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Author Topic: True 3D mandelbrot type fractal  (Read 383265 times)
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twinbee
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« Reply #60 on: June 08, 2009, 06:30:50 PM »

That's actually pretty good. If you have the CPU cycles to spare, I wouldn't mind seeing the zoomed in sections outlined by the green boxes. Chances are we'll see whipped cream, but at least it won't be UHT wink

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David Makin
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« Reply #61 on: June 08, 2009, 06:44:55 PM »

When I get home I'll post the UPR for it here - mmf.ufm:Solid-3D Quaternions is the formula which I have updated to the UF formula database but there are still issues with the compiled zip files not updating so if you want to get the update you'll probably have to download the updated version of mmf.ufm manually from:

http://formulas.ultrafractal.com/cgi-bin/formuladb?changes

For anyone impatient, once you've got the updated mmf.ufm just select the Solid-3D Quaternions formula and then set the fractal type parameter to "Alternative q^2+c" smiley

Note that with the formula UF's zoom does not work correctly if the viewing rotation parameters in the formula are not zero - you need to set the "x, y, z" target parameters etc. by hand.
« Last Edit: June 08, 2009, 06:50:12 PM by David Makin » Logged

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« Reply #62 on: June 08, 2009, 11:39:01 PM »

Hi all, here's a UPR for the main minibrot - visually centred at -1.76 on the x axis (not the one twinbee outlined but the slightly larger one further from the main cardioid).

