Welcome to Fractal Forums

Hi all.
I followed the original thread since before the /. linking, and first of all you guys are amazing.

So since the original thread is now locked I'm making a new one to continue the quest for the true 3D/4D equivalent of the mandelbrot set.


>This isn't a mandelbulb discussion<

I guess no one wants to continue this?

When I have time I'll post all the links to the other objects discovered while we were searching :)

Welcome to the forum. Take a look in the Mandelbulb forum - there are quite a few discussions there to find the real McCoy.

Aka, the Mandelisk ;)

Most attempts so far are displayed on Paul Nylanders pages including links to their origins and various renders:

http://www.bugman123.com/Hypercomplex (http://www.bugman123.com/Hypercomplex)

I guess this goes here, though the idea is based on the Mandelbulb method, just using different angles:

UF iteration code (magn precalculated):

            magn = sqrt(magn)^@mpwr
            th = @mpwr*atan2(zri)                       ;az
            ph = @mpwr*atan2(imag(zri) + flip(zj))  ;ax
            sz = sin(th)
            zri = magn*(cos(th) + flip(cos(ph)*sz)) + cri
            zj = cj - magn*sz*sin(ph)
            magn = |zri| + sqr(zj)

Reslts in these:

z^2+c:

(http://www.fractalforums.com/gallery/1/141_28_11_09_12_59_44.jpg)

z^8+c:

(http://www.fractalforums.com/gallery/1/141_28_11_09_1_06_21.jpg)

Hi all, I think I found *it*, or at least maybe at least a better bet than the Mandelbulb, first animation (z^2+c):

http://www.youtube.com/watch?v=9Oy4VLzndto (http://www.youtube.com/watch?v=9Oy4VLzndto)

More later....

OK, maybe there's still something better to find, this version has "correct" cross-sections in both XZ and XY planes for even powers>=2 but the odd ones are slightly off.
Anyway here's the iteration loop calculation as programmed in UF (all variables real except zri and cri which are complex, magn precalculated on loop entry):

            magn = sqrt(magn)^@mpwr
            th = @mpwr*atan2(zri)                      ;az
            ph = @mpwr*atan2(imag(zri) + flip(zj)) ;ax
            r = @mpwr*atan2(real(zri) + flip(zj))    ;ay
            sx = sin(ph)
            sy = sin(r)
            cy = cos(r)
            sz = sin(th)
            cz = cos(th)
            zri = magn*(cz*cy - sx*sy*sz + flip(cos(ph)*sz)) + cri
            zj = magn*(sx*cy*sz + sy*cz) + cj
            magn = |zri| + sqr(zj)

Note that |zri|+sqr(zj) is the square of the magnitude.

Edit: Apologies I overestimated how good the 2D cross-sections where - I thought some deviations from the "correct" outlines where errors in my clipping routine but it turns out that the cross-sections are close to being correct, but slightly disturbed in one way or another :)

Here's the "non-trig" version using complex and real - this can be used to get the "non-trig" all-reals version for the integer powers (zri,ztemp,zjk and cri are complex and the rest are real):

            magn = sqrt(magn)^@mpwr
            ztemp = (real(zri) + flip(zj))^@mpwr
            zjk = (imag(zri) + flip(zj))^@mpwr
            zri = zri^@mpwr
            ztemp = ztemp/cabs(ztemp) ; cy, sy
            zjk = zjk/cabs(zjk) ; cx, sx
            zri = zri/cabs(zri) ; cz, sz
            zj = magn*(imag(zjk)*real(ztemp)*imag(zri)+imag(ztemp)*real(zri)) \
                 + cj
            zri = magn*(real(zri)*real(ztemp)-imag(zjk)*imag(ztemp)*imag(zri) \
                        + flip(real(zjk)*imag(zri))) + cri
            magn = |zri| + sqr(zj)

Here's how I came up with the "Mandelview":

The usual way of transforming a scene for viewing in 3D is using the rotation matrices for example:
 
| cz -sz  0 |
| sz  cz  0 |
|  0    0   1 |
 
and
 
|  cy   0   sy |
|   0    1   0  |
|  -sy  0   cy |
 
and
 
|  1   0     0  |
|  0  cx  -sx |
|  0   sx  cx |
 
The reversal of the sign of the sines in the Y angle rotation being the "correct" method.
Now we can transform space using a combination of these matrices in any order, we want the order where the z is first, so either z*y*x or z*x*y.
I tried z*x*y first, i.e.
 
| cz -sz  0 ||  1   0     0  | = | cz   -sz.cx  sz.sx |
| sz  cz  0 ||  0  cx  -sx |    | sz    cz.cx  -cz.sx |
|  0    0   1 ||  0   sx  cx |    | 0        sx       cx   |
 
| cz   -sz.cx   sz.sx ||  cy   0   sy |  = | cz.cy-sz.sx.sy   -sz.cx   cz.sy+sz.sx.cy |
| sz    cz.cx  -cz.sx ||   0    1   0  |
| 0        sx        cx   ||  -sy  0   cy |
 
For the equivalent of the "Mandelbulb" method we only need that first row assuming you multiply like this (as I was taught):
 
|x y z|| a b c | = | x.a+y.d+z.g  x.b+y.e+z.h  x.c+y.f+z.i |
         | d e f |
         | g h i |
 
Because our vector to be transformed is |1 0 0| (pr more accurately |magnitude 0 0 | :)
 
Anyway I tried: magn*(cz.cy-sz.sx.sy, -sz.cx, cz.sy+sz.sx.cy) which gave a combination of Mandelbrot and Mandelbar cross-sections if I remember correctly.
Now a Mandelbar is just a Mandelbrot with a sign change so I just played around with the signs of the terms with sines until both cross-sections were Mandelbrots and ended up with: magn*(cz.cy-sz.sx.sy, sz.cx, cz.sy+sz.sx.cy) which is totally incorrect if applying "normal" rules - hence my comment on the animation regarding a secondary "imaginary" level since the "incorrectness" involves a sign change.

Afterwards I tried the z*y*x order and got a modified version of that which produced the correct cross-sections for the z^2+c version but still had the problem with the odd powers and in addition only one of the cross-sections was correct for each of the even powers above 2.

Edit: I have since found that some of the cross-sections are only close to the "correct" versions - I thought the difference was a problem with my clipping :)

I think the "Mandelview" gives the most interesting "power morph" for a 3D Mandelbrot so far, here's 2 to 10 and back:

http://www.youtube.com/watch?v=xgHr-aVUV_4 (http://www.youtube.com/watch?v=xgHr-aVUV_4)

Apologies for it being a bit fast, I didn't want to spend too much CPU time on rendering animations as I'm still trying to find a "perfect" 3D Mandy :)

Another alternative method of getting a true 3D Mandelbrot I've been considering is a 3D version of trigonometry using an idea that I've never actually seen described or used but I think is sound (I know I'm not rigorous enough for a proof):

Consider a right-angled tetrahedron (which is an object that can by defined simply by (x,0,0),(0,y,0) and (0,0,z) it's fairly obvious that any two such objects are similar in the same way as triangles if the ratios of the *areas* of the sides stay unchanged.
Considering this I thought that maybe we could define a specific right-angled tetrahedron by any two known ratios of areas - for example if we define the areas of the sides as xy, xz and yz for the small sides and xyz as the base (=="hypoteneuse") then if we know xy/xyz and either xz/xyz or yz/xyz then we have defined a set of self-similar tetrahedrons.
Is this correct ? If so it probably gives an alternative method of defining rotation in 3D and squaring, multplying etc. once we have a version of Pythagoras for tetrahedrons relatng the areas.

Edit: OK I know - make that the ratios of the square roots of the areas :)

Here's a full 360 of the z^2+c "Mandelview":

http://www.youtube.com/watch?v=V2c2SInnDNw (http://www.youtube.com/watch?v=V2c2SInnDNw)

The "Mandelview" cross sections are not as close to being "correct" as I at first thought, I initially thought the difference was a problem with my clipping algorithm but have since found that some of the cross-section are only similar to the "correct" versions.

The rotation matrices of the mandleview are analogous to the transform i have designed so it is interesting to see that someone else has thought along similar lines.

Your idea of a solid pythagoras formula is also very interesting.

I have to say that since Bellini described them these operators have inspired so much of modern utilitarian math as well as pure inventiveness, but underlying all of this is the fundamental power of iteration.  am exploring a fundamental revision of math that recognises this fundamental in all math and beyond maths into perception.

Hi David,

The areas of the faces of your tetrahedron are A_1 = xy/2, A_2 = yz/2, A_3 = zx/2 and something a bit more difficult to compute for the last face. But the tetrahedron is completely determined by x,y,z, and provided you don't allow negative value, knowing the areas allows to recover x,y and z. For instance x = sqrt(2A_1 A_3/A_2). So it's really equivalent to consider the sides of the tetrahedron or the areas of its faces. I'm not sure how you want to use this...

For the mathematically inclned, we can think of the trick for squaring vectors as a "covering" of the 2-sphere S^2 by another 2-sphere, ie a map from S^2 to S^2 such that the preimage of each point contains two points (or n points for the power n Mandelbulb). We saw that all the solutions found so far (involving doubling the angles in spherical coordinates) produced disconinuous maps. There is a reason for this. In the mathematical litterature, a covering is always a continuous applications, and we may wonder if there exists coverings of the sphere by the sphere. The answer is no. Connected coverings over a space X are classified by the subgroups of the fundamental group of this space ( http://en.wikipedia.org/wiki/Fundamental_group ), see theorem 1.38 here:
http://www.math.cornell.edu/~hatcher/AT/ATch1.pdf
The fundamental group of the sphere is trivial, so it admits no non-trivial connected coverings.

Edit: There can be branched coverings of the sphere by the sphere, see for instance
http://www.jstor.org/stable/2046685?origin=crossref
But I couldn't find any analytic formula. I'll think a bit more about it.

Best,

Sam

With respect to the ratios of the roots of the areas of the sides of the tetrahedron I was basically thinking that in the same way that a complex number is magnitude*(cos(theta)+i*sin(theta) then getting an equivalent to pythagoras for the areas of the sides of a tetrahedron would give us a "3D triginometry" such that our 3 valued number is magnitude*(side1/base + i*side2/base + j*side3/base), with side 1 being sqrt(x*y/2), side2 sqrt(y*z/2), side3 sqrt(x*z/2), as you say the area of the base is not so simple (I haven't tried to derive it, I was hoping someone would know the best formula for it for a right-angled tetrahedron in tems of x,y,z).
Exactly how to handle signs correctly in "3D trig" would be interesting :)

the last area would be
x²
or
y²
or
z²
 if I did everything correct (which actually shows your pythagoras in some kind of way: x²=y²=z² ;) )

and the volume is
abs(x*y*z)/6

>>In the mathematical litterature, a covering is always a continuous applications, and we may wonder if there exists coverings of the sphere by the sphere. The answer is no.
I haven't read the link, but as far as I can see the answer is yes. The simple map that doubles the longitude on a sphere is continuous (C(0) continuity). Any two points an arbitrarily small distance apart will end up no more than twice that distance apart. Perhaps you mean it is discontinuous in some other way.

Yes, the matter is a little bit more complicated than what I explained. The map you're refering to is continuous, indeed, but it has two branch points, that is two points which have a single preimage, at the north and the south pole. Such a generalized notion of covering is called a "branched covering". The theorem cited above is valid only for genuine coverings which do not have any branch point.

If we allow branched covering, then the one you mention is the only one of degree 2 (up to continuous deformations such as moving the branch points around). It is not very interesting from the point of view of the Mandelbulb, because it acts trivially on the coordinate parallel to the axis of rotation. For higher powers, associated with branched covering of degree larger than 2, there might be more interesting possibilities, I haven't looked into it yet.

