END OF AN ERA, FRACTALFORUMS.COM IS CONTINUED ON FRACTALFORUMS.ORG

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)

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

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...

Oh... 4D Fractdimensional profit? what an interesting concept
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..

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...  
(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...........................................................................
............................................}

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
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....

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.