Code:
MMF-4DMinibrot {
::cRN/Qjn29l5WvtOuRc83DQ+OY4+SLOnjOSzYJZvLIQ3tBFoAnsAb3+cDolps1e0tKJHHnP9d
  ouYzLTO5hEyhD/zRD5wfBS5dysBZ5P94DrWNUMUqErf+5/5qvsazTPXUXsvrZY9qLFHGOJgw
  wVnUFHPNIiiDXVKvq66FRry7kVqeBmq1YsjA2tKrTdoYoXs+J5rFHW9s87F1/cyX3+VSmdrf
  8hxpPusZy2himax6fVm99jdNnrPseVTrMrY4qIiW0K1wpmDiqzlDFty++HfoS22WUfca6q6B
  VnI8rkjyj1i4wHfIvpj8WOOel8tCtDQ4qWVX2JV23FN55ryLKV16gddVVew58q1avJx6uOmD
  G+yf0UWcAf63PLp5XThY/6VtvkPlxEr/lStZ5QxrqV/v/L8psRBafp/is96bi6m52vb0+ip9
  rG21tvP9LvPNUeZR7Ur9nPKuq6pWZN1/5kthOZdvx8KGyOJW/2nv+53/8Fdo2epTgUeZsV0Y
  Kq9FKxkX01PcbiVy6DXFPT/WVq3v1rmsos58gYjeVe7q4LRBpJjze4dRYA9XKhr/dkAmE51C
  1FxmAtxcREMFdZ0eOdAhi5zDldTBdpKfg6qlMcaqd6jUaTRjTnaNsf+heoaxVtgNtWelv/2j
  QeXT9ghr7pl2y3+MJd0+WX9ZmOlOhMN/bmok0oPVNvq0CMGGUOR3eO9feg6P63t5SGg5JqXr
  2TKS8J370zMMQnEmsPv0zZ05tOjRMjgZlWCCzuZ0JDZZpeVm04gag20Eb0ztvS2tEv1N7Lql
  dzn1o2HUtUx84WU2JZ9R19DCT1Z/L6YtUXROtJ2+qsc5sju7yT/9Rh7JyZHADHGu2qoTXHrL
  GOfQRGP3rolVfTzvUWuasZ/cpDN0wpOV/pmyDznxv1NMIWHjF10KukR1Foi/ziLTaMKY0Y5+
  UbQsO8v/XC/23wE6vxpJhrXGCnjT6+Aalvvn+GVk1PMP4VzOvP3hae5ezsGqcpT8N9JZ6ep5
  Dl7rimvcpc/PR3o8CdjR+vWteesgXf5oqW1Vkpvl7+cI70kkkv7rmPyOauq5Alzeq41VRJ30
  oXpOQbm4m4EDFGvXeut++KK4TDTvZqSSHJqaFRGTR+KtrTHG+N6qTZ5i1ix7W/WRtS2tY7QR
  myaqHOXVdVYEndtnmv/S3X9W79o5osqSevud0/ilN3xulNXI/v3/ElUbJHSu9TqxkLVvqK7F
  bvlPO1choD9Grfep8Y49ClbJlQnsSXf494kSPm9qJFM3i0aGNXhdTwIPBjsEMyOhXTKAw9HU
  tmgrmgnmgtKVtgVcS9iCALRRXRxZRtOxgWxKaJKKQMI1S0NMZ0NeB7GLR3YJK1LJYbasTaN2
  Naj9UN2S1Y30asA3Fg2hbirqJeqm4mYTsC3EBsNIyOzmykES9EO1KcTtUNVEHkAu5gtuR7WP
  R3KADR1jHbnF2Ki2Fks1ONszV4dcHF2Zp9OzqTS4dCMJIOxOTExVhF5XiRmM1e0DnAnstNYj
  z+Xkf1WEXoTW3Yqv2JwM6JLQcAG5lzj8q9i8L+ITbsPkM6k9KQlg0/JkbCC5SQo/Kg3vMTLv
  2D7gVvEUBZYwO3c0GvngNs5Idln5ao9yMe1rwGBEslJL5VeG5XfSmQwJNp9KEjsXkYxGIIy8
  OA4HQMhPgYC8ETgjYCsETwgYC+ETghYCsETghYCuETwmYCOETwkYC+ETwmYCOET4HSMBHiJ4
  TMBGiJwQMBTiJYRMBXiJ4SMBGiJYSMBXiJ4SMBXiJwQMBHiJYRMBPiJ4SMBeiJYSMBLiJ4RM
  BGiJwQMBTiJYRMBWiJ4SMBGiJYSMBXiJ4TMBXiJwQMBHiJYRMBfiJwQMBGiJYSMBLiJwRMBX
  iJwQMBTiJ4TMBGiJ4SMBeiJYSMBHiJwQMBOiJwRMBLiJwQMBGiJ4RMhPgYCWETwjYCfAxE8I
  mAHxEcJmgHxE4ImAHxE4ImgFxEYJmAHxE8ImwHQMBbiJwQMBeiJ4RMBOiJ4RMBOiJ8BET8HQ
  MxPgYi8ETkjYisET0gYi+ETkhYisETkhYiuET0mYiOET0kYi+ET0mYiOET8HSMRHiJ6TMRGi
  JyQMRTiJaRMRXiJ6SMRGiJaSMRXiJ6SMRXiJyQMRHiJaRMRPiJ6SMReiJaSMRLiJ6RMRGiJy
  QMRTiJaRMRWiJ6SMRGiJaSMRXiJ6TMRXiJyQMRHiJaRMRfiJyQMRGiJaSMRLiJyRMRXiJyQM
  RTiJ6TMRGiJ6SMReiJaSMRHiJyQMROiJyRMRLiJyQMRGiJ6RMxPgYiWET0jYifAxE9ImIHxE
  dJmoHxE5ImIHxE5ImoFxEZJmIHxE9Im4HQMRbiJyQMReiJ6RMROiJ6RMROiJaTMJjF19FaA3
  f9tPf9vN/OnlV7LU1jvketLHKyzP3r0HZiHf18qszlyOqPO3tv4dFVRt86/L1Uup4tc8jj9P
  +0/+T/xJ5hmL9zvH/J7G+of90TTfsLoz0hL9QRy9Obmpz9jCOWW31Ms8C3zfJvOSQPS1DP+w
  0D30XyT/lhyJqa5EV15bdlVu24DdJ/SeTH+ElOaOP4LRdTt6xHO2JPoTTjj1X10Mca8BS/9E
  ej4y63AenYTSMMbKi2smMGFtNJc2KsdEsONQ8udxR7imHCTxZ7b3FmGGmuYf72Fhi3GvLNa8
  g28Y72tMWSaaKERHlm/yhmBKxpXizlvrIETu+/xgBG9y
}