I should add that I'm suggesting considering it specifically in terms of the areas because in 3D we're essentially dealing with the surface of a sphere rather than the circumference of a circle :)

Methinks you made a mistake somewhere with the area of the base :)

>>  The map you're refering to is continuous, indeed, but it has two branch points,
So does the 2d mandelbrot, where i=0... so branch points aren't necessarily a bad thing.
>> the one you mention is the only one of degree 2 (up to continuous deformations such as moving the branch points around). It is not very interesting from the point of view of the Mandelbulb
True, but most mandelbulb attempts don't just double-cover the sphere, they quadruple cover the sphere. Afterall, in order to get any patch on the sphere to double in width and in length, you need to quadruple-cover. That's why the standard mandelbulb doubles the longitude and also doubles the latitude.

Doubling the latitude is kind of problematic since it isn't a strict 2->1 mapping... but it isn't the cause of the whipped cream effect... I'm not sure what causes that... but its might be because doubling the longitude actually leaves a whole semicircle untouched... but that's a guess.

>> So does the 2d mandelbrot, where i=0... so branch points aren't necessarily a bad thing.

What do you mean by i = 0 ? Recall that we are forgetting the radial direction. In the 2d case, we have a covering of degree two of a circle by a circle, without any branch point.

>>True, but most mandelbulb attempts don't just double-cover the sphere, they quadruple cover the sphere.

That's right. So investigating coverings of higher degree might be interesting.

>> In the 2d case, we have a covering of degree two of a circle by a circle, without any branch point.
The circle at i=0,r=1 doesn't move under the squaring operation (doubling the angle)
The poles (and in fact all the way down the greenwich meridean) don't move when you double the longitude.

I'm just saying that the poles on a sphere aren't really different to this point on a circle... are they? So such points aren't necessarily a bad thing.

A branch point is a point whose preimage has a single element instead of 2, it has nothing to do with the point being left fixed by the map. In the case of the circle, you can check that each point has exactly two preimages, these are the two square roots of the corresponding complex number when you embedd the circle into the complex plane.

I don't know if branch points are a bad thing or not, whatever this means, I'm just saying there are none in the case of the double covering of a circle by a circle, and there are two of them in the case of a double covering of a sphere by a sphere.

But there is actually a much, much better way to think about all this. Consider first the usual Mandelbrot set. We can see the complex plane as a sphere after adding a point at infinity
http://en.wikipedia.org/wiki/Riemann_sphere
Now the map z -> z^2 is a realization of the double covering we spoke before. The two branch points are 0 and the point at infinity. But this is not just a random double covering, this covering is conformal (it preserves the angles).
http://en.wikipedia.org/wiki/Conformal_map
The higher power mandelbrot sets correspond similarly to coverings of higher degree.

Now here is what I could consider as a mathematically meaningful generalization of the Mandelbrot set. Just as the complex plane can be turned into a 2 sphere by adding a point at infinity, we can "compactify" the three dimensinonal Euclidian space R^3 into a 3-sphere by adding a point at infinity. Now suppose that we have a covering of the 3-sphere by the 3-sphere which
- is conformal
- has a branch point at infinity. (To ensures that the point at infinity stays there and that we get a map from R^3 to R^3.)
I've no idea if such a thing exists... While there are a lot of conformal maps in 2 dimensions, they are much more rare in 3 dimensions. I think that the property of being conformal is crucial if you want to avoid the "whipped cream" effect. The map currently used in the Mandelbulb is not conformal.

Very interesting stuff.
"A map of the extended complex plane (which is conformally equivalent to a sphere) onto itself is conformal if and only if it is a Möbius transformation"
Are you saying that the Z^2 operation is a Mobius transformation? (visual here http://en.wikipedia.org/wiki/File:Mob3d-elip-opp-200.png)

There are some things going click in my head here... the mobius visual looks a lot like electromagnetism, and it has been quoted that mandelbrots have been observed in electromagnetic fields (hence a 3d mandelbrot likely exists in some form). Also the mobius looks a lot like a torus, which is a shape that can go to double cover with no branch points (I think).

>>  Now suppose that we have a covering of the 3-sphere by the 3-sphere which
>> - is conformal    
>> - has a branch point at infinity.
Isn't that the quaternion multiply? Hmmm... I'm definitely out of my depth here.

i am also pretty sure that all conformal xforms in the complex plane must be moebius xforms (and i don't see how z^2 can be represented using [az+b]/[cz+d]). one of the few sticking points from reading indra's pearls :)

btw guys, i hope that we can also gather principles of "nice looking fractal transforms", not specifically aimed at producing "mandelbrot sets". why this is the "holy grail" i can't understand... there are many other nice (better?!) fractals out there.

The Moebius transformations are the global conformal transformations of the Riemann sphere. But there are also "local" transformations which map open sets conformally onto open sets, but which might have poles at some isolated points. in particular, any holomorphic function of z (such as z^2), is locally conformally invariant.

You can test these claims very easily: take a square gird, transform it by any holomorphic function and check that the lines (which are now curves) still intersect orthogonally.

Oh yes, I did...
I kinda used the dotproduct instead of the crossproduct :S

It's

That makes more sense :)

I guess that's correct because that's the result I got too after looking on Wiki :)

Pretty simple

To get the area of any triangle in R³, simply do the crossproduct of two side vectors and half the result.
The sign afaik also tells something about the direction...

and for a tetraeder's volume, take a sixth of the dotproduct of one vector and the crossproduct of the two others.
here also the sign tells something about the direction... I read about it but kinda forgot....

So, we have the areas of the sides x*y/2, x*z/2 and y*z/2 with the area of the base as sqrt(x^2*y^2+x^2*z^2+y^2*z^2)/2 which gives us 3 ratios equivalent to sine and cosine in 2D:

r1 = x*y/sqrt(x^2*y^2+x^2*z^2+y^2*z^2)
r2 = x*z/sqrt(x^2*y^2+x^2*z^2+y^2*z^2)
r3 = y*z/sqrt(x^2*y^2+x^2*z^2+y^2*z^2)

I started going round in circles trying to see if this helps get a "correct" method for a 3D Mandelbrot and finally hit upon the following idea:

Consider the 2D complex iteration as follows:

1. Take a 2D coordinate = (x, y)
2. Convert to complex = (x + i*y)
3. Raise it to a power (e.g. 2) using complex algebra = (x^2-y^2 + 2*i*x*y)
4. Convert back to 2D coordinates = (x^2-y^2, 2*x*y)
5. Translate by a 2D offset = (x^2-y^2+cx, 2*x*y+cy)

So for the 3D version (using r1,r2,r3 above) we could try:

1. Take a 3D coordinate = (x,y,z)
2. Convert to the new 3D format = (r1, r2, r3) (edit. i.e. to the area ratios using the formulas above for r1,r2,r3)
3. Convert to an algebra format - e.g. cut down quaternion = (r1 + i*r2 + j*r3)
4. Raise to a power in the relevant algebra and convert back to new 3D format = (newr1, newr2, newr3)
5. Convert back to 3D coordinates = (newx, newy, newz) !!
6. Translate by a 3D offset = (newx+cx, newy+cy, newz+cz)

Obviously step 5 in the above is problematic since it needs newx, newy and newz in terms of newr1, newr2 and newr3 (with the signs correct).
Also the translation could be done after step 4 but I'm betting on the above producing more interesting results.

I don't really get your point.... what's the difference to the current method?

Btw, sin/cos/tan also are defined on the unit-circle, so it might help to look at the unit-sphere :)
tan would be 2D (defiing a plane)
and sin and cos would either also be 2D or some extra values would get added to that.
- or there would not only be tan and cot but a third thing, making it 1D again...
or something like that^^

I think, a 3D variation should both work in a rightangle tetrahedron and in a sphere, as both the right triangle and the unit circle help to add acuracy (arg()-function which is quadrant-dependent... - similarly the sphere's arg would be octant-dependend, so to say... [and probably 2D rather than 1D as spherical coordinate systems show anyway]) and simplify formulae (sin²(x)+cos²(x)=1 - for the 3D equivalent it would be sin²(x)+cos²(x)+___²(x)=1 )

Apologies I guess I wasn't clear enough - in the step "Convert to the new 3D format" I should have explained I meant convert to the area ratios using the formulas for r1,r2,r3 in terms of x,y,z.
This way the 3D transformation is performed on the areas rather than the distances.
Also if you look at r1,r2,r3 then r1^2+r2^2+r3^2 = 1.

ah, got it :D
Really a nice idea...
Any results so far? :)

Just about to try it - at least ignoring problems with the signs when converting back from newr1,newr2,newr3 to newx,newy,newz - if anyone has any bright ideas on how to get the correct octant for newx,newy,newz please let me know....

OK, it's pretty useless, at least it is without handling the octant problem and divide by zeroes correctly.

so... we gotta figure it out :)
that's probably a bit more tough...

the octantproblem most likely needs the definition on the unit-sphere to be solved...
(as the quadrant problem of argument needs the unit circle)

and getting along with zeros.... maybe the rule of l'hospital or how it's called can help here?
which solutions turn out just before or just after 0?
(I hope it's clear what I mean^^)

Actually i have just worked out that this strand is where i should post my musings for the  math suggestions toward a 3d mandelbrot. A mandelbulb as it is now called is a beautiful thing but not what i have been thinking about.

http://www.fractalforums.com/theory/transform-for-r3/ (http://www.fractalforums.com/theory/transform-for-r3/)  

This is my transform. I wonder if you could roughly render it David.

http://www.fractalforums.com/theory/formula/ (http://www.fractalforums.com/theory/formula/)

And this is my exploration guide. Be sure to look at the following posts as i complete the guide with regard to a handedness term which affects all 3 cartesian components.

I do not mean to distract your thinking but you guys who are writing about conformal maps and covers from could explain what you are thinking about. Say i do read wikipedia i still do not know what you are thinking about here. There is in fact no loss of generality issues here where we are being specific to one method or another.

In a fractal i would expect discontinuities at boundaries where one region juxtaposes with another. The mathematical horror which called these things monsters must now give way to the realisation that their beauty and artistic worth allied with computers means the computational swamp no longer need fill us with dread.

I like your train of thought.   :)

Regarding your "vx" formula, I have also wandered down rabbit holes thinking about similar approaches.  If we boil the 2D Mandelbrot down to its core, it is a process where do this, over and over:

z := z²
z += z₀

I tend to think about things like "what does it mean, conceptually, to 'square' a number?"

It's simple enough with reals:

1) Take a (1D) line of length x.
2) Take another line of length x and lay it perpendicular to the original line.
3) Take those two lines as sides of a square, and the area of that geometric shape is the "square" of x.

Extending that concept to 3D is quite natural (just form a "cube" instead of a "square"), but complex numbers aren't quite as easy to visually "square."

Squaring a complex number amounts to a rotation about the origin and a scaling of magnitude.  The term "square" doesn't have such a nice geometric analog any more.  Looking at the number being squared in polar form is sometimes more revealing...



In two dimensions, what it's saying is:

1) Take the distance to the point from the origin and square it (using the word 'square' here in the real sense).
2) Take the angle to the point measured from the +x axis and double it.
3) At the point described by this new distance and angle is your new, "squared," complex point.

Now we immediately run into the big question...  How do we extend that concept into three dimensions?

Things like this keep my mind very busy.  :)

I've often found it annoyingly inconstistent that squaring is applied when dealing with both 2 dimensions and 3 dimensions - it makes more sense to me that to extend complex z^2+c to 3D triplex equivalent we should actually consider z^3+c rather than z^2+c :)

to bad we are not dealing with areas/volumes, if it would be a volume i would suggest cubing for squaring in 3 dimensions ...   :police:

I tried around with the area-based sine a bit but I don't have a CAD-program which would allow me to actually define the whole thing for areas...

