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There is also a 4D version of spherical coordinates.
The little film belows shows the 4D (degree eight) Mandelbulb, projected into 3D when the value of the fourth coordinate changes from -1 to +1...



http://www.youtube.com/v/8sojcs5yhHI?fs=1&hl=nl_NL&rel=0

(done in Ultrafractal)

This looks pretty cool :D

Could you do 4D transforms on this and project it in different ways? :)

that was cool! it would be good to magnify slightly as it decreases in size.....

As to transformations in 4D, here is a rotation in 4D around the x-y plane, which changes the shape of the bulb.
(the rotation around the vertical axis is just a simple rotation in 3D)

http://www.youtube.com/v/Ktqm_vYy5K8?fs=1&hl=nl_NL&rel=0"

Amazing JosLeys, would love to see more of this...  O0

Very nice videos Jos. 

Looks really nice :)

Very nice videos, I like the style : simple yet rich graphics to show some mathematical features.

Is this the birth of the 4d power8 bulb?
Great!

If this is the birth of the 4D bulb, I'm afraid the baby looks just like it's 3D relatives...

The 4D version does offer some new views.
Here is a fourth-power Julia, that I do not think can be reproduced by the 3D formulas. (the Julia constant has a value for the fourth coordinate)

Maybe your 3D versions have some parameters to get these 4D variants, i can't do them.
Nice Julia, btw.
I am still eager to find a minkowski 4d variation, so how much must i pay or beg to get the formula  :D

Those 4D mandelbulb are great! ;D

It is possible to render full 4d objects (not just slices) by projecting all 4 dimensions onto the view frustum...
It would be good for viewing these things as for instance you would never see disconnections in a set that is fully connected.
You sometimes see what looks like self intersection, but that is just overlap due to viewing angle.

Jesse, you asked about the formula. Here it is :
(it's basically the same as my 3D code)

Ahhh, a version 1.6.2. coming up Jesse? ;)

Ah, thank you!
I already experimented with more sine/cosine options in the 3d case, but nothing very special found yet.

Maybe i can make an extern formula out of it, but it would be somewhat slow with the 4 point DE...


PS: looking at the code, i assume that RRdz and Rdz is the same variable?

  Congratulations on your discovery, although temper it with the knowledge that like most great discoveries, it's been around for a while:

  The 4d, and higher (I made up to 10d way back in January), have been around since close to the rediscovery of the bulb (someone else was working on them around March of this year, and in fact I think the topic was mentioned in the original thread, although I wasn't aware of any implementation before mine).  Here is a 6 dimensional bulb video (crappy, didn't figure out coloring methods yet) I posted on youtube January 24th of this year:

http://www.youtube.com/watch?v=9Jzfdr1Zh6Y (http://www.youtube.com/watch?v=9Jzfdr1Zh6Y)

  Anyways, there are VERY awesome pseudo-julia that can be created using the correct coloring methods, etc.  with the 5d bulb.  I posted a thread over in the images forum with a few images from a 5d pseudo-julia.  Each iteration creates a great variety of patterns as one zooms straight into the fractal (the pattern changes each iteration as you zoom deeper... it's amazing, although the pseudo-mandel variety r2 modes 1-4 have some variety as well). 

  Also, for the both of you (Jesse and JosLeys) I'd like to point out that the complex triplex, quadplex, and quintplex (untested, but assumed) math versions I developed are slightly faster for cpu implementations (not sure about gpu implementations).  As I've a crappy gpu I can't compare the complex triplex... versions to trig versions on a gpu (many of which I guess to have trigonometric optimizations, but maybe not?). 

  For example, for the degree 9 (z^9) 4d bulb @ 10 iterations (.2 k value), the trig version took 75.813 seconds and the complex quadplex version took 47.453 seconds:  in other words the trig version takes 159.8% longer for the same exact image.  This adds up, especially when you render larger images (than these 500x500 test images)...

  Now the difference is lower for less iterations... so when you increase iterations (deep zoom) the quadplex version really starts to pay off (at 5 iterations, there is only ~ +120% time for the trig version). 

  500x500 zoomed in:
  21 iter trig: 2:26.172     vs.     21 iter quadplex: 1:21.109   =  180.2% difference!!!!

