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Author Topic: Quadray Sets  (Read 16337 times)
Description: Quadray representations of the mandelbrot set
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DarkBeam
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Fragments of the fractal -like the tip of it


« Reply #30 on: April 29, 2011, 05:52:39 PM »

@Aexion: Very nice formulas! thanks for sharing and welcome to FF!

I've found an interesting thing: My "Mandelbrot on real numbers" patterns are on the thin surface of the 6-bulbs version!!  huh?

This is the M3D implementation:

<Quoted Image Removed>

And this is from my method for generating real number mandelbrot patterns:

<Quoted Image Removed>

I couldn't get the same coloring, but I added some skew to match the other image and comparing the patterns similarty.

Any idea why is this happening??



What is your formula? So we can make a confront wink
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Kali
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« Reply #31 on: April 29, 2011, 06:44:00 PM »

Luca, I guess you didn't see the link in my post. Here it is: http://www.fractalforums.com/mandelbrot-and-julia-set/mandelbrot-on-real-numbers-t5375/

Anyway, I tried several things in that thread, but the image here is using inner coloring (exponential smoothing, no bailout) of this simple 2D iteration:

x=x*x+cx
y=y*y+cy
z=sqrt(x*x+y*y)

(z is for coloring)



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DarkBeam
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« Reply #32 on: April 29, 2011, 07:21:51 PM »

Okay, now look at Aexion formula, that is almost identical

x=x*x + 2*z*w + cx (if I remember right)... (at z=0, z*w = 0 so it's identical to your fmla wink )
aexion cx= a linear combination of x,y,z

etc etc grin
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Kali
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« Reply #33 on: April 29, 2011, 07:31:26 PM »

I guess you are right  cheesy Thanks!
I was too lazy for trying to figure out that  embarrass

So I own a thin slice of this formula! I want a 0,0000001% of the profits aexion could make out of it!

I should call my lawyers...

rolling on floor laughing
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DarkBeam
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Fragments of the fractal -like the tip of it


« Reply #34 on: April 29, 2011, 07:39:02 PM »

I guess you are right  cheesy Thanks!
I was too lazy for trying to figure out that  embarrass

So I own a thin slice of this formula! I want a 0,0000001% of the profits aexion could make out of it!

I should call my lawyers...

rolling on floor laughing


Less than 0,0000001%, since his cx is a linear combination of cx and cy, yours is plain cx Order in the court !! tease
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Aexion
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« Reply #35 on: April 29, 2011, 09:26:22 PM »

I guess you are right  cheesy Thanks!
I was too lazy for trying to figure out that  embarrass

So I own a thin slice of this formula! I want a 0,0000001% of the profits aexion could make out of it!

I should call my lawyers...

rolling on floor laughing

Oh... 4D Fractdimensional profit? what an interesting concept smiley
DarkBean is right in the fact that when of the four dimensions, in that precise point of the formula, the multiplications by 0 cancels the linear combination and it turns into your formula. Have you visited the MiniBrots that lies on the line along the set? They could give interesting patterns too.. smiley
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knighty
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« Reply #36 on: April 29, 2011, 10:03:37 PM »

Hi, These quadray sets are very nice. I like particularly the six spacechips about to collide grin. Thank you Aexion.
Couldn't resist to give it a try. Here are attached fragmentarium scripts. (try the sign parameter).

* quadray01-07.txt (1.09 KB - downloaded 333 times.)
* quadray01-jacobian-02.txt (1.36 KB - downloaded 193 times.)
* quadray04-03forV0.8.txt (1.82 KB - downloaded 405 times.)
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Jesse
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« Reply #37 on: April 29, 2011, 10:54:01 PM »

My first tries of a general quadray transform before some 4d formulas were a bit disappointing, not that good as Aexion's suggested formula.
But it seems that the rotated 4d box likes it...  smiley
(You need the _QuadrayTransform2 formula from the 'NewFormulasSince1697_3.zip' update:
http://www.fractalforums.com/index.php?topic=6061.msg28944#msg28944)