I at best have two programs:

blender - not a CAD program, so if you don't know python well enough (which is the case for me), probably pretty useless and
GeoGebra - limited to 2D...

operating on areas is not too easy it seems...

one thing that isn't clear yet is, how does the "rightangles tetrahedron"-angle look like: it's neither a plane angle nor is it an opening like with cones, which could easily be meassured one way or the other...
how to define that double-90° angle and the surrounding 3 more double-angles?

btw: the definition of area-based sine also uses standard norm sqrt(x²+y²+z²)=1, so it wont work to use the third power either...

though, you could in general experiment with odd formulae... here for instance the real and imaginary part of z in a base-i norm unit sphere, if x and y vary between -2pi and pi...

http://www.wolframalpha.com/input/?i=3Dplot+real%28%28-x^i-y^i%2B1%29^%28-i%29%29%2Cx%3D-2pi...2pi%2Cy%3D-2pi...2pi
http://www.wolframalpha.com/input/?i=3Dplot+imag%28%28-x^i-y^i%2B1%29^%28-i%29%29%2Cx%3D-2pi...2pi%2Cy%3D-2pi...2pi

It should be remembered that Pythagoras (although used for distance calculations) actually relates areas rather than distances :)

I think the problem lies in the fact that i is a unary operator defined on the plane, and j and k have followed suit. To define a 3d operator is perhaps as straight forward as

i on a = ia and iia = -a and iiia =-ia and iiiia =a

i on (a,b)= (-b,a) ii(a,b) = (-a,-b) iii(a,b) = (b,-a)  iiii(a,b) = (a,b)


i on (a,b,c) = (-c,a,b,)  ii(a,b,c) = (-b,-c,a)  iii(a,b,c) = (-a,-b,-c,)  iiii(a,b,c) = (c,-a,-b,) iiiii(a,b,c) = (b,c,-a) iiiiii(a,b,c) = (a,b,c,)

We then use z in the same notation for triplex but we need a triplex transform for z²


In we use a matrix transform  0  -1 and a definition (a,b)(c,d) = (ac - bd, bc +ad)
                                                1   0


can we generalise this to punch out the above definitions for i on the triplex?

@David Makin
that then would mean, that the plane-version would act on cubes....

that way, there indeed would be a ....

I was thinking of maybe something in terms of x^2*y, y^2*z and z^2*x and/or x*y^2, y*z^2, z*x^2 :)

I guess, trying that geometrically first would be simpler than directly doing the math...

If I remember correctly Pythagoras' formula was based on a geometric proof.

Yup, it was.

That's why I guess, the 3D variant might need the same first, if it's valid at all...

though, by now there are thousands of proofs both geometrical and mathematical, afaik :)

For I designed a transform based on the transform in the following way:

(a,b,c) vx (d,e,f)  

(ad - be - cf, ae + bd, af +cd, bf + ce) using as a manipulation /construction space.

This i reduce to

(A,B,C,D) =(ad - be - cf, ae + bd, af +cd, bf + ce)

Using pairs from the construction bracket in the transform i obtain 6 building blocks

1 (AA - BB, 2AB)

2 (AA - CC, 2AC)

3 (-CC - BB, 2BC)

4(AA + DD, 2AD)

5 (DD - BB, 2BD)

6 (DD - CC, 2CD)

THE unary OPERATORS i and j are used to inform the manipulations so that i² = j² = -1 and (ij)² =+1.


Now my intention was to rotate the planes xy, xz, yz by this construction and i assumed that was what was happening until i rechecked the construction principles. The yz plane is not the same as the other two planes with the unary operators i and j operating on the axes. Under the transform the yz plane is sheared to the xij plane whatever that is. It may be a vortex surface.

so the first constructed transform is mistaken in two counts. The manipulations were faulty and i will show the correct manipulations; but the design was mistaken as it was not tranforming to a map of geometrical space.


The expansions are as follows for the right handed form

AA = (ad)² + (be)² + (cf)² - 2abde - 2acdf + 2bcef

BB = (ae)² + 2abde + (bd)²

CC = (af)² + 2acdf + (cd)²

2AB = 2(a)²de + 2ab(d)² - 2ab(e)² - 2(b)²de - 2acef - 2bcdf

2AC = 2(a)²df + 2ac(d)² - 2ac(f)² - 2(c)²df - 2abef - 2bcde

2BC = 2(a)²ef + 2bc(d)² + 2abdf + 2acde

Now the first posted construction was based on combining blocks 1,2,3

supposedly giving

*{AA - BB + AA - CC, 2AB + BB - CC,  2AC + 2BC}

=> {AA - BB/2 - CC/2, AB + BB/2 - CC/2, AC + BC }.                                                    


 [ in fact it should be {AA - BB/2 - CC/2, AB - BB/2 - CC/2, AC + BC } due to an error in the original formulation of block 3]

So clearly (when i expand it) my original manipulations were wrongly copied from page to page to screen.

But now i realise i have not combined like with like and so have to construct the following transform from blocks 1and 2 which i fear will be even less interesting than my mistaken one

{AA -BB/2 - CC/2. AB, AC}

AA = (ad)² + (be)² + (cf)² - 2abde - 2acdf + 2bcef

BB = (ae)² + 2abde + (bd)²

CC = (af)² + 2acdf + (cd)²

2AB = 2(a)²de + 2ab(d)² - 2ab(e)² - 2(b)²de - 2acef - 2bcdf

2AC = 2(a)²df + 2ac(d)² - 2ac(f)² - 2(c)²df - 2abef - 2bcde

2BC = 2(a)²ef + 2bc(d)² + 2abdf + 2acde


So the corrected vx is as follows

(a, b, c) vx (d, e, f)  = ((ad)² + (be)² + (cf)² - 2abde - 2acdf + 2bcef -((ae)² + 2abde + (bd)²)/2 - ((af)² + 2acdf + (cd)²)/2,
                                       (a)²de + ab(d)² - ab(e)² - (b)²de - acef - bcdf + ((ae)² + 2abde + (bd)²)/2 - ((af)² + 2acdf + (cd)²)/2,

                                               (a)²df + ac(d)² - ac(f)² - (c)²df - abef - bcde + (a)²ef + bc(d)² + abdf + acde)

This gives  
   (x, y, z)² = (x, y, z) vx (x, y, z) = (x⁴ +  y⁴ + z⁴ - 4x²y² - 4x²z² + 2y²z²,

                                                     (2x³y - 2xy³ - 2xyz² + 2x²y² - 2x²z²,

                                                                (2x³z -  2xz³ - 2xy²z + 4x²yz   )

I would like to see if that makes alot of difference to the rendering please David if you have the time.

AA = (ad)² + (be)² + (cf)² - 2abde - 2acdf + 2bcef

BB = (ae)² + 2abde + (bd)²

CC = (af)² + 2acdf + (cd)²

2AB = 2(a)²de + 2ab(d)² - 2ab(e)² - 2(b)²de - 2acef - 2bcdf

2AC = 2(a)²df + 2ac(d)² - 2ac(f)² - 2(c)²df - 2abef - 2bcde

2BC = 2(a)²ef + 2bc(d)² + 2abdf + 2acde

The left handed formulation requires the following

DD = (bf)² + 2bcef + (ce)²

2AD =  2abdf + 2acde - 2bc(e)² - 2bc(f)² - 2(b)²ef   - 2(c)²ef

2BD =2ac(e)² + 2(b)²df + 2abef + 2bcde

2CD = 2ab(f)² + 2(c)²de + 2bcdf + 2acef


Now from the 6 blocks i can construct 3 linked formulations abbreviated as followa

{AA - BB/2 - CC/2, AB, AC}                           RIGHT HANDED

{(AA + DD - BB - CC)/2, BC + AD}                 A MIRROR PLANE  at to the xy plane but here plotted in xy to have a look

{DD - BB/2 - CC/2, CD, BD}                          LEFT HANDED


THIS STRUCTURE i think might be interesting even if the mandelbrots are not. Particularly the plane as it may contain reflections of details in the brots not visible in the 3d brots.

Any way is any one interested as i am in seeing what this system looks like? I will expand it and find the vx for (x, y, z) if you are.

David you said that the cut down quaternion would lathe and i think paul shows such a rendering, so can you explain what is happening in these renderings? The first is a 3d representation of z=z² +c. The second and third if it shows are

z= z²- 2yzij +c which should be the same as

real(z)²-imag(z)²-imaj(z)² +2*real(z)*imag(z)*i+2*real(z)*imaj(z)*j+c

which is .

I don't see how "-2yzj" is the same as "+2xzj" ?

Did you remember to avoid variable corruption ?

I mean you can't do:

zri = zri^2+cri
zj = 2*real(zri)*zj + cj

You have to do (for example)

zj = 2*real(zri)*zj + cj
zri = zri^2 + cri

Hi all, I apologize for not keeping up with this (or Mandelbulb discussion), but I hope this thread is an OK place to post an interesting 3D mandelbrot set-like render I ran into.

(http://www.hevanet.com/bradc/ComplexLogisticBifurcationDiagram3D.png)
http://www.hevanet.com/bradc/MiscMathStuff.html (http://www.hevanet.com/bradc/MiscMathStuff.html)

It is probably too chaotic overall, but I thought looking at the bifurcation diagram was interesting.

Hey I recognize that picture :)

hmmm....

looks much like a solidified Version of a rotated Buddhagram by Melinda Green...

Or the mix of a Mandelbrot and a logistic map (which are closely related anyway) :)

Your train of thought prompted me to explain that the identity =

Is a logarithmic transformational one which euler derived based on a naperian construction process for logarithms. Napers construction for logarithms is by definition 2d. To extend it to 3d therefore you would need to define a construction of logarithms in 3d . You may also need to derive a 3d equivalent to the sine function. In any case it is an interesting project and may lead to a better understanding of so calle 3d

I put the formulae in as described in my post.

Also i have used a standard expansion with i*j not defined by myself but by the programme.
This is not your programme it is by terry w glintz.

@ David

can you write z^2+c in terms of real(zri), zj and presumably zi ?

Is zj = imag?(zri) and zi = imag(zri) ? in your programme?

In my program complex z^2+c would simply be zri^2+c, however for the triplex zri (complex) is the r and i part of the triplex and zj (real) is the j part, so the (-sine) Mandelbulb (trig version) in UF code could be simply:

magn = sqrt(|zri| + sqr(zj))
th = power*atan2(zri)
ph = power*asin(zj/magn)
magn = magn^power
zri = magn*cos(ph)*(cos(th) + flip(sin(th))) + cri
zj = -magn*sin(ph) + cj

where |zri| is x^2+y^2 if zri is x+i*y.

Thanks dave.

I see what is happening in terry glintz programme now. Although he uses Quaternion or hypernion math to shape the mandlebulb i can write a bespoke formula for the iteration. I am not sure how he plots the result but i do not get a lathed mandlebulb for the above equation.

so i got the lathed mandelbrot finally! Beginners mistake really. So here are some 3d ones using terry glintz programme.

