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Here is a summary of some different 3D Mandelbrot set formulas for your convience. I have tried to include a second order and 8th order rendering for each formula presented here. Additional formulas are welcome.

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Hi Paul, you might want to try normalising that last one to 1 before multiplying by the magnitude - if the magnitude of the trig is zero then use (1,0,0) multiplied by the magnitude.

Thanks for all these.  That is the sort of summary I requested back in the original thread.  This will really help others looking for formula info.

This is the method that I used to create my 3D versions

(http://4.bp.blogspot.com/_RtmvIFH1wRs/Sw8a4t8knHI/AAAAAAAAAEE/Jm0tnxd_hj4/s1600/doughowto.jpg)

Doug, I don't know enough math to follow that; could you turn it into a formula so I can see how new x,y,z values are created?

Ted

Here's a link to that question that I've all ready put into a post

http://www.fractalforums.com/theory/has-anyone-tried-this-formula/msg9198/#msg9198 (http://www.fractalforums.com/theory/has-anyone-tried-this-formula/msg9198/#msg9198)

(http://mandelcousinPower3)
(http://mandelcousinPower3top)
These top two aren't a different formula, they're using the second method down in the posted list, but I don't think even powers are appropriate for this type as it makes the iteration discontinuous in space. Power 3 is my favourite, it would be interesting to get some more detailed shots of this. I'm fairly sure it is fully connected, and shows a lot of variety.
(http://mandelPower2)
(http://mandelPower2side)
These bottom two from the third formula down in the table. The iteration should be continuous in space at odd and even powers, its certainly a bizarre and interesting power 2 shape.

Possible typo in the definitions at the top of this thread (image http://www.fractalforums.com/theory/summary-of-3d-mandelbrot-set-formulas/?action=dlattach;attach=524;image): r is defined as the nth power of the original vector length, and then the definition of phi uses r in the denominator. I think this value in the denominator should actually be the original vector length, not its nth power.

Thanks for pointing that out. I made the correction.

I am getting confused as to why I cannot reproduce the exact same shapes of 3D mandelbulbs when I plug above formulas into a voxel generator. I do apply the exact same iterative formulas in a spherical coordinates, yet the voxel representations of the bulbs look distinctly different (especially apparent for power 2).

I am beginning to think that maybe a 3D raymarching approach based on Hubbard Duady does not really give the "correct" result. Could anyone verify this hypothesis or prove it wrong?

Couple more formulae for you paul.

(x,y,z)² = ()  left hand version.

(x,y,z)² = ()  right hand version.

Do you want the correct vx versions as well?

  I've added 4 new formulas, starting in post 114 of this thread:
http://www.fractalforums.com/3d-fractal-generation/truerer-true-3d-mandelbrot-fractal-(search-for-the-holy-grail-continues)/msg12604/#msg12604

  They definitely don't strictly stick to the accepted Mandelbulb format, but they are all 3d fractals (not simply "regions inside level sets for 3d fractals" (quoted from an unnamed source, haven't asked permission to use their name)  like the previous formulas I posted in the thread).

  Images are in the thread.  As it is, some of them are pretty cool and have really neat fractal symmetry (I like Type B (not Mandelbulb B) the most).  Zooming into the point (front) of type B higher order Z^8++ is pretty fun.

  If you think they are appropriate for this thread, go ahead an import them.  May have a mod move them over here and put a link to this thread from that one.... or whatever.

  matt

In case someone is interested, i put together a site, http://sites.google.com/site/fractals3d/home (http://sites.google.com/site/fractals3d/home), for what I feel is a more geometrically correct multiplication, operating on geodesics.

--Ingemar Eriksson

Can you show us a high resolution render of your z^2+c version ?

Nice, can you post here the formulas?

I'll try to do it in the coming days. I assume you refer to power 2 mandelbrot and not julia set?

I'll post the code snippet as well.

Yes, I meant the Mandelbrot :)

Hello Ingemar, your version looks very much as one i stumbled upon experimenting, a one i THINK is the same as Tglad mentions earlier in this thread (cosine method), here:

(bottom two images)

http://www.fractalforums.com/theory/summary-of-3d-mandelbrot-set-formulas/msg9207/#msg9207

Thing is that the one in this image has a assymmetric top/bottom in power 8. BUT the power 2 looks identical to yours.
Any formulas?