Edit:

Animation now here:

http://www.fractalforums.com/gallery/?sa=view;id=684
« Last Edit: June 09, 2009, 01:21:54 AM by David Makin » Logged

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« Reply #63 on: June 09, 2009, 02:38:01 AM »

Hi all,

I guess you all probably noticed that that "True 3D" Mandy looks closer to a quaternionic Mandy than a normal hypercomplex Mandy ?
Well just by changing two signs in the 4D multiplication table gives you an alternative "True 3D" Mandy but this time closer to hypercomplex than quaternion i.e. squarry rather than roundy:



http://makinmagic.deviantart.com/art/quot-True-3D-quot-Mandy-Squarry-125272102

The only changes to the multiplication table I posted earlier are that i*j = j*i = -k instead of +k.
In z^2+c this just gives w = 2*(x*w - y*z) instead of w = 2*(x*w + y*z).
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lycium
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« Reply #64 on: June 09, 2009, 10:38:55 AM »

excellent work dave, the earlier "roundy" (hehe) mandelbrot looks pretty much exactly like what i'd expect a 3d mandelbrot to look like!

especially cool is that you're using distance estimation to intersect the surface, so you get very fine solid details with a relatively short computation; in my algorithm i mark cells as containing surface once i've probabilistically intersected it, so i'll miss a lot of details at first and as the render progresses it gets more accurate (hence i do a kind of warmup before the proper render). horrible tongue stuck out

looks like you've got this one licked, congratulations are due for your fruitful directed efforts.
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lycium
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« Reply #65 on: June 09, 2009, 10:43:00 AM »

Ideally it should use a distance estimator method but that requires the derivative and as far as I know the only way of getting that for the number system used is the full derivation of the derivative of 4 functions of 4 reals, which I was too lazy to calculate smiley
Using the dstance estimator one could get a lot more detail in the smooth areas without losing it in the complicated areas (also rendering would actually probably be more efficient).

ah whoops, i thought you were using distance estimation for your recent renders. regarding the analytic derivatives, there's always maple (which can output c code for functions)! i've used it in the past to do analytic integration of functions over the sphere against spherical harmonic basis functions, it works a treat and seems pretty indifferent to the complexity of the function at hand.
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David Makin
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« Reply #66 on: June 09, 2009, 11:38:47 AM »

Ideally it should use a distance estimator method but that requires the derivative and as far as I know the only way of getting that for the number system used is the full derivation of the derivative of 4 functions of 4 reals, which I was too lazy to calculate smiley
Using the dstance estimator one could get a lot more detail in the smooth areas without losing it in the complicated areas (also rendering would actually probably be more efficient).

ah whoops, i thought you were using distance estimation for your recent renders. regarding the analytic derivatives, there's always maple (which can output c code for functions)! i've used it in the past to do analytic integration of functions over the sphere against spherical harmonic basis functions, it works a treat and seems pretty indifferent to the complexity of the function at hand.

cheesy Imagine how good it's gonna look using DE - or alternatively solid based on iteration density.
I think I'll have a go at trying it using DE this evening, it's not just the derivative that put me off though - at the moment I haven't a 3D formula of my own geared towards DE, maybe I'll try modifying a private copy of Ron Barnett's 3D Quat formula.
When playing with the 4D multiplication tables I noticed that these "True 3D" formulas mathematically speaking are most closely related to the standard hypercomplex method, for z^2+c they are a family such that:

new x = x^2 - y^2 - z^2 +/- w^2
new y = 2.0*(x*y o w*z)
new z = 2.0*(x*z o w*y)
new w = 2.0*(x*w +/- y*z)

Where o is +/-, this gives a family of 8, though for z^2+c it's 2 identical sets of 4. Of the 4 I've added 3 different non-hypercomplex types to my version of mmf.ufm:Solid-3D Quaternions and will upload it to the UF database this evening.