Some more variations.

hi, i found it useful to take the geometrical view:
in the picture, it is shown, how we can interpret the 2d and 3d mandelbrot algorithm, means, the squaring of a respective number z1. I used this to write some programs, varying the way, how a z1 will be "squared". If we preserve in one plane the pattern, shown at top left, we will get in this plane the original 2d set. Its trajectorial field consists of logarithmic spirals, working together with the constant vctors added, as shown at the right above. Now, what is the best corresponding field of trajectories in 3d? The respective field when squaring quaternions is rotating the complex plane around the x-axis until z1 will lay in this oblique plane, then it will be squared in the rotated plane as usual. The resulting trajectorial field looks in some direction not so rich, therefore the quaternion-M-set is a little bit disappointing. Other approaches are shown below: in "gedatou", (implemented in QUASZ by T.Gintz) i took one point (zero)on the line, representing the north pole in the coordinate system, which represents a flattened uni circle in the green rectangle, squaring is done by doubling the distance from zero.(in the shown ccord-system, the z-axis will represent the log(|z|), so doubling from zero will cause squaring of the distance(in the "real coordinate system)). I also wrote "rings of fire"(implemented in QUASZ as well about 10 years or so ago), which uses the way, which is used in the wonderful mandelbulb(great work-congratulations), it used preferably only the squaring-mode,(was experimenting with higher multiplications of angles, because f.e. tripling the angles, the north pole would not be projected to the equator, as it will be by squaring, but to the opposite pole-but i usually did not triple .. the distance-and-in my beloved basic-programs-fast and furious-the resulting objects did look like the multiples we get when doing higher powers in the 2d-sets-but i was not far from it! Anyway, the mandelbulb objects are looking great, i wonder how they are producing the fine structures? In the mandelbulb-algorithm a squaring(or respective taking to higher powers) is done by doubling in the shown("flattened") coord.-system the distances of any points z1 from a point on the equator. While in "gedatou"(the name a combination of german"gehirn"-(brain), because often the resulting objects resemble to brains(and this not just happening, i wrote some posters and papers about fractal neural nets, using the rich connectivity of fractals) and Fatou-), the z1-points will be sent from one pole on a journey around the meridians, while these are rotating around the polar axis, the points in the mandelbulb-algo will be sent from the equator alon the meridians, which are as well rotating around the polar axis. best regards (I am not right sure, whether the picture will be shown, -hope best)

some might enjoy a picture, made with "gedatou"-seems to be too large to be uploaded here or to the gallery-as  said in the reply above, it is related to the mandelbulb-as well as to the quaternion-squaring-done with chaos pro,
the (very provisionary-you have to scroll down little bit)link:http://trajektorulm.tr.funpic.de/index.htm (http://trajektorulm.tr.funpic.de/index.htm)
the formula as:
a1 = x1 * x1 + y1 * y1 + z1 * z1;
 //a2 = a1;//: REM  new distance if squaring only
a1 = sqrt(a1);
loa1=log(a1);
a2=exp(power*loa1);
yioi1 = y1 * y1 + z1 * z1;
 yioi1 = sqrt(yioi1)         ;
//alpha1 = angle(x1, yioi1);
alpha1 = arg(x1+1i* yioi1);(would be in C++
alpha1=atan2(x1,yioi1))// if not, then atan2(yioi1,x1)
gamma1 = arg(y1+1i* z1);

alpha2 = power*alpha1;
gamma2 =power* gamma1                ;                                           
x2 = cos(alpha2) * a2               ;                                         
yioi2 = sin(alpha2) * a2        ;                                       
y2 = (yioi2) *cos(gamma2);                                       
z2 = (yioi2) * sin(gamma2) ;
  :tease2:

Hi thomas and welcome. I am trialling terry glintz programme and am impressed with its usefulness. Your diagram enlarges ok when clicked on. Terry says he is bringing out an updated version with the mandlebulb formulae in built soon. For me the ability to explore variations of formulae quickly is the most useful aspect of his programme. I do not understand yet how zplot works but it seems to give very good renditions of the mandlebulb just in 3d using i and j unit vectors. Anyway your exposition is welcome.

Here is gedatou from terry glintz excellent quasz trial version.

Using 
   z= x^2-y^2-z^2+2y(x-z)i+2zxj,z=z^7+c

and also

    z= x^2-y^2-z^2+2yxi+2z(x-y)j,z=z^7+c

Gives the following.

hi jehovajah, thank you for your reply, the pictures are loking very interesting, will later try to understand, what is happening in your formula. Found, on can download the pictures attached in original size, by clicking on the blue link left below, so i only attach the additional trajectorial fields, shown for points on the unit sphere, for programs quaternions, gedatou and mandelbulb. Would be interesting to see, how the trajectories will wander throughout the unit sphere in the formulas, you showed. In gedatou, multiplying "alpha"(latitude, corresponds to phi in mandelbulb) with higher values gives a mandelbulb-like aspect, more than if we change alpha(identical with "theta" in mandelbulb-longitude). The journey around the poles enables the vectors, to stay around zero, because the points z^2 will lie often on the opposite half of the unit sphere, so adding the orriginal vector will end up in most cases within the unit sphere and then taking the distance to a high power will end up very near to zero, so the next addition has a great chance to stay within the unit sphere. This seem to be a basic mechanism how the mandelbulb structures evolve. With power 2 the distances will not be so small after squaring the numbers within the unit sphere, so they may leave more often the uni sphere after addition of the original vector c. (How can one change the inbuilt formulas in quasz? And btw-trying to upload a picture to fractalforums, i often get the message: error 500, internal server error-anybody knows, how to manage that? The large picture with the combined trajectorial fields is therefore for afficionados available at a provisionary site: please scroll down a little bit:
http://trajektorulm.tr.funpic.de/index.htm (http://trajektorulm.tr.funpic.de/index.htm)

some of your spirals? This is twinbee formula in quasz trial at .0001 really small value and 12 iterations bailout 6.

this is gedatou with power 3 (odd powers will save he complex plane as in the mandelbrot-al
gorithm
it is iterations 7, with higher iterations not looking really great, nevertheless interesting
(your pictures show complete fractals, interesting how the rectangular grid you got, what is the really small number/value(power n in z^n ? The spirals are ways, we could imagine, that points will wander, if not taking z=z ^2 + c  directly, bt going continuously from z to z ^1.1, z ^1.2, z ^1.3 etc. the fields of spirals, we get by his give us a good impression an information, whether the fractal resulting will be interesting) ;)

Just trying to code twinbee formula in 150 characters or less lead to this!

Found here http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/

and the simplification here http://www.fractalforums.com/3d-fractal-generation/true-3d-mandlebrot-type-fractal/30/

Whoops my mistake i see it is karl131058 who did the simplifying sorry.
Anyway because the variables in quasz are quad variables and i have used them like real variables i do not know if the intended outcome has been realised. More likely some unintended consequences of quad algebra have been introduced. This is the formula used.
s=imaj(z),g=x#*x#,h=y#*y#,m=s*s,f=g+h,e=g-h,d=f-m,if(abs(y#)<.001),z=g-m+0*i
> -2*s*sqrt(g)*j+c
> else z=d*e/f+i*2*d*x#*y#/f-2-s-sqrt(f)*j+c endif

You have to visualise that this is all on one line!!
so the s,d.e.f.g,h and c are all quaternions.In addition they have not been properly initialised apart from c that is. Terry has posted some updates to the windows platform yesterday but not the mac yet. The trial version has limitations obviously but it is so intuitive and sweet to handle. The grid structure appeared in a region that looked like spots. See
http://www.fractalforums.com/mystic-fractal-programs-gallery/mandelbrotin-3d/ 

Thanks for that info on the spirals.

Ok, so now i am getting somewhere in understanding how this mandelbrot works.
.

This is the mapping of the mandelbrot  m set in the set A of the tensors mapped by the following tranformation onto a set B of the tensors by the following procedure. B needs to contain a set |B| which is all those elements b of B such that |b|<2 .Elements of m map exactly into |B|. The restriction on |B| can be altered but this alters the set m.

I have described the mapping, the second necessary part is the transform. The transform is an iteration mod(n) on a function of z,c in A in which z is a fixed value in A which is squared and c is a variable in A which is varied at each iteration period (mod(n)) and added at each iteration. During each iteration period (mod(n)) the result of each manipulation becomes the new z so z is determined by the result of each iteration, Thus it is a feedback variable determined by the function form and the iteration rules. The rules state that at the end of each iteration period (mod(n)) z returns to its fixed value while c can represent any element of A, however during each iteration period c is fixed while z is determined by the outcome of the function manipulation. So to write z=z^2+c is hardly adequate and essentially misleading.

The mapping and the transform together allow one to comprehend what is happening in this cybernetic feed forward feed backward system.

As artists will appreciate the set m is sculpted from the set A by the transform And the Restriction on set B . By making |B| the mapping of A under the transform and requiring it to be one to one we obtain set m as the residue or the sculpted form of the whole process.

De nombreux lecteurs connaissent probablement l’image de gauche qui est l’ensemble Mandelbrot en 2D. Pour chaque nombre complexe c, on considère la transformation du plan complexe définie par z z^2+c. Partant de z0=0, on itère cette transformation, c’est-à-dire qu’on construit la suite zn avec zn+1=zn^2+c. L’ensemble de Mandelbrot est l’ensemble des c qui sont tels que la suite zn ne tend pas vers l’infini quand l’entier n tend vers l’infini.

So the transformation can be written in french which is marginally better than this



Thinking about the transformation sculpting, this is controlled by the restriction on B . With the restriction in place we can think of the rotation as positioning the sculpting tool and the expansion and translation as chipping off a bit of a sphere if the iterated z breaks the restriction.  Also the iterated z traces a path which may be an orbit or some other path. If we map A to A then the paths mapped out will trace the path and depth of the sculpting tool's cut if those points lie on a path that breaks the restriction.

The current mathematical construction of geometrical space uses two mod(4) unary operators defined on a plane. It is possible to define a unary operator for geometrical space using one mod(6) unary operator. I have not done so yet but i have plans to investigate this. The fact that we use an even mod(2n) set of unary operators is so that we can preserve the unary operator sign on each of the axes in the plane and this essentially is the same as requiring orthoganality in the axes.

We combine the two mod(4) unary operators  so that one pair of axes coincides with each other while the other pair are orthogonal to each other. So if the pair of axes in the x direction coincide then the orthogonal pair are arranged in the z and y directions respectively. This construction however is a restricted set for a so called 3d space, as we only have two planes of rotation these being xy and xz. That is to say that the unary operators mod(4) are rotating about the z axis and the y axis respectively.

While this is sufficient to describe the position of any point in 3d space it is not sufficient to describe the rotation of any point in so called 3d space. It is clear to me that i need a third unary mod(4) operator to describe the rotations in the yz plane about the x axis. This means that since the axes of the third operator will coincide with both the y axis and the z axis i need only to describe how the unary operator flips between the other 2 operators in these directions bearing in mind sign .

For the moment using v for this operator,

             .

The question now is how to represent this in a way that does not confuse it with so called 4d space or quaternions.

One thought occurs and that is to use a similar construction to my definition of vx but with the unary operator for the yz plane rotation being v.

Here's a polynomial mandlebrot type image. z=z+c+c^2+c^3+c^4+c^5+c^6. I am going to be referring to polynomials soon.

  Guys, found it:

r2=sqrt(sx^2+sy^2+sz^2);
theta=atan2(sqrt(sx^2+sy^2)+flip(sz));
phi=atan2(sy+flip(sx));

r3=r2^2;

nx=r3*cos(2*theta)*sin(2*phi);
ny=r3*cos(2*theta)*cos(2*phi);
nz=r3*sin(1.5*theta);

  Apparently the Z axis rotations should be .5 less than the x/y rotation.  Must keep same xy rotations, or maybe not (perhaps other thetas should be changed as well, don't know yet, for now, here are some pics of the current format (2 theta for x and y, 1.5 for z, also I used same magnitude for z, so perhaps it should be set to 1.5 as well... we will check, won't we?):


  First images attached.

If we set:

nx=r3*cos(mag*theta)*sin(v*phi);
ny=r3*cos(mag*theta)*cos(v*phi);
nz=r2^mag*sin(mag*theta);

with mag=1.5, we don't seem to get the foam problem (much).  Of course, increasing details a bunch from far away will still create foam, but it doesn't look like we get way too much.  Apparently not.  It just seems to shrink the fractal, which seems logical considering that we get smaller fractals when we increase iterations in 2d Mandelbrots.  Still get a bit of a mess at too high number of iterations when zooming in (lots of details means lots of stuff), and I've got a bunch of clipping problems with the renderer I'm using, which may be due to exceeding its floating point limits (don't know its extended capabilities, and don't really want to find out until I have a DX11 card to play with).    

