True 3D mandelbrot type fractal

[twinbee]

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

[David Makin]

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"

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.

[David Makin]

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
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  8hxpPusZy2himax6fVm99jdNnrPseVTrMrY4qIiW0K1wpmDiqzlDFty++HfoS22WUfca6q6B
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  2D7gVvEUBZYwO3c0GvngNs5Idln5ao9yMe1rwGBEslJL5VeG5XfSmQwJNp9KEjsXkYxGIIy8
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  QMRTiJaRMRWiJ6SMRGiJaSMRXiJ6TMRXiJyQMRHiJaRMRfiJyQMRGiJaSMRLiJyRMRXiJyQM
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  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

[David Makin]

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).

[lycium]

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

looks like you've got this one licked, congratulations are due for your fruitful directed efforts.

[lycium]

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
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.

[David Makin]

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
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.

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).

[cKleinhuis]

in fact it gets more interesting

can you create an image with ~250 iterations ? 

[bib]

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

[lycium]

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 

[cKleinhuis]

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

two 3d brots? 

[David Makin]

in fact it gets more interesting

can you create an image with ~250 iterations ?  

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).

[David Makin]

What is more useless than a 3D brot?

Art ?

[David Makin]

What is more useless than a 3D brot?

Art ?

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
.
.
.
.

[lycium]

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