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The YZ symmetrized Juliabulbs I developed a long time ago (in 3-d fractal years) were only continuous with seeds (a,0,0). Recently, I figured out how to get continous (a,b,0) and (a,b,c), but the big price one pays is losing all of the spirals. Also, making the fractals continuous is not the same as making them smooth (= continuous derivatives), so they still look a bit fractured.

Here, I'll show some of the "Mandels" with (a,b,0).

The code snippet for image 1:

   if (y*y<z*z) {d=y;y=z;z=d;}
   if (y>z) {x=-x;}
  
   r = x*x+y*y+z*z;
   x1 = (x*x-y*y)*(1-z*z/r);
   y1 =   (2*x*y)*(1-z*z/r);
   z1 =  2*z*sqrt(x*x+y*y);

   x=x1+a;y=y1+b;z=z1;

For image 2:
 Replace the (y>z) test with (x*y<0).

Here's some Julias of the XYZ-symmetrized quaternion with the discontinuities removed:

That is a great improvement of the julia possibilities, msltoe.  Though the z continuity is only because you don't add the c value to z, or is there another formula where you can use the c component (adding to z)  in julias?

Two pics, one from the minibulb on top (or on the downside in your images) of the non-julia bulb, and a julia seed of afair (0, 0.4, 0):

Jesse: Thanks for taking a look at this new find. I noticed the bulbous thing too but realized it was a lot like my earlier Juliabulbs. I was hoping for the antennae sticking out.

The following one statement should remove discontinuities for  (a,b,c) seed Julias and the M-set version:
  if (z<=y) {z=-z;x=-x;}

Top view and back view:

These are beautiful & the zoom Jessee did looks very promising for interesting details.
Nice to see it with the 'whipped cream' removed!

One more for good measure. A close up, top-view, of the (a,b,c) version. Near c=0 seems to be about the best.

Thanks msltoe, that works for all julias.
Unfortunately the nice minibulb vanished in the new version, you can't have everything...

Yes the a,b,c variant seems to lose a bit of the nice variants, although I think there may some similar bulbs hiding very close to c(or z)=0.
Here's the complete formula for the XYZ variant:

  if (x*x<z*z) {d=x;x=z;z=d;}
  if (x*x<y*y) {d=x;x=y;y=d;}
  if (z*z>y*y) {d=y;y=z;z=d;}

  if (x*z<0) {z=-z;}
  if (x*y<0) {y=-y;}

  x1=x*x-y*y-z*z;
  y1=2*x*y;
  z1=2*x*z;

  x=x1+a;y=y1+b;z=z1+c;

And a picture:

i would not believe that, until i zoomed a bit in and found something similar, looks like one can spend some more investigation:

ah, a fourth formula...
think i need an extra folder only for msltoe formulas in the next version  :dink:

interesting, the 3 first conditional symmetries seems to be made optional and the continuities still remain, the pictures are without them - cheesy bulb and a zoom-in that reminds somewhat on the suns surface, just for fun:
 

@ Jesse

The second pic looks really cool! Can you render a high resolution version of it? At least full HD? With the whole thing inside the picture?

It would so much make it into my wallpaper gallery!

I do not have the original parameters of the 'wildsun' image, but you could use these as starting point with the new program version 1.6.8:

Mandelbulb3Dv16{
K.....9/...o0...w....22...UACmOQ3BnazSZAp2RZzNzD.xsukiUufzfsLF/FgluuzcN907R0wvsj
................................tHa.qMFOo/2........A./..................y.2.....
................/M.0/....6Ut1...D1....E5.....6.yVYBRMPhD/..........c./...w1.BnAH
z.URBOhD....0..........wz.......................................................
........qta1CZXFfvfeMu8pE5onyEPFiCNxubfDll75dYLjdu1NS1w0L8ylyuKAOmn3VjiDZGW/PKOb
EvXqHplzSxguyUbsx8F9BIgD......oV0.....................sD..kz....................
.............................UWRa4.iqNO.6Rbd/UxRa4.OsNO.sXbd/UMSa4..............
..........................UF.k3.W....k6...EE....70....2....B....D/...2HB...UJl1I
...U.adGfwTa8h0d8jAn6/....6.nc..z8yz.wloWHC0esxjtNZq/OpVuz1...........EA8..IsY0.
HMdMN9PJKz1KdHh18A7uz0........../6U0.wzzz1................................E.0c..
zzzz.................................2U.8.kzzzD.................................
/6U0.wzzz1...................................AN6S.......eemod03.zzDU.EJJQlqc....
...mxBN6S.......QsrYVs/.......bTH4W5.......lyBN6S.......MwrYVs/......kqTH4W5....
...kzBN6S.........kYVs/.eemod03.IJ3LgB8.wyrYVs/.................................
E....2....E.....I....A....EHnl4RjJqItp4B.IaQ.........................2E./k.1A...
...................................................sz.........yD................
................................................................................
........................}