  I'll have to wait for those of you with better computers (DX11 rendering capability) to really zoom in.  


  So, once again, to be clear:
r2 =       sqrt (sx^2 + sy^2 + sz^2)
theta =  atan2 (sqrt (sx^2 + sy^2) + flip(sz))  // flip (sz) means multiply sz by i
phi=       atan2 (sy + flip(sx))

magnitude= r2 ^ 2

nx= magnitude * cos (1.5 * theta) * sin (2 * phi)
ny= magnitude * cos (1.5 * theta) * cos (2 * phi)
nz= r2^1.5      * sin (1.5 * theta)


  So far, the pictures look clearer with all 3 theta values + z magnitude set to 1.5, leaving phi values at 2.

  The pictures below are an example.  The clearer one is with all set to 1.5, although there are interesting details in the 2 at lower iterations (probably due to the distortions it introduces, and I like the way it (2 value back side of Mandelbrot) looks a bit more organic, but you can see for yourself).  

  The 8th order Mandelbulb, and the rest seem to be clearer with z value of exactly .5 less like the 2nd order, although we need further tests with better rendering engines.  They are all (2nd, 3rd... 8th) a LOT more symmetric.  

  Makes me wonder about the time dimensions I've been adding, if they should be 1/3 for 1st time dimension, 1/4th for second, or 1/4th for first, 1/16th for second..... but it's late and I need food.  

Alright, here are the TRUE 3d mandelbrots with 1.5 for all values (1.5 * theta, radius^1.5 for z).

Looks cool! But it is clearly sculpted flatter than previous versions and has sacrificed a lot to have the Mandelbrot border.

Yeah, 

  I was also told that it would have 4x90 degree symmetry over the real axis, although I don't entirely agree with that assessment.

  I was doing something similar to what you did earlier in this thread (adding an additional angular increment to the one axis) and thought why not try out +/-.5 angle differences. 

  Looked good at 2 in the morning.  Now... looks similar to a bunch of other bulbs. 

Somewhere online there's a sketch of what the "true 3D" Mandelbrot *should* look like - it was linked I think somewhere in the original discussion thread but I just looked and couldn't find it.
When I say "should look like" I just mean it's exactly what most folks would expect/want it to look like based on the 2D version i.e. it's purely aesthetically based ;)

  I think I saw the sketch you mention somewhere around here, perhaps in the original thread (someone mentioned it was similar to the tetrabrot, if I recall correctly).  

  I guess I could see someone expecting something along those lines, assuming 1 axis= 0 we should get a 2d Mandelbrot on the other 2 axes plane, but I don't imagine there being something that simple when we add an additional dimension.  I'd think that we wouldn't be able to expect to use the same angular rotation that we use with the x and y components, apply the same rotation to the sqrt(x^2+y^2) and z components and get anything like a regular 2d Mandelbrot on the z/x plane.

  Here is a triple Mandelbrot, bailing out for complex z = x+iy, y+iz, x+iz, iterations z=z^n + relevant pixel components.  Looks weird and picassoesque.  I am sure somebody has tried this before, however.  I suppose someone should try bailing out between z= sqrt(x^2+y^2) + i*sqrt(y^2+z^2).... as well.


 

  Yet another 3d type of fractal.  Doesn't look like a Mandelbrot, but it looks like we could maybe slice it at a certain point and get a Mandelbrot 2d cross section.  It's reflected everywhere, so... weirdness.  More reflection than expected.  Maybe hybridize it with mandelbulb type formulas later.  Seems to be reflected over every axis, probably because of the way I did it.  <-- that looks right

  I like to set bailout really high (1e30), to make bulbous stuff, but I find a bailout of around 64 seems to be a good sweetspot, 128 is nice too.  

Some pictures at the bottom, here is an animation through certain angular increments (and magnitude as well, not really sure actually, I did change the iterations at some point, and you can see the pop):
http://www.youtube.com/watch?v=iPAnK8I0-lU (http://www.youtube.com/watch?v=iPAnK8I0-lU)

In addition to the x,y,z pixel components I created these:
pixelw=sqrt(pixelj^2+pixeli^2);
pixelv=sqrt(pixelr^2+pixelj^2);
pixelu=sqrt(pixelr^2+pixeli^2);

 in the above pixelr= x axis component, pixeli= y axis component...

r2=sqrt(sx^2+sy^2+sz^2);

phi=atan2(sw+flip(sx));     //  flip means change from real to imaginary
tango=atan2(sv+flip (sy));
theta=atan2(su+flip(sz));     // I used theta for the z component instead of the x

nx=r2^n*sin(v*phi);      //  I like to be able to adjust the magnitude
ny=r2^n*sin(v*tango);  //  and angular rotation of these guys
nz=r2^n*sin(v*theta);  //  next step is to do it for each individual
nw=r2^mag2*cos(mag*phi);   //  component, as altering any should give you
nv=r2^mag2*cos(mag*tango);  // a different fractal
nu=r2^mag2*cos(mag*theta);

sx=nx+pixelr;
sy=ny+pixeli;
sz=nz+pixelj;
sw=nw+pixelw;
sv=nv+pixelv;
su=nu+pixelu;


I do bail<bailout check.  So is set bail in the iteration component:

bail=abs(sw)^bail1*abs(sx)^bail2+abs(sv)^bail1*abs(sy)^bail2+abs(su)^bail1*abs(sz)^bail2;

  abs(x) = absolute value of x  (you probably know that)
  bail1 is the wvu component exponent, bail2 the xyz components...

  It doesn't seem to matter how you add/subtract bail between bail1 and bail2 if the xyz and wvu rotation (and maybe mags) are equal.  However, when the rotation/mags are different, picking different values for the bail changes things.
  The first 2 pics are examples of this: the one has bail1=4 bail2=-6, the other is bail1=3 bail2=-3 and every other fractal component is the same: I'll explain my picture name format in a later post, I like to include iterations, bailout, and all the other components with the pics.

  I haven't done julias of this new type yet.  Never got too familiar with julias, but you guys know the coding difference: static values instead of the pixel values added every iteration.

  Sorry about the sloppy writing.  Brain tired.

i know, it gets you like that. Anyway welcome to the thread. Your palette reminds me of those rocket ice lollies you used to be able to get.yummy!
Reflection and the mandelbrot might be a useful avenue to pursue especially as you are getting the rotation and extension through the v and n variables, but can you control the axes that express reflection? You may need to do this to approach a sculpting of a 3d mandelbrot.

  Thanks jehovajah,

  I'm thinking we can accomplish that by taking a hybrid version between the mandalball (from mandala and mandelbrot/bulb and its spherical tendencies in higher rotations) and the mandelbulb/mandelbrot x/y.  

  Do the standard x,y things from mandelbulb/mandelbrot, but do the mandalball z axis equations (which generate reflections).  Gonna try it in a few hours after this current animation is done computing and I've done all that I have to get done.  Maybe after dinner I will have a few minutes.

  Can't get caught up now, so I'll try and just leave this message, in case I forget what I was going to try.. I'll probably see this again.  :D


  Hahaha, mid work out I thought: maybe we should differentiate pixelw, pixelu, pixelv components:
for pixelw, if pixeli and pixelj are the same sign, leave as is.  If they are the opposite:
pixelw=sqrt(abs(pixeli^2-pixelj^2))

  So basically, if pixeli*pixelj<0 ....   (do the same for other components)
  Of course, the later equations might need to be modified as well.  Back to the work out.  Try not to think about fractals.  :D

  Allright, here is yet another one with the newest formula type, except I added signs to the w,v, and u values.  The thing is, instead of having the reflection over the x axis, I took into account whether z and y are positive or negative.  That might be the equivalent of taking the original 2d mandelbrot and having different values generated if y is positive or negative (no reflection over x axis): haven't thought about it enough yet, so this may be a hasty incorrect conclusion.  At very low iterations, it has complex repeated details.  We can increase repetition by using the original pixel assignments instead of the new ones, or simply making the new ones less complex (perhaps simplify it to what I said in the previous post: if pixeli*pixelj < 0 do the one thing, if not, do the other).

  We've got a very interesting beast here.

  I set values with  (need to look this over, feel free to):
   /* pixelw assignment  */

      if (pixeli<0 && pixelj<0) {      
             pixelw=0-sqrt(pixelj^2+pixeli^2);
      }
      else if (pixeli<0 && pixelj < abs(pixeli)) {
         pixelw= 0-sqrt(abs(pixelj^2-pixeli^2));
      }
      else if (pixeli<0 && pixelj >= abs(pixeli)) {
      pixelw= sqrt(abs(pixelj^2-pixeli^2));
      }    
                 else if (pixelj<0 && pixeli < abs(pixelj)) {
      pixelw= 0-sqrt(abs(pixelj^2-pixeli^2));
      }
                 else if (pixelj<0 && pixeli >= abs(pixelj)) {
       pixelw= sqrt(abs(pixelj^2-pixeli^2));
            } else {
       pixelw=sqrt(pixelj^2+pixeli^2);
      }
      
/* pixelv assignment  */

      if (pixelr<0 && pixelj<0) {      
         pixelv=0-sqrt(pixelj^2+pixelr^2);
      }
         else if (pixelr<0 && pixelj < abs(pixelr)) {
            pixelv= 0-sqrt(abs(pixelj^2-pixelr^2));
      }
         else if (pixelr<0 && pixelj >= abs(pixelr)) {
            pixelv= sqrt(abs(pixelj^2-pixelr^2));
      }  else if (pixelj<0 && pixelr < abs(pixelj)) {
          pixelv= 0-sqrt(abs(pixelj^2-pixelr^2));
      }else if (pixelj<0 && pixelr >= abs(pixelj)) {
          pixelv= sqrt(abs(pixelj^2-pixelr^2));
      } else {
         pixelv=sqrt(pixelj^2+pixelr^2);
      }
      
/* pixelu assignment  */

      if (pixelr<0 && pixeli<0) {      
         pixelu=0-sqrt(pixeli^2+pixelr^2);
      }
         else if (pixelr<0 && pixeli < abs(pixelr)) {
            pixelu= 0-sqrt(abs(pixeli^2-pixelr^2));
      }
         else if (pixelr<0 && pixeli >= abs(pixelr)) {
            pixelu= sqrt(abs(pixeli^2-pixelr^2));
      }  else if (pixeli<0 && pixelr < abs(pixeli)) {
          pixelu= 0-sqrt(abs(pixeli^2-pixelr^2));
      }else if (pixeli<0 && pixelr >= abs(pixeli)) {
          pixelu= sqrt(abs(pixeli^2-pixelr^2));
      } else {
         pixelu=sqrt(pixeli^2+pixelr^2);
      }

Here is a youtube of the old formula (pixelw=sqrt(pixeli^2+pixelj^2) rather than the above complicated sign assignments), which is basically a 3d mandala animation.

  You can use the old formula to make pretty mandala type things by setting bail1 and bail2 to opposite signed values at the same time having high bailout, low iterations, and I think you need to vary the wvu and xyz rotations.

  Of course, the new formula is just freaky.

http://www.youtube.com/watch?v=n59GN4yEbT4 (http://www.youtube.com/watch?v=n59GN4yEbT4)

  Here are 4 more pics from the new formula.  Order 3 and order 4.  Order 3 is rather boring.  Order 4 cooler.

  Haven't even altered the bail settings, or really messed with rotations a lot yet.  Although, I have been using only wvu bail=^2, and haven't even done xyz bail (I wonder if its totally different now that things are signed?).