Mandelbulb3Dv16{
O.....Y/...g2...w....26....jduVL.36C.1.......02E0WSnRKAIIzfjwXaNMAMyzgV7tyg6Faxj
................................5BrPXo4E3z1........Y./..................y.2.....
................/Q.0/....6.y....23...2E2.....AhfX0Nio/sD/M.........c./...w1.BnAn
z.EnAncD1E..0..........wz........................................y1...sD...../..
.w1...sDJUG42JHPPy9icZSJKImKwsyf1xUlrNgDhz.puXyfpuvfJf8LQJ.azOtJoGyCtdsjWmnwgtMf
1v1XPnitvYbWzqGzae3L3UtD......2q0...T0................sD.2E.0...................
..............................MRo8.4qFf.kO5h0UoRo8.srFf.sV5h0.FSo8..............
.....................wzzz1.U..6.P....M2...EB....m0...M4....F....8/...I1....UJl22
...U.e6e4zjeYFnz8Welzf8No.6.1c..zzCg.MQizccZWGyD7EdGkxmWcz1..........2k.8.klLS4.
QvM93P58iz9GMmnWK2zwz0........../6U0.wzzz1................................E.0c..
zzzz.................................2U.8.kzzzD.................................
/6U0.wzzz1...................................kdKd.EHVM/.UcFk982.cxo7.ogCn0eM.g5P
D/ErKRtUc/kF.V1.TO4bOZ0.B3W3.ESSQeJ8.oI6K.EovRxdJ.EHVM/.FlrpbK/.B3W3.sITLTO3.oI6
K..kzldKd.EHVM/....crI0.zzFoTu2.yzngi84.xzZX.45.................................
E....A....E.....I....2....kLFJLMY7LMtFZQVtqQaxaQh71.............................
...................5.1........zj................................................
................................................................................
.....................2.....3....4....wZIjFLMoJ4BY/..n/..........................
Ak.1Ak.1................................Y.A.............................Y.2.....
................................................................................
................................/....E/...U/..../7oPsNJMmZrIX34PZF1N............
............5.../.........................2........sz.........zD................
...wz...........................................................................
............................................}


* QuadrayRotBox.jpg (232.02 KB, 800x700 - viewed 363 times.)
« Last Edit: April 29, 2011, 11:11:35 PM by Jesse » Logged
Jesse
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« Reply #38 on: April 29, 2011, 11:13:32 PM »

A closer look at the old temple...


* QuadrayRotBox2.jpg (249.17 KB, 800x700 - viewed 374 times.)
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Syntopia
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« Reply #39 on: April 30, 2011, 12:18:06 AM »

Many thanks for the answers and ideas on coloring and raymarching, Jesse, Aexion and Makin!

Knighty, your second Fragmentarium sample keeps track of a running Jacobian to estimate the distance - I haven't seen that before - it seems to work great. How did you get the idea for this? Is there any formal math behind it?

There are many interesting bits of information in this thread - and in the rest of Fractal Forums. Has there ever been a discussion about starting a Wiki to summarize all this information - e.g. overviews of different formulas, number systems, software implementations, different approaches to raymarching and DE-calculations, coloring systems, recurring fractal motives, and so on?

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DarkBeam
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Fragments of the fractal -like the tip of it


« Reply #40 on: April 30, 2011, 12:23:03 AM »

Yay, that amazing quadray iz badass, will go and take a look cheesy
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David Makin
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« Reply #41 on: April 30, 2011, 04:48:37 AM »

Many thanks for the answers and ideas on coloring and raymarching, Jesse, Aexion and Makin!

Knighty, your second Fragmentarium sample keeps track of a running Jacobian to estimate the distance - I haven't seen that before - it seems to work great. How did you get the idea for this? Is there any formal math behind it?

