is plain nasty, with 50 terms! :hurt: http://www.wolframalpha.com/input/?i=expand+(a%2Bb*i%2Bc%2Bd*i)^5+-+(a%2Bb*i)^5 (http://www.wolframalpha.com/input/?i=expand+(a%2Bb*i%2Bc%2Bd*i)^5+-+(a%2Bb*i)^5) Code: 5 a^4 c+5 i a^4 d+20 i a^3 b c-20 a^3 b d+10 a^3 c^2+20 i a^3 c d-10 a^3 d^2-30 a^2 b^2 c-30 i a^2 b^2 d+30 i a^2 b c^2-60 a^2 b c d-30 i a^2 b d^2+10 a^2 c^3+30 i a^2 c^2 d-30 a^2 c d^2-10 i a^2 d^3-20 i a b^3 c+20 a b^3 d-30 a b^2 c^2-60 i a b^2 c d+30 a b^2 d^2+20 i a b c^3-60 a b c^2 d-60 i a b c d^2+20 a b d^3+5 a c^4+20 i a c^3 d-30 a c^2 d^2-20 i a c d^3+5 a d^4+5 b^4 c+5 i b^4 d-10 i b^3 c^2+20 b^3 c d+10 i b^3 d^2-10 b^2 c^3-30 i b^2 c^2 d+30 b^2 c d^2+10 i b^2 d^3+5 i b c^4-20 b c^3 d-30 i b c^2 d^2+20 b c d^3+5 i b d^4+c^5+5 i c^4 d-10 c^3 d^2-10 i c^2 d^3+5 c d^4+i d^5 Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: Kalles Fraktaler on April 18, 2016, 09:32:20 AM Because delta and reference are encapsulated by the abs function for the real value, so there is no reducing easily available. Edit: No, I just made some typos, it will work Anyway, here is a test zoom of the 4th power: https://www.youtube.com/watch?v=z-HbUHNhbGg Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: stardust4ever on April 18, 2016, 07:32:00 PM Holy cow! Thanks dude... :dink:
:wow: EDIT: In other news, here's a short 9th Order B.S. (!) zoom video I made in Fractal Extreme yesterday using my own compact float point code: https://www.youtube.com/watch?v=TeEEyeuBA-U Code: 9th Order Burning Ship [nested] Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: Kalles Fraktaler on April 18, 2016, 09:38:46 PM Holy cow! Thanks dude... :dink: Cool, looks like an owl.:wow: EDIT: In other news, here's a short 9th Order B.S. (!) zoom video I made in Fractal Extreme yesterday using my own compact float point code: https://www.youtube.com/watch?v=TeEEyeuBA-U Code: 9th Order Burning Ship [nested] Outzoomed it looks like almost the 9th power mandelbrot :) Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: Kalles Fraktaler on April 19, 2016, 11:13:47 PM https://youtu.be/R9yme1WAGW8
Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: stardust4ever on April 20, 2016, 12:51:07 AM :thumbsup1:
Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: quaz0r on April 20, 2016, 08:13:56 AM :thumbsup1:
Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: stardust4ever on April 21, 2016, 02:51:45 AM Some side by side large details I rendered of the West Miniships in 4th order and 5th order Burning Ship / Quasi Burning Ship fractals. The Quasi and standard Burning Ship fractal variants share a striking similarity along the west needle, regarding placement of dendrites and minis.