  Here are the pics:

  And here is one with 2 xyz rotations and 3 wvu rotations, bail1 (wvu) = 1 bail2 (xyz) = -1

  Notice the demonic face on the right, and the angelic statue on the left.  Probably simply due to clipping (if you've looked at some of the 180 degree rotations I posted above, you can see that parts get clipped out due to my background detection).

One last round of pics of the new type tonight. 

  I extended it to 4d, and as you can imagine, it took a bit to write out and double check the +/i assignments for the variables.

  Extending the equations is the simple part...  But when you see the code for sign assignment.... sheesh.

  Here are a couple (not many, very tired, gotta eat some food and crash) of the new 4d variety.  Just changed the times a bit.

  I've gotta clean up the code, remove extra fractal test types, extra variables that were added, etc.  I think tomorrow I can, but maybe I'll be busy.

  Gotta decide whether to keep the extreme variation in signs, or make them simple and reduce fractal variety.  At the moment, the 2 sides of the fractal are just about day and night.  Both have interesting features, so it would be a shame to loose a quadrant of unique activity just to have a symmetric fractal.  On the other hand, some people like symmetry, but still desire more variety than the mandalaballs from the videos.  Maybe make 3 different varieties.. or more.  The basic knowledge is here in the thread, and I'll post the extended dimensional version for others to pick over in a bit. 

  Here are the 4d pics, at various (probably similar times), 2 of the standard 2 rotation (2nd order) at t1~.2, and I think 1 is what I've dubbed "mandala mode" (a checkbox in my parameter window).  Nothing pretty.  Just a demo of the additional possible weirdness.  A side note: takes longer to calculate these, so... these are lower resolution, I am on a laptop, after all. 

  From what I've been able to tell, we can have symmetry at the sacrifice of details included in the fractal.  I've done a bunch of quick and dirty math, and ended up with various results <-- that's pretty specific, ehh?

  We can make many specific symmetric fractal types, each with a quadrants detail repeated through all quadrants.  

  For a quick and dirty new look to the mandalas, reverse the sign from

pixelw= sqrt(pixeli^2+pixelj^2)    to...   pixelw= 0-sqrt(pixeli^2+pixelj^2)    // I wrote 0- because the software I am using does not recognize leading -

and do the same for pixelv and pixelu...

Another thing you could do is change the sign within the magnitudes of the mandala:
pixelw= sqrt(pixeli^2 + pixelj^2)    to...   pixelw= sqrt(pixeli^2 - pixelj^2)
which gives you another variety.

  Of course, you can combine the two as well (making the new magnitudes negative), or divide them into quadrants as I did in post 90.

  So, for now, it is time to clean up code, set up switches, or types (mandala +/-, mandala magnitude subtraction +/-, quadrant +/-....).

  Some input might help, or if a moderator takes these posts to a new thread, or maybe we should just start a new thread from the beginning (although it seems to be related to this threads topic).

  And here is a new type, I took the angles by squaring the x,y,z axis components, used the negative of mandalamode pixelw,pixelu,pixelv

pixelw= 0-sqrt(pixelj^2+pixeli^2)   
pixelv= 0-sqrt(pixelj^2+pixelr^2)
pixulu= 0-sqrt(pixelr^2+pixeli^2)


theta=atan2(su+flip(sz^2));  <--- squared z axis component
phi=atan2(sw+flip(sx^2));     <--- squared x axis component ... so on and so forth
tango=atan2(sv+flip (sy^2));

  Here is a simplified assignment for pixels w, v, and u, which is much easier to extend to higher dimensions:

pixelw=pixelj*abs(pixelj)+pixeli*abs(pixeli);  //abs(variable) means take the absolute value of the variable
pixelv=pixelr*abs(pixelr)+pixelj*abs(pixelj);  // don't know if there is a squaring function that preserves sign, or this
pixelu=pixelr*abs(pixelr)+pixeli*abs(pixeli);  // could be even simpler....

  if (pixelw<0) {
  pixelw=0-sqrt(abs(pixelw));
  }   else {
  pixelw=sqrt(abs(pixelw));
  }
  if (pixelv<0) {
  pixelv=0-sqrt(abs(pixelv));
  }   else {
  pixelv=sqrt(abs(pixelv));
  }
  if (pixelu<0) {
  pixelu=0-sqrt(abs(pixelu));
  }   else {
  pixelu=sqrt(abs(pixelu));
  }

  this is just for the complicated assignment ones.  Making a video that I will add to this post later.

  Matt

  Here is the link to the video.  Not exciting, just a bailout reduction video to show some details (by bailout amount) in a low iteration count new fractal type:
http://www.youtube.com/watch?v=lAny0QDBBRQ (http://www.youtube.com/watch?v=lAny0QDBBRQ)

I suppose it depends on the compiler and such, but in some cases pixelj^2*sign(pixelj) might be faster.

  Cool, I'll check and see.  

  Another thing I thought of is modifying the equation, leaving out calculating nw,nu,nv, and the pixelw,v,u values altogether and after:
sx= nx +pixelr;
sy= ny +pixeli;
sz= nz +pixelj;

doing:

sw=sy*abs(sy)+sz*abs(sz);
if (sw<0) {
sw= 0-sqrt(abs(sw));
} else {
sw= sqrt(sw);
}
.... same for sv and su values.  I wonder how that will look.  :D  For later though, got to do tons of stuff tonight.

  Ok, here are a couple of images with the above modification.  Note that I also set angles with

theta=atan2(su+flip(sz^v1));  //changing v1 from 1 to 2 for the following images made relatively boring fractals.. interesting
phi=atan2(sw+flip(sx^v1));    // seems to make slightly "dark" looking fractals...
tango=atan2(sv+flip (sy^v1));  // the demon head fractal was a power 2 one as well

Are you kidding!  I see them in my sleep!!!.   :laugh:

Anyway well done with what you are exploring so far. I am busy writing up polynomial rotations so i have missed a day, and my what i have missed. Will comment when i have more time but looks like the search is on!

  I'm wondering if these fractals are even Mandelbrot extensions into 3d.

  In 2d we rotate x values against y values, which, if we take this formula down to 2d we would be doing.

  In 3d we rotate x and sqrt(y^2+z^2), so in 2d we would rotate x and sqrt(y^2).. x and y.  I suppose this requires paper and pen, or at least a wxMaxima window so I can look at the interactions directly.

  That ideas thrown into the mix, so here is some more from the latest addition to the new fractal formula family, although it's the same exact fractal as above, I did an incremental bailout increase because I like the fractal and wanted to see it "grow".  I need anti-aliasing, and I set background detect to "Extremely Soft" which reduces calculation time but causes the clipping you see in many of the images I post- in other words, there are hard edges and a few noticeable clips.

http://www.youtube.com/watch?v=UWgfAK4HmqA (http://www.youtube.com/watch?v=UWgfAK4HmqA)

  In the pic below, I set background detect (also altered magnitude (not rotation) by +.5, so that explains the slightly different features) to standard instead of extremely soft like in the video above.  I think this may be enough to redo the above video and eliminated clipping, but maybe not.  For now, I have tons of stuff to get done, so barring an exciting idea, will be away for a bit.

  matt

Interesting pictures,
here´s another one, maybe we can take these attempts and efforts as some sort of brainstorming,
leading to nowhere or to an unexpected end:
In this, it is the mandelbulb algorithm, but now not adding the vector c as usual, but the vector c´,
the vector some sort of conjunctive to c, in relation to the equatorial plane, in which the original mandelbrot structure will shine up.
In that plane, the conjunctive vector will cincide with the original vector c-
it was just trying-

Looks good, so can you render alternate views?  Which algorithm are you using? Can you define your conjunctive to the equatorial plane?

@Mbenesi: can you control the reflections in the northerly directions? Maybe stop reflecting in the horizontal axis, so the bottom part can develop and be looked at in 3d?

sorry, wrote a long reply, an error occured, now a shorter version:
with conjugated i mean the original vector c with x1,y1,z1 will be conjugated by taking -x1 instead of x1.
(In your programs it might be y1 or z1, your coordinate system is oriented differently, the vector will be symmetrical to the plane, for which phi is always zero in the mandelbulb algorithm, it is the equatorial plane, in which the mandelbrot structure will occure)
In the attached picture, the vector z´, which will be added at the end of each iteration, is reflected to this plane, but the determining its angle theta and phi, phi will be changed to 0.5, at last, the original value of phi will bechanged to -0.5*phi, torn towards the equatorial plane. But then also conjugate the value of the point znew, which we get as the result of squaring the triplex number(doubling theta and phi, squaring the length of this vector), so the result will be no more simple locodromic like spirals cut more trajectories in form of a "bretzel" an 8 .
the complete formula(working in fine chaospro) also attached ;)

viewing from a side

 I suppose eliminating one of the variable assignations might work (any of the wvu pixel assignments), or at least altering a couple of them.  The variable assignation methods assign sign values to quadrants of the fractal.  Perhaps we can eliminate sign values, perhaps we can eliminate differences in calculated magnitude (instead of magnitude of sqrt | pixeli * |pixeli| + pixelj* |pixelj| | which results in more differences in the various quadrants).

 I figured out that the new type is definitely not reducible to a 2d Mandelbrot.

  A trigonometric 2d Mandelbrot's equations look like this:
r2=sqrt(bail);
theta= atan2 (x+flip(y));
a=cos(n*theta)*r2^n +pixelr;
b=sin(n*theta)*r2^n +pixeli;
bail=x^2+y^2;

  The new type reduces to this:

r2=sqrt(sx^2+sy^2);

phi=atan2(sw+flip(sx));
tango=atan2(sv+flip (sy));

sx=r2^n*sin(n*phi)+pixelr;
sy=r2^n*sin(n*tango)+pixeli;
      
sw=r2^n*cos(n*phi)+pixelw;
sv=r2^n*cos(n*tango)+pixelv;

Which actually combines 2 2d Mandelbrot calculations into one beast.  If we take:

r2=sqrt(sx^2+sw^2)   or  r2= sqrt(sy^2+sv^2)

we get a standard 2d Mandelbrot rotated 90 degrees between the 2 formulas.  We don't get anything great looking (at the 2d level) if we calculate the combined Mandelbrots at high iterations.  Some neat low iteration shapes emerge, but nothing of the higher iteration variety:  these aren't standard escapetime fractals with higher iterations leading to more replicated details emerging like in the 2d Mandelbrot or 3d Mandelbulb.  If we combine the 2 magnitudes by applying them to the specific variables they are associated with, we get a quasiMandelbrot reflected diagonally over a quasiMandelbrot.


  Extending the quasibrots up to 3d doesn't look promising.  Get rather boring images, more like quaternion fractals than some type of 3d Mandelbrot.  Perhaps using a combination of Mandelbulb and new type would give us something interesting.  

  So, I happened upon the definition of the word "fractal" in the wikipedia article, while trying to find out about the formula I've been using.

  Apparently it is not a fractal due to its lack of self replication at the smallest level.  I don't know the term for what the formula generates (specific term, it generates geometric objects, obviously). 

  Not even sure if it is appropriate for the fractal forums? 

  Anyways, a couple last pictures of it, hopefully Vector pursues his method and hits gold, I might try to mess around with this formula and tweek it until I can get actual fractals out of it. 

 

 @ mbenesi.  Fractal i define as being the product of an iterative procedure, so these are fractals no matter what. Would it help if you thought of this as an object sculpted out of a solid sphere reflected into four axes?   What we are trying to achieve is the most analogous form to the 2d mandelbrot that is as aesthetically interesting and fascinating at all levels of magnification. The formula for the procedure that sculpts this object is sought for a space we have no proper reference in, and so proceeds by analogy. To say that the solution is still out there is misleading. We have a solution, but it is not as aesthetically pleasing as some of us would like, However the many beautiful sculptures you and others have produced are not to be belittled.