There are many interesting bits of information in this thread - and in the rest of Fractal Forums. Has there ever been a discussion about starting a Wiki to summarize all this information - e.g. overviews of different formulas, number systems, software implementations, different approaches to raymarching and DE-calculations, coloring systems, recurring fractal motives, and so on?

I don't know if it's where Knighty got the idea but before Jos Leys showed that the "simple" "analytical" derivative worked for Daniel White's Mandellbulbs and we were still rendering it using either brute force or my old deltaDE method I suggested that a "correct" analytical DE would require use of the Jacobian and had a play around with the idea, but my maths wasn't up to it - then as I said Jos found the simple method worked fine (even though it doesn't necessarily follow due to the non-standard nature of the Triplex calculations). I also pointed out that using the full Jacobian for the running derivative may even give useable analytical DE for completly arbitrary 3D+ formulas e.g.

newx = x + exp(y) - 5* cos(z) + cx
newy = sin(x) + cos(y) + cy
newz = cosh(x) + sinh(y) - z + cz

Not that that particular choice is likely to produce anything of merit !
I got the idea for using the Jacobian originally because I had used it to allow arbitrary (2D) Newton formulas in a ufm for Ultra Fractal i.e. formulas for Newton fractals involving two real functions of two real variables.

Christian was planning a book or some such but he hasn't mentioned it for a while - at the moment the only related Wiki is the one for Ultra Fractal:

http://ultrafractalwiki.fractalforums.com/

Failing a book then an ff wiki is a great idea, if someone is prepared to put the time in to create and maintain it wink
Less time-consuming to maintain but potentially as useful would be if there was a good cross-referencing plug-in for the forum's engine....
« Last Edit: April 30, 2011, 04:50:20 AM by David Makin » Logged

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knighty
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« Reply #42 on: April 30, 2011, 05:29:11 PM »

The informations about the DE methods (and other stuff) are so scattered here that I can't really tell where exactly I got the Idea  embarrass. One thing is sure, it was here at fractalforums. As far as I know, David Makin was the first who had the idea of using jacobian to do DE of "arbitrary" 3D+ fractals (see the monstrously big thread about the mandelbulb wink) but it seems to me that things were not clear on how to use them...
My reasoning is quite simple and straightforward. The "standard" escape time fractal formula is of the form:
z[n+1]=F(z[n])+c.
z can be a complex number, a quaternion, a vector or a triplex...
The standard formula for DE (when infinity is a superattracting point) is:
DE=0.5*|z[final]|*log(|z[final]|)/|D(z[final])|
(D() is the derivative operator)
Keep in mind that this is an approximate formula. See:this book for the maths involved that I could'nt (fully) understand hurt.
now in case z is a vector, the natural derivative is the jacobian. Taking the formula as it is, it is "natural" to use the norm of the jacobian. That's it.
Of course, there are several ways to compute that norm. I've used the infinity norm here for it's simplicity. The "best" norm would be the maximal singular value...Sloow!  evil...
The jacobian could of course be computed numerically just as the gradient.
I don't know if the use of the norm of the jacobian is mathematically corret, but if this is the case, it would give better insight on why the other approaches, especially the scalar derivative DE, work.
...Well! I think that I should start a new thread about this subject instead of polluting this one.

It would be great to have a wiki... Hard hard work but worth it smiley
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knighty
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« Reply #43 on: April 30, 2011, 05:43:46 PM »

Back to the quadray sets:
It seems to me that its mainly a 3D slice of a highier dimensional fractal. So why not use other directions for the slices? One way to do it would be by applying a rotation to the 4D (or 8D) variables before the loop.
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DarkBeam
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Fragments of the fractal -like the tip of it


« Reply #44 on: April 30, 2011, 06:05:12 PM »

Hi, These quadray sets are very nice. I like particularly the six spacechips about to collide grin. Thank you Aexion.
Couldn't resist to give it a try. Here are attached fragmentarium scripts. (try the sign parameter).

Hey please an image cry
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