J

Ingemars image:
http://7745431616379112976-a-1802744773732722657-s-sites.googlegroups.com/site/fractals3d/home/real%20ingemarbrot%20-%20power%202%20-%20maxit%2010%20-%201200x1200.jpg?attachauth=ANoY7crTz4OffVZae10r0dMa67y0CQhWZkxkNmh3WqI8pwd0940yXIc6I1jLPWYVXiOFVl86-_4kXZNYJ_qscuLNXJeot9D3PQLeqOW8C8dAJikvW7tviEBmWGaXMTz5LcIonrBW1snHZE4zNhUOX6G6Rc78NW_hp7-eQPFu_znhc3UvYPZPRorxO6qum3ZbmA5_vCpNtT8lmZdmD8IUbccTP02nuy8AQopwTtvtBhxhH-hFlOHB9eC7QXiQfjKImwt6fzWA9XYP&attredirects=0

And here is my render (with modified Subblue pixelbenderscript)

Hello all,

I have lurked here for way too long so decided to join the party.

After experimenting with the triplex algebra and alternate formulas for a while now, I have a few things to contribute.

Firstly, maybe this topic should be called "Alternate triplex algebra" as the variations are based on modifying the maths behind the triplex algebra and not the formula z=z^p+c.  After implementing most of Paul's listed variations at the start of this topic I tried a few others;

1. "IQ variation"  This comes from Inigo's page here http://iquilezles.org/www/articles/mandelbulb/mandelbulb.htm.
2. Negative IQ.  Same as IQ but with a negative z compent on the triplex (same as negative SIN compared to SIN)
3. Negative COS.  Same as Daniel's original COS version with a negative z.
4. Negative Rucker.  Again, same as the Rudy Rucker variety with a negative z.

They all give unique results and new bulbs to explore.  There probably are a whole bunch of other variations to explore and render.  If anyone has other ideas for other variations please share.

Some example images of the variations of the power 8 mandelbulb and other formulas here here
http://softology.com.au/gallery/gallerymandelbulb.htm I also have links to my flickr galleries with zoomed in sample images.

The more complex formula varieties do lead to more disconnected and "dusty" fractals (which is logical as it is like comparing the 2D Mandelbrot to the more complex versions of the mandelbrot formula), but when zooming in they do give some unique new looks.

Jason.

Thank you for that summary, some interesting variations there.

If you could add the power 2 renderings aswell it would be great.

One thing I do miss is renders of top vs bottom of the non symmetric power 8 bulbs. I have found it out for myself now, but it took a while, would be nice to have here in your list bugman!

Thank you for your new variations Softology

The "Mandelbulb Power 8 Reversed Order Positive Z" looks very interesting in pow 8!!!

Looks a bit like hobolt:s riemandelettuce, is it the same formula/coordinate system?

http://www.fractalforums.com/3d-fractal-generation/riemandelettuce/

(http://softology.com.au/gallery/mandelbulb_power8_reversedorderpositivez.jpg)

I hope it is OK I linked the image here...

The variations I show are all based on the usual triplex algebra at the start of this topic.  The listed algebra is the SIN version;

SIN

theta=arctan2(z.y,z.x)
phi=arcsin(z.z/radius)
costh cosph
sinth cosph
sinph

And the other variations have different trig when calculating theta,phi and the x,y,z components of triplex z when doing cartesian to polar conversions, exponents and multiplication.  From what I can see the riemanlettuce uses a different system entirely and creates a more discinnected surface.

-SIN

theta=arctan2(z.y,z.x)
phi=arcsin(z.z/radius)
costh cosph
sinth cosph
-sinph

COS

theta=arctan2(z.y,z.x)
phi=arccos(z.z/radius)
costh sinph
sinth sinph
cosph

RUCKER

theta=arctan2(z.y,z.x)
phi=arctan(z.z/z.x)
costh cosph
sinth cosph
sinph

Reversed Order

theta=arctan2(z.y,z.x)
phi=arcsin(z.z/radius)
costh cosph
sinth
-costh sinph

Reversed Order Positive Z

theta=arctan2(z.y,z.x)
phi=arcsin(z.z/radius)
costh cosph
sinth
costh sinph

IQ

theta=arccos(z.y/radius)
phi=arctan2(z.x,z.z)
sinth sinph
costh
sinth cosph

Negative IQ

theta=arccos(z.y/radius)
phi=arctan2(z.x,z.z)
sinth sinph
costh
sinth cosph

Negative COS

theta=arctan2(z.y,z.x)
phi=arccos(z.z/radius)
costh sinph
sinth sinph
-cosph

Negative Rucker

theta=arctan2(z.y,z.x)
phi=arctan(z.z/z.x)
costh cosph
sinth cosph
-sinph

The variations all seem to work with formulas other than z=z^p+c, so finding new variations leads to a whole bunch of new fractals to explore when using multiple formulas.