While you reflect on what you have been producing remember that you have 2 strains of ideas here and masses of ways to explore. This can seem overwhelming, so narrow your researches down to explore each part of what you have produced. You have some fascinating stuff, and some novel approaches which are inspiring. :dink:

Thanks,

  Here is a low resolution animation I made with the new type I have been working on.  Hopefully I'll work with it, extend it to higher dimensions, clean it up, and perhaps combine it with the other approach and see what that gives us. 

  Here it is.  Definitely interesting variety of details.  If you can watch it in slow motion, it is interesting to see the changes that increasing angular rotation can cause.

  http://www.youtube.com/watch?v=LIJIsc5oA9M (http://www.youtube.com/watch?v=LIJIsc5oA9M)

beautiful animation :D
I'd also consider that a fractal in a way...
it self-organizes after repeating the same rules a multiple times... :) (If it would diverge or converge against a single point, it wouldn't be a fractal, I guess...)

Maybe you could do some box-counting... It most likely will have a fractal dimension :)

Thanks, gotta learn about box counting.  Don't know how easy it is to do with what I have, maybe do it in Maxima.

  For some reason, I don't think these objects are "fractal" over  standard space, rather I think they might have fractal properties (of self similarity) over rotation "velocity" (what you multiply the angle by). 

  matt

  Ohh, here is an "old" one, rendered with anti-aliasing enabled.  Looks better, I think:

  Here is an additional animation of the new type.  It's low resolution, so a lot of details are hard to pick out, but you can see some of the variety that can be found with rotation & magnitude variation.  Also, for some reason, it kept losing color when I started the animation, and I was in the middle of watching a movie with the woman so I just let it go after the third try.

  In other words, it's gray, in addition to being low resolution. 

  http://www.youtube.com/watch?v=T_Ws7f5kPkc (http://www.youtube.com/watch?v=T_Ws7f5kPkc)

  Here is a second one, same exact formula, a few starting parameters different, which makes for a different base pattern.  More like a flower.  In this one, colors worked AND it's a slightly higher resolution.  Maybe a little less clipping issues as well.

  http://www.youtube.com/watch?v=7QH0cQIvK4I (http://www.youtube.com/watch?v=7QH0cQIvK4I)

Absolutely fascinting! The animations alone have a purpose to illustrate electron probability shells! One could even illustrate gravitational collapse. Anothwe could be a form of big bang explosion and the expansion of the universe! big ideas i know for what you have been tinkering with,but i mean to inspire you!  :o

Hi Benesi,
the animations really look very well and have a certain beauty,
i sometimes(as most of You also will have) got Julia objects, with a very caleidoscopical aspect.
Indeed, each circle(or little bulb) on the periphery of a Julia set is a picture of the circle, which is defined by
the circle with radius Bailout(or it can at least be interpreted as such a picture). So, the Julia sets consist(as far as i understood with limited mathematical base) of lots of mirrored pictures of a starting circle. So in nhyperbolic geometry, there are Julia sets, got only by consecutive mirroring at the hyperbolic lines(the orthogonal circles representing those straight lines).
Maybe, a 3d M-set could be constructed by those mirrors, which, very sadly, would be no planar mirrors, but f.i. spheres as in the case of the hyperbolic geometry-pictures. I think it has been done to some extents at the very beautiful, well known, video of Quasi-Fuchsian Kleinian groups(i hope, it is named correctly by me). Have You got an idea, what Your formula is doing? Which way a point will be projected by it? ;)
 

 Hi all,

  I suppose I could find out which way a point will be projected since they are only going through a few iterations to reach the complexity I desire (4-6 for the most part, some are higher).  

  That's only 4-6 rotations and additions of the original value, so I could write out an algebraic equivalent (for each number of iterations/ bailout settings).  From there we could map the direction the points will travel and perhaps find their acceleration; which is related to the angular velocity (change in angle multiplication) of the point in space.  Of course, it's a little more complex than simple accelerations.  :D

  Working on an old formula trying to find out about it (did others work on this?  I suppose I should read the whole original thread):

r2=sqrt(bail);   // I set the "original" bail before this loop, avoiding an additional variable for bailout
                    // and I also set the original sx,sy,sz as pixelr,pixeli, and pixelj respectively before starting the loop
                    // pixelr is the x component, pixeli the y, pixelj the z, it's easy to make a julia of this...

theta=atan2(sx+flip(sqrt(sz^2+sy^2)));   //flip means multiply the value by i
phi=atan2(sy+flip(sqrt(sx^2+sz^2)));
tango=atan2(sz+flip(sqrt (sx^2+sy^2)));
         
         
sx= r2^n * cos (n*theta) +pixelr;  // n is the number of rotations and magnitude exponent... just like 2d complex equivalent
sy= r2^n * cos (n*phi)    +pixeli;
sz= r2^n * cos (n*tango) +pixelj;
         
bail=sx^2+sy^2+sz^2;

  You only get regular "bulbs" for odd powers of n (or mixing values....).  Powers 5, 9, 13 .... seem to be the best for whatever reason, although I was playing with 11 recently.  No great images.  Basically lots of fractal type replication, but more like chasing a 2d mandelbrot set (you have to seek out the 3d equivalent of a "seahorse valley" instead of being able to zoom in anywhere in the fractal).  There is fractally goodness along certain areas, but I haven't been able to go to deep with ChaosPro, get lots of clipping issues, etc.

  I'm thinking the fractal is more symmetric at powers of 1+n*4 because these are the powers of the 2d mandelbrot set that are most symmetric.  The spikes seem to be in areas that correspond to the bulbs breaking off the equivalent 2d mandelbrot, and the structured parts are in the corresponding valleys between the 2d mandelbrot bulbs.

  I did notice that you can travel off too the side of the main pattern after zooming in a bit and find slightly different side patterns: keep in mind this is only at ~19-23 iterations, not 100-1500 iterations like the 2d mandelbrot when you start finding crazy new patterns in the valleys.

  I also have lots of troubles with clipping (hit the "adjust" button a bunch of times and wait for the image to recalibrate, hoping that it hits the right range... patience is required) when I start getting up into "high" iterations (23 or so) as the adjust button apparently hits a maximum (both numbers are "0" after iteration 19 or 20 instead of being .00000001 and .0000000001 or whatever).  

  If you do explore the version: below is a z^9 and z^13 version, including an extreme zoom of the z13.  You can see what I call escape (it's the opposite, I know) spikes, where the fractal's area is actually larger (so the spikes are the part of the set that DOESNT escape, but...) , and you can see the axes where fractally stuff happens.  Apparently the 'escape' spikes are attractors that distort the fractally areas, although you probably need to zoom into the fractally areas a bit before going towards the escape spikes or you will simply enter an escape zone, which is pretty boring.  

  I also included a maximum zoom (for my software, you will see the weird lines that start appearing in the graphics and notice some clipping issues) z^13 (only 16 iterations for the high z^n value reaches max zoom detail).  You can keep on zooming into the detailed areas (not the escape spikes: the escape spikes appear between the details, even when you keep on zooming in) that look a bit like the eye thingies that pop up in mandelbulbs.  I assume you can zoom into the tentacle thingy coming out from under the escape spikes as well, but my software is maxed out and will introduce crazy-ass artifacts if I zoom more.  

  If you stay on the central point of one of the bulbs from the main figure, you can zoom (until my software maxes out: perhaps forever) and keep on finding the same bulb as a little bulb within the bulbs around it.  I did this for the extreme zoom z^13 one, but traveled off to the side of the central bud and started selecting deviations from the main.  If you deviate you can find replicas as well, just like in normal 2d mandelbrots.

  I also included a couple of zooms of the batcave in z^9 in the post below.

Batcave zooms.  For this z^9 run I haven't run into my max resolution yet, even though I'm at higher iterations (I think I went up to 24, maybe higher) than when I maxed out for the z^13 run (16 iterations). 

  You can keep zooming into the thing, or pix another nexus of activity and zoom into that for a slight variation.

well, that one's clearly fractal :)

Yeah.   Yet another 3d Mandelbrot variety.  At least this formula is fractal, instead of just a pretty pattern generator.

  Wonder how it would look in someone's better raytracer?

  Allright, yet another new formula type:

r=sqrt(bail);
theta=atan2(sk+flip(sz));
phi=atan2(sy+flip(sx));

  Set bail as x^2+y^2+z^2 to initialize.  Initialized sx,sy,sz,sk as corresponding pixel values.  pixelk= sqrt( pixelr^2+pixeli^2), as usual pixelr is the real or x component, pixeli is imaginary or y component, pixelj is the z component of the pixel...

sx=2^n*sin(n*phi)*cos(n*theta) + pixelr;
sy=r^n*cos(n*phi)*cos(n*theta) +pixeli;
sz=r2^n*sin(n*theta)                +pixej;
sk=r3*cos(zfold*theta)              +pixelk;   //  I did make an if statement in my formula so I can switch this to -pixelk instead of +pixelk

bail=sx^2+sy^2+sz^2;   //also check if bail < bailout value, if so, bail...

Here is an incremental zoom out from a deep zoom little knob with lots of details, first one is higher res/iterations but same as the second one.

(http://www.fractalforums.com/gallery/1/1170_05_02_10_9_35_23_0.jpeg)
(http://www.fractalforums.com/gallery/1/1170_05_02_10_9_35_23_1.jpeg)
here comes the zooming, zoomed out a bunch between images, usually with a little spot towards the center as the old image.
(http://www.fractalforums.com/gallery/1/1170_05_02_10_9_35_23_3.jpeg) (http://www.fractalforums.com/gallery/1/1170_05_02_10_9_35_23_4.jpeg) (http://www.fractalforums.com/gallery/1/1170_05_02_10_9_36_47_0.jpeg)

(http://www.fractalforums.com/gallery/1/1170_05_02_10_9_36_47_1.jpeg) (http://www.fractalforums.com/gallery/1/1170_05_02_10_9_36_47_2.jpeg) (http://www.fractalforums.com/gallery/1/1170_05_02_10_9_36_47_3.jpeg)

(http://www.fractalforums.com/gallery/1/1170_05_02_10_9_36_47_4.jpeg) (http://www.fractalforums.com/gallery/1/1170_05_02_10_9_39_05_0.jpeg) (http://www.fractalforums.com/gallery/1/1170_05_02_10_9_39_05_1.jpeg)

Had to reset because I was in a bad location.  The blue nob at the top is where I zoomed into (or out from).

(http://www.fractalforums.com/gallery/1/1170_05_02_10_9_35_23_2.jpeg)

  Anyways, the new fractal type has some potential.  The other new type is really fractally as well, and needs more exploration.  But I keep on playing with the formulas... ....  

Ok, the second order (z^2) thingy of this is nuts.  It's got all these twisted patterns, flowers, etc.  

  Just set k=-k to see it (I have a thing in my formula to change scale and change k's sign, setting it to -k is wierd, ultra detailed, and weird also).
Zooms generally have increased iterations, just to bring out more details (and I'm not trying to render tons of details until I find something anyways)...