(did applied a softening filter afterwards)

Oh sorry, I meant MslToe-julia1.jpg. And I do not have any rendering software installed except povray. I thought it would only be 3 clicks for you. However, if you did not save your parameters, never mind :)

You could download mandelbulb3d in the download section for free, unpack the whole archive, start the program, copy the text parameters below, then press the 'open txt' button and render it on your own.  Of course the program is for win, but i cant post images bigger than 1200 pixels and you should also do something for it...  :dink:

Mandelbulb3Dv16{
K.....k1...Q6...w....2A...kLi8gX1zN..1.......U.E6QQ8uYGRxyXD4hW5lLxkzuafVPYpFByD
................................onQRjZCwoz1........A./..................y.2.....
................/MU0/....6.p..../0....kC.....YCKWXHKJjnD/..........c./...w1.OaNa
z.EnAnYD....02...........cNaNaNaNaxD............................................
........CzkCqINHZx96JUxE8lxLzCSdfke8W/pjKFdGLy9IUxHd1Kyz58cNz0QSSyh34KmDLA8Vyt3Z
Bx9gQwe84qWDzO97evy0FeqD......o1......................sD..kz....................
.............................UWRa4.iqNO.6Rbd/UxRa4.OsNO.sXbd/UMSa4..............
...........................U..6.C....o0...EB....Y/...g2...kB....8/....2....UJl12
...U.OXRkyDM6/nzeSwpz5qEx.6.lc..zzzz.cVoK/nl2xvjQvM93P58iz1...........U28..OfE0.
aFBqwnQahznj.u68Yrpuz0........../6U0.wzzz1................................E.0c..
zzzz.................................2U.8.kzzzD.................................
/6U0.wzzz1...................................UTVR.kz.f2.ee0U.06.kzDw.EJJsnwF.UDn
7/.mxVTVR.kz.f2.Qs5y3q/.z1gG..bTsLM5.wDk8/.lyVTVR.kz.f2.Mw5y3q/.z1gG.kqTsLM5.wDk
8/.kzVTVR.kz.f2....y3q/.ee0U.06.IJ3yAT2.wy5y3q/.................................
E....2....E.....I....2....EHnl4RjJqItpaA.IaQ....................................
...................wz.........zj................................................
................................................................................
........................}

It says "Error, formula option is not valid."
Doesn't this text configure the formula?
Which one do I have to select?

I can't find a way to enter the formula above, nor to translate the formula above to an m3f file...

Oh boy, I'm feeling like a stupid user. If I annoy you to much just let it be, I can understand it :P

Mircode: the copied parameter text worked for me. Make sure you are using the latest M3D v.1.6.8 code as he just added my formulas.

-mike

Oh boy, what a stupid user  ;D

kidding, just make sure to copy the whole text including mandelb... and the brackets, if it is still not working with program version 1.6.8 you have to specify the directory where the m3f formula files from the archive are stored in by clicking on the "INI DIRS" button + restart.
You should have a 'MsltoeSym2' formula in the formula window under '3D'.

I configured everything via INI DIRS and restarted, still didnt work. But manually selecting Mstltoesym2 helped and I could finally render it. :D

http://dl.dropbox.com/u/11704339/MsltoeJuliaBig.png (http://dl.dropbox.com/u/11704339/MsltoeJuliaBig.png)

 I tried a slightly different approach a few months back, which resulted in a nice continuous (not in the derivative sense :)  ) fractal for z^2,6,10...:

  x1= (x*x-y*y-z*z);
  r=(x*x+y*y) ^ -.5;

  if (y>abs(z))
      { y1= 2* x * (y*y-z*z) *r }         else
      { y1 = - 2 * x * (y*y-z*z) *r }

  z1 =- 4 * x * y * z * r

  Didn't mess with julias of it too much however.