I would be interested in any other variations other people have tried using the above style changes.

Jason.

Hi Jason and all.

Actually there are 9 reasonably sensible angles you can choose from to use in a Mandelbulby-style manner - with tangents:

   Type 1:         x/y           ,            y/z           ,          z/x     -> or the other way up
   Type 2: x/sqrt(y*y + z*z),   y/sqrt(x*x + z*z),   z/sqrt(x*x + y*y)
   Type 3:         x/m           ,            y/m          ,          z/m       where m is sqrt(x*x + y*y + z*z)

Also you can mix and combine these in multiple ways e.g. using 2 as in the Mandelbulb or using 3 or maybe even 4 !!
You can combine 2 in the normal Mandelbulb way or you can combine them say in ways based on normal 3D viewing transformations or.....

I've tried a number of alternatives in my wip3D formula for UF:

http://www.fractalgallery.co.uk/MMFwip3D.zip (http://www.fractalgallery.co.uk/MMFwip3D.zip)

When I get chance I'll post more details on the ones I've tried in this thread but I'm currently not getting enough time to work on improvements, let alone on adding documentation !!
If you're OK reading UF code then you should follow the ones I've tried if you look at the formula (wip3D5), though there are precious few comments and some of those may not be correct (after cutting/pasting). I found that in the cases I've tried a simple adaption of the Mandelbulb analytical DE method works but in some cases required extra scaling.

How do I use these variations with the usual triplex algebra?  What part of the calculations do they replace?

To make my previous summaries more complete.  This is for the Nylander Positive SIN version;

(converting cartesian to spherical polar)
radius=sqrt(z.x*z.x+z.y*z.y+z.z*z.z)
theta=arctan2(z.y,z.x)
phi=arcsin(z.z/radius)

(z^p)
radius=power(radius,p)
theta=theta*p
phi=phi*p
z.x=radius*cos(phi)*cos(theta)
z.y=radius*cos(phi)*sin(theta)
z.z=radius*sin(phi)

Which parts of the calculations do I need to change to implement your variations?

Please all remember that I'm approaching this from the point of view of finding interesting fractals to render and ignoring all but basic maths :)

The simplest ones to try are simply replace phi or theta with any 2 of the 9 angles I gave the tangents for, otherwise do the same calculation with the normal modifications available such as reversing the signs etc. Obviously some combinations make more mathematical sense than others :)

I forget exactly what else I did (I'd have to check my own code) but I think one method I tried was using the first 3 angles i.e. from tangents x/y, y/z and z/x giving th, ph and de and combining the sines/cosines as follows:

newx = r*cos(th)*sin(de) + cx
newy = r*cos(ph)*sin(th) + cy
newz = r*cos(de)*sin(ph) + cz

Although I'm not sure if that's what I did without checking, I do know that the above has a problem in that it does not maintain a unit scale from the trig calculations so to avoid the problem I renormalised the trig part before multiplying by the new magnitude (r) and adding the constants - if the trig part was (0,0,0) then I used newx=r+cx, newy=cy and newz=cz.

Just tried another idea for getting a "true 3D" Mandelbrot.
Basically I thought for r,i,j then treat r,i and r,j as two complex numbers, raise both to a power, combine the real parts of the results somehow and use the imaginary parts as the new i and j respectively, there was an issue with respect to magnitude but that's simply fixed by scaling the result, here's the UF code I came up with (magn is the magnitude of the triplex on entry):

              magn = magn^@mpwr
              zjk = (real(zri) + flip(zj))^@mpwr
              zri = zri^@mpwr
              zj = imag(zjk)
              x1 = real(zri)*real(zjk)
              if x1>0.0
                zri = x1/sqrt(abs(x1)) + flip(imag(zri))
              else
                zri = flip(imag(zri))
              endif
              x1 = sqrt(|zri| + sqr(zj))
              if x1>0.0
                magn = magn/x1
                zri = magn*zri + cri
                zj = magn*zj + cj
              else
                zri = cri
                zj = cj
              endif
              magn = sqrt(|zri| + sqr(zj))

The result is more similar to the "true 3D" attempts using the unit multiplication table variations than the Mandelbulb so I was going to discard the method completely but I decided to try applying the generic Mandelbox-style square folding and the result was extremely interesting....will post a result in this thread - about 20 minutes to go.

