An old formula revised

[M Benesi]

  Well, I messed around with a new method of doing the 4 dimensional version of this formula.  It added some interesting characteristics, unless I've forgotten about there existence in the year or so since I've played with this particular fractal.

  Anyways, I took something from my "Cheater's  Brot":  I basically formatted the pixels so that instead of the fractal being aligned with the xyz magnitude, the fractal is initially aligned with the x axis.  From here, I simply used the x-axis as the 4th dimensional input- the other 3 axes being shifted by 45 degrees one way, and the "magic angle" the other way (magic angle is an NMR thing.... and happens to be of mathematical use (surprise surprise) as well).  

  Since the other axes were shifted, this prevents the x axis pixel value from being double implemented, although technically it is inputted twice- one of the times it is shifted.  This opens up possibilities for other fractal types as well:  we can take the total magnitude (or "complex" magnitude*) and align it to the x axis by shifting by the magic angle, then 45 degrees.  We can then use this value as a 4th dimensional input that is related, but not the same as, our other input values.

  anyways....  Some images of the z^2 4d.  Some new formations, similar to old ones, but with some additional details (click the first too enlarge):


  

[M Benesi]

  And here's the place at the back of the z^2 (elephant valley of this particular formula) that I find particularly interesting.  It's Mayan, Incan, or Aztec looking.  Interesting that simple mathematical formulas produce images that recall a very mathematical people..



And here it is with only "2 spokes" (same formula, just... arranged differently):


makes nice Julias too  (almost all click to enlarge):

[KRAFTWERK]

Beautiful bulbs Mr Benesi!

[M Benesi]

thanks 

[Alef]

Interesting fractal. One more version of Holly Grail;)

[M Benesi]

  Yeah...  

  I like this next combo even more.  I combined the rotated 3d (three!!!) version, in Julia form, rotated it back to mandy coordinates, calculated the mandy part, added in the ROTATED x pixel component (well, for most of them... it looks tighter that way) without any julia components, rotated the thing back to Mag xyz coordinate system, and... well.. you'll see.

  I didn't post any, but if you cut into the fractal, there are interior caverns that are incredibly detailed as well!  This is a neat one, although it is calculation intensive!!  lots of compute cycles.  Wonder how the 4d combo will turn out? 

  All of these are z^2 julias at -.5 for x,y, and z.... click to enlarge... maybe.


  the first one is with non rotated x pixel value, the zoom into the ball is with rotated x pixel


  The last one there shows you that there are stalks off of the ball, that is actually a tiny ball in front of a ball, in front of a ball, down a bit.. etc.. etc..    Rotating and combining works wonders...  

[KRAFTWERK]

Now, this is getting more and more interesting!

 

[kram1032]

It looks a whole lot like what people might imagine the holy grail to be like.
How does the MSet of this particular version look like?

[M Benesi]

  The Mset from this is a bit screwed up.  Doesn't look way too great. I've figured out what I believe will be considered a canonical grail 3d Mset, but can't implement it in Chaospro without doing tons of unnecessary work, this one is a LOT easier.  Once I find a decent program that lets me calculate by x-planes, I think we'll have the mset.  If you're curious about my hypothetical mset, we can start a new thread- but I can't render it in ChaosPro.

  Update:  Mset is good.  Just have to use 1/2 pixel values or it gets too crazy.   Very similar to the julia, although a bit more wild.  Playing with higher n z^n for a bit.  Will get back to z^2 later...

  One problem with the Mset- too much detail increase per iteration... probably have to play with bailout.  In addition, for whatever reason, Julias require lower resolution than Mandies to be crisp (because the pixel seed is used less!).  Also, adding in the y and z pixel values stretches it in places, not as bad as a regular mandy, but it's not as uniform and fractally.

  I'm having the best luck with julia type pixel initiation, setting the first part's julia seed to -.5,-.5,-.5  (I'll explain this later, if anyone is going to try and implement the formula), then adding in 2/3rds (.66) of the x pixel component after the second calculation.  I picked 2/3rds because I like it the most... for now, also I believe that the length of the Mag x XYZ fractal is about 1/sqrt(2) shorter than the mset... so 2/3rds is close... probably should change it to sqrt(.5), ehh??

  If you use the total x component, you get the stalky thing that Kraftwerk commented on.

  So. here are a couple more-  the first one cracks me up, and has some grail like features- it's on the main stalk leading down the -x axis.  Funny as hell...  the next ones are from the elephant valley section, the first being a zoom into the frilly part to confirm it doesn't behave like a Mandy and lack cool fractal details (it has details), the second being a zoom into a hole off of elephant valley, took a left turn, zoomed into another hole, and started to find twisty interconnected stuff at iteration 9!! (well, much more zoomed, and more iterated- will post the bit of twists (image 4) when it's done calculating).



  Ok.. here is that 9 iteration one.  There are 2... same image, different coloring... dunno... I think the bluish is better, you can see how the stuff is interconnected here in the elephant valley...



  Now, you can set it to the actual x-axis pixel value, and get this nice stalk thing at the end, and I assume you can find the spherical parts back towards where seahorse valley is.  The x-pixel/sqrt(2) one has slightly stretched spheres towards the stalk, but once again I assume they are more spherical back towards the valley.  X-pixel*.66 has spheres right up the stalk... but it doesn't have those stalky stalks... soooo...   The last one is a deeper zoom into the elephant valley from earlier.



Stalk:

Somewhere in elephant land:

[kram1032]

Really nice but what about the whole set?

[M Benesi]

  here....  The last 3 are the same image, with the yz angle stretched differently for pixels.  Standard has THREE sections, corresponding to x, y, and z values (just like the original mag vs. xyz).  Keep in mind the partial Mandies (x- axis only) look better at first glance (less chaotic), so I'm posting them instead of the full Mandy (x,y,z).
the side, elephant valleyL:


  The pointy end, different yz angle stretches (for pixels):

[M Benesi]

  Here is a comparison of the stalk with the plane old x axis component, followed by the sqrt(2)*x axis component (which will be up shortly).  Keep in mind that with either of these, as you travel away from the negative x axis, you start running into less "stalky" more spherical parts of the fractal.   This means it might be unnecessary to use the sqrt(2) x axis component- you can just travel towards the positive x axis and have the more spherical outcroppings.



  Here are a couple in which I did the Mandy before the mag xyz.  Works as well, but has more of a Mandy shape.  The flat mandy variety doesn't work that well, still using the one I consider the most faithful to the 2d Mset. 

  Of course, you need to do the CORRECT coordinate transform between the 2 formulas to get it to work correctly.  This is probably why nobody has done this in Mandelbulber or the other formulas yet.  There are a couple other little tricks as well.  anyways-  here are the images.

  First one is standard, the second one with my z^2 improvement switch (from an old thread).  I like the standard more, not necessary to add details or size to an already detailed fractal.

  Did I mention these are all z^2? 

[DarkBeam]

When you finished mangling leave the formula to us

[miner49er]

Does this have the regular Mandelbrot set as a 2D cross-section?

If so then...wow!

[kram1032]

this certainly is darned close. I do see some quaternion-like whipped cream elements, but for once there is more detail than cream. (As far as I can tell)