High resolution zoomed in areas of the west needle in the 4th and 5th B.S. and Quasi B.S.: http://sta.sh/01e6u2eank9n The 3rd order version of the Quasi Burning Ship for comparison. Notice how more tightly compacted the dendrites are in the 3rd Quasi B.S mini: http://sta.sh/0aiensqbdgu In the 3rd order version of the Quasi Burning Ship, there also exists a northward facing needle that strikingly resembles that same region in the Buffalo 3rd. You can see the northward facing needle in this high resolution zoomed out capture of the entire set: http://sta.sh/01hrnckb4dvc Also for comparison, I have done zoomed in renders of the west needle for the Buffalo and Celtic fractals in the 4th and 5th orders: http://sta.sh/029v7h94cu9p Note that the 4th order fractals exhibits 8-fold symmetry and the 5th order fractals exhibit 10-fold symmetries with regards to the dendrites. The Buffalo and Celtics are indeed very "bulby" fractals when getting into higher orders! :D Generalized abs() formula for any power (arbitrary exponent): Burning Ship: Absolutes the value of Zr and Zi before the exponent. Buffalo: Absolute the value of Zr and Zi after exponent. Celtic: Absolutes the value of Zr after exponent. Quasi Burning Ship: Absolutes the value of Zr and Zi before the exponent, AND absolutes the value of Zi after the exponent.* *For Quasi Burning Ship, it is not always necessary to absolute Zi or Zr before the exponent, if Zi or Zr only appears in the imaginary portion of the expanded equation. For even powers, the Quasi Burning Ship is the Partial Buffalo (Imaginary Only), which takes the imaginary side of the Buffalo and the real side of the standard Mandelbrot. For odd Powers, the Quasi Burning Ship is a hybrid combining the real side of the Burning Ship and the Imaginary side of the Buffalo formula. For 2nd order only, the Burning Ship and Quasi Burning Ship are the same fractal. Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: Kalles Fraktaler on April 22, 2016, 04:42:41 PM If you can put up with not being able to go deeper than e300, I can upload a new version with these new fractals included?
Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: TheRedshiftRider on April 22, 2016, 10:10:43 PM Great, I guess depth will not be an issue, just being able to explore them would already be nice.
Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: stardust4ever on April 23, 2016, 04:13:06 AM If you can put up with not being able to go deeper than e300, I can upload a new version with these new fractals included? Go for it. Well, e300 is a still a hella lot better than the 43 zooms I'm currently getting with my Fractal Extreme sampleplugin hacks! :gum:IS there a reason why does it fail beyond e300? Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: Kalles Fraktaler on April 23, 2016, 09:35:25 AM Go for it. Well, e300 is a still a hella lot better than the 43 zooms I'm currently getting with my Fractal Extreme sampleplugin hacks! :gum: The reason is that e300 is the limit of the exponent of the hardware datatype 64-bit double. IS there a reason why does it fail beyond e300? One can extend that to e600 by multiplying and dividing the values, "scaling", but that gets too complicated for anything beyond cubic Mandelbrot, at least for me. Beyond e600 I use the dll that has the formulas in 80-bit long double compiled with gcc. But I have to uninstall and install to switch between 32 and 64 bits. I will eventually do it though to be able to make also the new formulas beyond e300. Beyond e4900 (which also could be extended for plain Mandelbrot with scaling though), the limit of the exponent of long double, the much slower custom datatype floatexp is used, i.e. a double and integer pair where the integer replaces the exponent of the double. The limit of the exponent is then some 2 billion :D Ok the exponent limit numbers above are not the exact limits but what I use, since there is some margin Anyway, get the new formulas from http://www.fractalforums.com/index.php?topic=23602.msg92220#new Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: stardust4ever on April 23, 2016, 09:57:50 AM e300 is plenty deep to get some decent zoom movies. Thanks for doing this!
:toast: Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: quaz0r on April 23, 2016, 09:58:22 AM its too bad cpu makers dont build in some functionality to let us use the vector units as native float128 / float256 / etc :fiery:
Title: Re: 4th AND 5th Order ABS Fractals Are Here!!! Post by: stardust4ever on April 23, 2016, 10:14:06 AM its too bad cpu makers dont build in some functionality to let us use the vector units as native float128 / float256 / etc :fiery: When FPGA caches become standard on new CPUs it will give software devs a new tool to play with much how GPUs are now used for work rather than play. Suppose someone writes an FPGA core with 1024 bit or higher calcs. Even if the core only runs at 400Mhz, it could do arbitrary far faster than a 4Ghz CPU would allow. And only programming instructions you need like add and multiply, you reduce the transistor count so you could duplicate the bignum core until it fills the FPGA cache.And not just fractals. Cryptography uses some pretty big numbers that would benefit from unlimited precision. SETI. Protein Bending. Prime search. Bitcoin farming. Look at 4k BluRay players using tiny ASICs to decode video. It would require an immensely powerful PC to play back said content at 60fps. |