Here you are, click the link if no image appears:

"Mediæval Hall"

(http://fc00.deviantart.net/fs70/f/2010/080/4/6/Medaeval_Hall_by_MakinMagic.jpg) (http://makinmagic.deviantart.com/art/Mediaeval-Hall-158035380)

  This is just a variation of the "Christmas Tree Fractal" formula from Paul Nylander's website (that I accidentally recreated).  Anyways, if you don't take the absolute value of the y pixel component, it is more "true" to the classic 2d Mandelbrot, in the sense that the 'distortion' (asymmetry)  introduced by the - x component in the classic 2d Mandebrot for even n (z^2,4,6,8....) is magnified by the y component's interaction with the z component.  I'd think that this was "the real deal", simply because it follows the same pattern of distortion as the 2d Mandelbrot, but it isn't as pleasing as something that replicates the look and feel of a 2d Mandelbrot for even z^n.

Here is the complex triplex version, which should give some insight into the mathematics going on (and it's about 2 times as fast on my CPU):

Trig version of above code, with abs of y pixel component (which introduces symmetry).

Hi everybody,

This is my first post in this forum and first participation to the quest for the MandelGrail.

Recently, I tried to see how changing the coefficients of a quadratic "multiplication" could modify the shape of a Mandelbrot 3D.

Here is the iteration function z^2 + c :
x = v * mx * v + cx
y = v * my * v + cy
z = v * mz * v + cz
where v = (x, y, z) and mx, my and mz are 3x3 symmetrical matrices, with coefficients mxij . So, this can be written :
x = mx11 * x^2 + mx12 * x*y + mx13 * x*z + mx21 * x*y + mx22 * y^2 + …,

Let us start with the « revolution Mandelbrot », defined by mx11=1, mx22=mx33=-1, my12=my21=1 and mz13=mz31=1 (other coefficients = 0).

(http://img163.imageshack.us/img163/507/mandelrev.th.png) (http://img163.imageshack.us/i/mandelrev.png/)

From there, we can slightly change the mij coeffcients, to see the effects on the shape. I will pass on all the coefficients that distort the shape in an « unfavourable » way (i.e. that clearly move us away from what a « 3D Mandelbrot » should be)
 
Here are some interesting results :
- changing my22, my23/my32, mz23/mz32, mz33 distorts the shape longitudinaly and creates a lateral « fractal spoke » in directions y and z. Watching from the front of the shape :

(http://img837.imageshack.us/img837/4848/spoke.th.png) (http://img837.imageshack.us/i/spoke.png/)

- changing my33 kind of « splits » the shape in the yz plane in an interesting way. Actually, when you combine the creation of a fractal spoke (on mz23/mz32) and this splitting (on my33), you get three spokes!
To make them identical and equally distributed around the x axis, the « best » parameters seem to be mz32=mz23=0,5 and my33=1/3 (or their opposites).

(http://img39.imageshack.us/img39/275/pow2b.th.png) (http://img39.imageshack.us/i/pow2b.png/)
(http://img571.imageshack.us/img571/7025/pow2.th.png) (http://img571.imageshack.us/i/pow2.png/)

From there I tried to create additional spokes but could not do that just by changing the matrices coefficients. Since we want some kind of symmetry around the x axis, I tried to see how the iteration equations looked in the following polar coordinates :
x =  r cos ph
y = r cos th sin ph
z = r sin th sin ph

This gives us, in the iteration equations :
x = r^2 cos 2ph + cx
y = r^2 cos th sin 2ph + 1/6 r^2 sin^2 ph (cos 2th - 1) + cy
z = r^2 sin th cos 2ph + 1/2 r^2 sin^2 ph (sin 2th) + cz

The first coefficients are the ones of the « revolution Mandelbrot ».

The second coefficients in y and z equations are similar to a Mandelbrot equation in the yz plane, multiplied by a coefficient (r*sin ph) (i.e. r projected in the yz plane).
This gives the idea to choose a different power for th, and write :
x = r^p1 cos(p1*ph) + cx
y = r^p1 cos(th)sin(p1*ph) + 1/3 r^p1 sin(p1*ph)*cos(p2*th) + cy
z = r^ p1 sin(th)cos(p1*ph) + 1/2 r^p1 sin(p1*ph)*sin(p2*th) + cz

with p1 the power for ph, i.e. along x (p1=2 if you want a shape similar to the classical Mandelbrot)
and p2 the power for th, for the shape in the yz plane. It is as if you « combined » two Mandelbrot figures with different powers, one along x and the other along y and z.
When you increase p2, the number of spokes increases in a way similar to the way the 2D Mandelbrot evolves when you increase its power.
(Below for p2 = 3, 6, 20)