Unfortunately I don't have Maple (or Mathematica or similar software).
« Last Edit: June 09, 2009, 11:43:19 AM by David Makin » Logged

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cKleinhuis
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« Reply #67 on: June 09, 2009, 11:50:05 AM »

in fact it gets more interesting cheesy

can you create an image with ~250 iterations ?  afro afro afro
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« Reply #68 on: June 09, 2009, 11:53:10 AM »

You guys are totally crazy! What is more useless than a 3D brot? I love that smiley
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lycium
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« Reply #69 on: June 09, 2009, 12:27:46 PM »

Unfortunately I don't have Maple (or Mathematica or similar software).

no problem, just head on down to your local university and find a physics/maths professor; they're always chomping at the bit for interesting problems to solve (refreshing break from developing solutions in search of a problem). a good heuristic is to ask the one with the wildest, most unkempt hair  afro
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cKleinhuis
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« Reply #70 on: June 09, 2009, 02:39:25 PM »

You guys are totally crazy! What is more useless than a 3D brot? I love that smiley


two 3d brots?  afro
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David Makin
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« Reply #71 on: June 09, 2009, 04:33:48 PM »

in fact it gets more interesting cheesy

can you create an image with ~250 iterations ?  afro afro afro

You could, but in order to render and see all the detail it would have to be phenomenally large (to get the difference between 249 iterations and 250 iterations as big as a whole pixel even in the densest areas).
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« Reply #72 on: June 09, 2009, 04:35:05 PM »

What is more useless than a 3D brot?

Art ? cheesy wink
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« Reply #73 on: June 09, 2009, 04:47:06 PM »

What is more useless than a 3D brot?

Art ? cheesy wink

smiley I can't resist the temptation :-

Climbing mountains
Racing fast cars
Having sex with no intention of having children
Looking for weapons of mass destruction in Iraq
Using sanctions to stop countries nuclear weapons testing
Watching TV
Sunbathing
Finding the solution to Fermat's last theorem
Calculating pi to more than 100 significant figures
.
.
.
.
« Last Edit: June 09, 2009, 04:52:32 PM by David Makin » Logged

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lycium
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« Reply #74 on: June 10, 2009, 01:17:27 PM »

smiley I can't resist the temptation
nor can i... what a strange, strange selection of examples! the over-arching theme in my replies is that one should try to see the world beyond one's own chosen set of values, and that just because something is useless to you or me, doesn't mean it's intrinsically useless.

Climbing mountains
some like to make fractals, some like to climb mountains.

Racing fast cars
much as i dislike racing, not all recreation needs to be intellectual.

Having sex with no intention of having children
this one is by far the strangest; there's nothing wrong with consensual sex, and the last thing our poor planet needs is more people.

Looking for weapons of mass destruction in Iraq
this serves as a convenient excuse for all sorts of dodgy manoeuvres... no moral explanation however.

Using sanctions to stop countries nuclear weapons testing
it should somehow be strongly disincentivised, and limiting the growth of a militaristically-inclined nation (e.g. north korea) is difficult to logically argue against.

Watching TV
that's a personal judgement, but i'm with you 100%. it's a first class, fully addictive and mind-altering drug. then again, so is alcohol, which is notably absent from this list...

Sunbathing
personal judgement, and i agree, but some people want to attract others; instead of blaming the effect, why not find fault in the cause - that people find tanned skin attractive.

Finding the solution to Fermat's last theorem
so some purely intellectual pursuits (making fractals) are definitely less useless than others?

Calculating pi to more than 100 significant figures
as above, try not to be so biased tongue stuck out
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