  NOTE:  Complex derivatives are fricken simple... and all these should pass Cauchy Riemann (http://mathworld.wolfram.com/Cauchy-RiemannEquations.html) conditions.  Derivative of z^n?  n*z^(n-1) 

   I'll start a thread for the pseudo-julias, later tonight or tomorrow.  Pretty busy as my brother is arriving in town tonight... but the discovery was very simple (and probably already exists somewhere else, as this thread shows, someone's probably already done it elsewhere in the forums before now). 

  I just take the starting x, y, and z values, subtract the new values and add in whatever julia seed value I was using.  To extend this to 5d, do the same thing with your starting x,y, z, k, and l values, subtract the new values and add in your julia seed.  Simply phenomenal....  Check the variety on this 11 iteration 2048x2048 image (done with x,y, and z values sandwiched between the k and l values, you can vary the placement of the 5 dimensions for different images):

http://lh6.ggpht.com/_gbC_B2NkUEo/TIc-2TiNOvI/AAAAAAAAAp8/lwUI1FxYTnw/11%20iter%202048x2048.jpg

Yes, sorry. Just a typo.

No problem, for my test implementation i did not need it anyway.
It seems to work so far, a quick test with a quaternion as hybrid...

(http://www.fractalforums.com/gallery/3/1127_08_09_10_11_21_36.jpeg)

Now i will try a boost instead of a rotation for 4d... (in the next time)

Just wanted to say that my experiment with a hyperbolic version for theta1 gave no good result.
But i am not an analytic mathematician, so multiplying the Arcsinh(w/R) with the power and using the Cosh and -Sinh out of it may make no sense at all?
The sense should be that our universe is basically a minkowski 4d spacetime, so i am trying to use the fourth dimension in a similiar way.

By mistake i found a way to inflate bulbs, so if someone's bulb needs some air or water just call me:  :no:

No need of air or water here, but if you find a way to fill them with Beer, let me know!!   :toast:

Now i know what the holy grail would be  :D

  You add in a 4th component to your magnitude without a rotation? 

Yep, i forgot to scale theta1 so the bulb turned into a spehere with higher cw values.

You are the 6, 7? dimension man, so no problem for you.
I just wish your images would not look like ultrasonic scans, so as they were pre-birth scans  :dink:

An deflating?

  I like the way they look... the contrast really makes them pop (to my mind's eye).  I made a similar "mistake" back when I first tried 4d bulbs... the same one.  You can do some neat stuff with it if you do it right however (the non-rotated magnitude component):

http://www.youtube.com/watch?v=K597mPrU61E (http://www.youtube.com/watch?v=K597mPrU61E)

Ok, the viewing habits may vary, for me it is sometimes hard to distinguish between the 3d structures and what is coloring.
I would like to see SSAO from chaospro, have you tried it? Should be fast, independently of the formula.
But thanks for the good work anyway!


That is neat, yep.
The hope is still to find a bulb with speheres on spheres... something different from the bulbs so far.

  I know it's a tangent, don't want to hijack the thread, but the structure of the coloring is really part of the magic of fractals.  You've got to consider that a 2d mandelbrot, just looking at the edge, without coloring all the different iterations, it's pretty boring.  With the 3+ dimensional formulas, you can't do anything but "look at the edge" (well, you can slice it, but that's not what we're aiming for), and one way to see the interesting fractal structure is to color it with various iteration values.


  The AO only mode looks a lot like the sample on the SSAO wiki page, so... yes.  Can mess around with that a bit, however I still think the real structure of fractals lies in the coloring schemes, and just like a 2d painting can be made to be 3d by perspective effects, 3d fractals can be made to pop out with coloring (instead of simply shading).  I think IQ (Inigo?) mentioned orbital coloring as a method of popping out the fractals a long time ago...  anyways..

  In the following 5d (not 4d) fractal, you can see that the method of coloring highlights mathematical "valleys" (shadows), pops out the 3d details, and creates a pretty texture to further enhance the look of the fractal. 

(http://lh3.ggpht.com/_gbC_B2NkUEo/TIxV8PcqjEI/AAAAAAAAAqE/yvP6_ljwgHA/dot%20294%20dot%203682.jpg)

   Yeah, but I don't think that will be "the" 3d Mandelbrot, as the apparent 3d extension of the 2d complex formula is the 3d complex formula I posted in the "3d Mandelbrot Formulas" sticky-thread.  It just makes sense that the 3d version has greater distortion due to the additional imaginary component... thus the distorted fractal dubbed the xmas tree fractal (by ???)...