(http://img820.imageshack.us/img820/825/pow3b.th.png) (http://img820.imageshack.us/i/pow3b.png/)
(http://img267.imageshack.us/img267/7627/pow3.th.png) (http://img267.imageshack.us/i/pow3.png/)

(http://img267.imageshack.us/img267/7910/pow6b.th.png) (http://img267.imageshack.us/i/pow6b.png/)
(http://img185.imageshack.us/img185/9694/pow6.th.png) (http://img185.imageshack.us/i/pow6.png/)

(http://img46.imageshack.us/img46/9674/pow20b.th.png) (http://img46.imageshack.us/i/pow20b.png/)
(http://img683.imageshack.us/img683/6729/pow20.th.png) (http://img683.imageshack.us/i/pow20.png/)

As you can see the p2=20 shape is quite hairy.

And yet,  still no sign of lateral spheres. Sorry Twinbee!

Hi Bugman. I have been experimenting with a two parameter mandelbulb, where the two parameters p and q multiply the angles theta and phi


This variation encompasses some of your special cases and provides more flexibility, because proper choices of p and q (positive and negative) can smooth out or roughen up mandelbulbs.

Some more examples here http://www.aurapiercing.com/gallery/main.php

Excellent work FrozenOwl. Love the vectors and the matrices to clarify the kind of products involved.

Hej Frozen Owl, Somehow I thought of the same concept as you independantly, but to make it a bit more complicated I also wanted the squaring to satisfy |(x,y,z)^2|=|(x,y,z)|^2. Working this out gives you the following set of equations which the mx11,mx12 etc have to satisfy:

since I wrote down the formula's before I was aware that you had the same idea so you have to translate, for example mx11=a00, my13=b13 etc.

a00^2 + b00^2 + c00^2 - 1 = 0
2*a00*a01 + 2*b00*b01 + 2*c00*c01 = 0
2*a00*a11 + a01^2 + 2*b00*b11 + b01^2 + 2*c00*c11 + c01^2 - 2 = 0
2*a01*a11 + 2*b01*b11 + 2*c01*c11 = 0
a11^2 + b11^2 + c11^2 - 1 = 0
2*a00*a02 + 2*b00*b02 + 2*c00*c02 = 0
2*a00*a12 + 2*a01*a02 + 2*b00*b12 + 2*b01*b02 + 2*c00*c12 + 2*c01*c02 = 0
2*a01*a12 + 2*a02*a11 + 2*b01*b12 + 2*b02*b11 + 2*c01*c12 + 2*c02*c11 = 0
2*a11*a12 + 2*b11*b12 + 2*c11*c12 = 0
2*a00*a22 + a02^2 + 2*b00*b22 + b02^2 + 2*c00*c22 + c02^2 - 2 = 0
2*a01*a22 + 2*a02*a12 + 2*b01*b22 + 2*b02*b12 + 2*c01*c22 + 2*c02*c12 = 0
2*a11*a22 + a12^2 + 2*b11*b22 + b12^2 + 2*c11*c22 + c12^2 - 2 = 0
2*a02*a22 + 2*b02*b22 + 2*c02*c22 = 0
2*a12*a22 + 2*b12*b22 + 2*c12*c22 =0
a22^2 + b22^2 + c22^2 - 1 =0

A friend of mine already made a render of one of the solution's I've found
The solution he used is
a12 = 2, b00 =- 4/5, b01 =- 6/5, b11 = 4/5, b22 =- 4/5, c00 =- 3/5, c01 = 8/5, c11 = 3/5, c22 =- 3/5 and all others are zero

It's rendered with mandelbulber (I just replaced one of the formula's in the program with my own) (http://leden.vslcatena.nl/~robert/mandelbulb/maartend.jpg)

Of course the set equation has infinitly many more solutions (but it's not very easy to find them). Of course when seeing the "squaring" as a map F from R^3 to R^3 we see that if M1 and M2 are orthogonal 3x3 matrices then if F has the property |F(x,y,z)|=|(x,y,z)|^2 then the composition M1 F M2 also has that property (i.e. |M1(F(M2(x,y,z)))|=|(x,y,z)|^2). This allows you to continuesly deform the thus obtained mandlebulb depending on the rotation matrices M1 and M2. Does someone know if there are already movies of using this idea to get a mandlebulb changing it's shape in the way I just described?

Last weekend my friend made a small video of the fractal formula I wrote down.

It can be found at http://leden.vslcatena.nl/~robert/mandelbulb/Derickx-Equations.mp4

thanks very much for this post bugman... the "math" goes over my head  :embarrass:... but I like pictures... this was very informative...