Zoomed into notch at bottom, you might be able to make out exactly where:
 (http://www.fractalforums.com/gallery/1/1170_06_02_10_5_08_36_0.jpeg) (http://www.fractalforums.com/gallery/1/1170_06_02_10_5_08_36_1.jpeg)
This is some neat bud thing somewhere in it (I got lost when I found it):
(http://www.fractalforums.com/gallery/1/1170_06_02_10_5_08_36_2.jpeg) (http://www.fractalforums.com/gallery/1/1170_06_02_10_5_08_36_3.jpeg)

Twisty buds zoomed in from left hand pic to right:
(http://www.fractalforums.com/gallery/1/1170_06_02_10_5_08_36_4.jpeg) (http://www.fractalforums.com/gallery/1/1170_06_02_10_5_09_48_2.jpeg) (http://www.fractalforums.com/gallery/1/1170_06_02_10_5_09_48_0.jpeg)

  Gonna add a top and side view in a bit.  Here's the top, side is calculating:

(http://www.fractalforums.com/gallery/1/1170_06_02_10_5_26_09.jpeg)

side:
(http://www.fractalforums.com/gallery/1/1170_06_02_10_7_13_03_0.jpeg)

rear:
(http://www.fractalforums.com/gallery/1/1170_06_02_10_7_13_03_1.jpeg)


  The fractal details are weird, I think it doesn't have the patterns that the higher orders z^3++ do.  You can zoom in on them, but they are pretty repetitive, even if they are slightly different.  Perhaps I'm just not looking at the right places.

  Here is a z^4 (working on a z^3, wonder how it will look... I'll post it):
(http://www.fractalforums.com/gallery/1/1170_06_02_10_7_53_21.jpeg)

Update: or not post it.  Pretty boring. 

nice stuff :D

i´ve got lots of distorted M-sets on screen, These look very interesting, will try them on my own too, seems as if you were taking three angles from the vector to the starting point z to all three axes, then doubling those, or multiplying by n, this seems different to the common approaches, at least at first look-carry on

Alright guys, here is another of 2 new formulas (yeah, I know... making lots of formulas, not exploring them a lot), but I really like the symmetry of this one.

   r=sqrt(bail);
        theta=atan2(sx+flip(sy));
        tango=atan2(sx2+flip(sz));

 sx=r^n*cos(theta*scalef)+pixelr;
         sx2=r^n*cos(tango*scalef)+pixelr;
      
        sy=r^n*sin(theta*scalef)+pixeli;
        sz=r^n*sin(tango*scalef)+pixelj;

sx=sqrt(abs(sx*sx2));   //  I have an if statement so I can switch between this one (which I like more) and
                                // this more traditional formula (which I find more boring):  sx=sqrt( (sx^2+sx2^2)/division variable);
                               //  check that... I changed the division variable to 1... and it is interesting for the z^9 version posted below

bail=sx^2+sy^2+sz^2;

  The first 2 are of the front of the 4th order version (I like the front's 4x90 symmetry<-- joke for MakinMagic through all z^n, but... higher is more intricate as you will see in the 9th order version).

  Keep in mind I think these are all low iteration (5), just to make quick demo images.

(http://www.fractalforums.com/gallery/1/1170_06_02_10_7_36_48_0.jpeg)
zoomed a bit (I think):
(http://www.fractalforums.com/gallery/1/1170_06_02_10_7_36_48_1.jpeg)
This is the same location as the last image above, just increased to z^9:
(http://www.fractalforums.com/gallery/1/1170_06_02_10_7_36_48_2.jpeg)
This next one is the z^9 from the rear.  These guys are really fun to zoom into.
(http://www.fractalforums.com/gallery/1/1170_06_02_10_7_36_49_3.jpeg)
z^9 from the rear, sx=sqrt(sx^2+sx2^2):
(http://www.fractalforums.com/gallery/1/1170_06_02_10_7_50_04.jpeg)

  I will tell you that I've zoomed into the center of the sx=sqrt(sx*sx2) mode ones pretty much as far as I wanted to go, and it was all fractally.  Time to check if the rear entry on the sx=sqrt(sx^2+sx2^2) one is as fractally/entertaining.


  It's like a hole with a bunch of squid suckers on the side.  Interesting, I guess.  :p  There may be weirdness in the suckers.

  Ok, here are two more 9th order images.  (z^9)

  Same exact location, but the first is sx=sqrt(sx^2+sx2^2):
(http://www.fractalforums.com/gallery/1/1170_06_02_10_8_02_43_0.jpeg)

and this one is sx=sqrt(sx*sx2)     //  you should see why I like this mode a bit more, the details.. the details

(http://www.fractalforums.com/gallery/1/1170_06_02_10_8_02_43_1.jpeg)

Here is another formula:
  
        r=sqrt(bail);
        theta=atan2(sx+flip(sy));
        tango=atan2(sx2+flip(sz));  // sx2,sy2, and sz2 are set to sx,sy, and sz respectively before iteration begins
        tau=atan2(sy2+flip(sz2));   //  bail is set to sx^2+sy^2+sz^2 before iterations as in the rest of my formulas
        
        sx=r^n*cos(theta*scalef)+pixelr;
        sx2=r^n*cos(tango*scalef)+pixelr;
        sy=r^n*sin(theta*scalef)+pixeli;
        sy2=r^n*cos(tau*scalef)+pixeli;
        sz=r^n*sin(tango*scalef)+pixelj;
        sz2=r^n*sin(tau*scalef)+pixelj;

sx=sqrt(abs(sx*sx2));  //more detailed mode, you do need to zoom into specific areas to get fractally goodness
sy=sqrt(abs(sy*sy2));
sz=sqrt(abs(sz*sz2));
  OR:
sx=sqrt((sqr(sx)+sqr(sx2))/fixedRadius);   // this is the "boring" mode
sy=sqrt((sqr(sy)+sqr(sy2))/fixedRadius);
sz=sqrt((sqr(sz)+sqr(sz2))/fixedRadius);

and of course:
sx2=sx;
sy2=sy;
sz2=sz;

  bail=sx^2+sy^2+sz^2.......

  This first one is to demonstrate this formulas potential.  It's the top of the front of the 9th order version in what I called "the more detailed mode".  Note that it appears to be sharp and pointy, but this is an optical illusion, it is actually bulbous and oozey (you need to look at it the right way and you will see this).  I'll have to locate it and post a less detailed image or alter lighting so you can tell.

(http://www.fractalforums.com/gallery/1/1170_06_02_10_8_08_55_0.jpeg)
Here is the front of the 9th order:
(http://www.fractalforums.com/gallery/1/1170_06_02_10_8_20_43_0.jpeg)
Same location, "boring" mode:
(http://www.fractalforums.com/gallery/1/1170_06_02_10_8_20_44_1.jpeg)

Here is the rear of the 9th order:
(http://www.fractalforums.com/gallery/1/1170_06_02_10_8_20_44_2.jpeg)
And here is a random 12th order location:
(http://www.fractalforums.com/gallery/1/1170_06_02_10_8_08_55_1.jpeg)

  There are some neat places in this one, but I find type B to be more pleasing to the eyes, except for some stark locations within Type C which... appeal more to my imagination and love of stark mountains rising out of cold glaciers.

In Type B, setting angular rotation to -2 instead of 2 looks a bit promising.  Doing a render of sqrt(sx^2+sx2^2) first, then sqrt(sx*sx2).

  The renders were slightly boring.  
First one is sqrt(sx^2+sx2^2) mode:
(http://www.fractalforums.com/gallery/1/1170_07_02_10_2_37_58_0.jpeg)
this one is sqrt(sx*sx2) mode (I like it more although it is lower z-resolution and lower iterations):
(http://www.fractalforums.com/gallery/1/1170_07_02_10_2_37_58_1.jpeg)


This z^4 Mandelbulb B (<-- the new formula in this thread) is pretty neat, although I need to remember to start rendering png(s) instead of jpg(s), as the quality is lower than the image in ChaosPro.  Probably another setting I don't know about, like the parameter tab that allows you to set z-axis resolution, which makes it so you can zoom more..

  It looks like there is a cup in the middle of it.  Not to mention the clusters of grapes at the top, the leaves, and the other clusters... They all look neat with pixelk set negative.
(http://www.fractalforums.com/gallery/1/1170_07_02_10_2_40_17.jpeg)
  

Here is yet another new formula and some images from it.  This one is a combo of Mandelbulb B (my xy-plane vs. z axis type) and type B that I call type B2 because of the extreme originality of the name.

  pixelw=sqrt(pixelj^2+pixeli^2)*scalef;   <-- pixelw is one of the variables we will be using in this one.
  //set:
sw=pixelw;
sx=pixelr;  <-x component of pixel
sy=pixeli;  <-y
sz=pixelj;  <-z

 r=sqrt(bail);
theta=atan2(sx+flip(sy));   <--it's like type B here
tango=atan2(sx+flip(sz));
whiskey=atan2(sx+flip(sw));   <-- then I do a play out of the Mandelbulb B handbook here
r2=r^n;                                 //  the difference is that I use sx generated with whiskey for all of them, instead of using the angle
                                           //  and applying it to sy and sz with trig functions, of course, also in Mandelbulb B we only have
                                           //  2 angles, one between sx and sy, and one between sz and sk... but you can re-read that one
                                           //  earlier in the thread

sx=r2*cos(whiskey*fixedRadius)+pixelr;   //fixedRadius is a SCAVENGED variable from another formula in this set of formulas
sy=r2*sin(theta*fixedRadius)+pixeli;        // you can just set it to n (the magnitude), but if you want to set angular rotation
sz=r2*sin(tango*fixedRadius)+pixelj;       //to a negative number you need to use different variables... and fixedradius is in a
sw=r2*sin(whiskey*fixedRadius)+pixelw;  //convenient location on my menu... soo... therefore I used it

  bail, once again:

bail= sx^2+sy^2+sz^2;   // if bail>bailout then bail....

Ok.  The images from this one are pretty neat.  I like them.  They seem to have decent variety, even at low iterations. 

Here ya is, some 9th order images (lots of zooms in the same area):
(http://www.fractalforums.com/gallery/1/1170_07_02_10_10_28_29_0.jpeg) (http://www.fractalforums.com/gallery/1/1170_07_02_10_10_28_29_1.jpeg) (http://www.fractalforums.com/gallery/1/1170_07_02_10_10_28_29_3.jpeg) (http://www.fractalforums.com/gallery/1/1170_07_02_10_10_28_29_4.jpeg)
Here be some 10th (all lower than 10 iterations from the same location in the front, just zooming more and more):
(http://www.fractalforums.com/gallery/1/1170_07_02_10_10_29_43_1.jpeg) (http://www.fractalforums.com/gallery/1/1170_07_02_10_10_29_43_2.jpeg) (http://www.fractalforums.com/gallery/1/1170_07_02_10_10_29_43_3.jpeg) (http://www.fractalforums.com/gallery/1/1170_07_02_10_10_29_43_4.jpeg)

Way to go ! Each different form has its own intrinsic interest and really shows that there is no standard 3d mandlebulb, because the extra degrees of freedom preclude that. When you get done it would be nice to see a catologue of the different types with the z²+c images then the interesting zⁿ+c images. Thanks Benesi. I am still figuring out how the generating programmes actually work using quaternion math and what limitations this puts on our freedom to render our conceptions,  :D if any.

  I was thinking about that earlier (due to a comment my father made):  Is there an infinite number of 3d Mandelbrot sets? 

  Rule 1:  The 3d set should be able to be reduced to the 2d set by elimination of the 3rd axis.
 
  In other words:  It should produce fractals that have exact correspondence to the 2d versions when viewed with z~0.

  If we only define 3d Mandelbrot by rule #1, we can make an infinite variety of 3d sets, all with different relationships between the z axis and the standard x-y axis manipulation of the 2d Mandelbrot.  This doesn't really lead to a satisfying definition, or a simple well defined standard for extension of Mandelbrot sets into higher dimensions.

  Setting the standard Mandelbulb's angles and rotations like this:

r=sqrt (sx^2+sy^2+sz^2);
theta=atan2(sqrt(sx^2+sy^2)+flip(sz));
phi=atan2(sy+flip(sx));

sx=r^n * cos (n*theta) * sin  (n*phi) +pixelr;
sy=r^n * cos (n*theta) * cos (n*phi) +pixeli;
sz=r^n * sin  (n*theta) +pixelj;                     //pixelr is x axis component of pixel, etc

Gives us a detailed elephant valley (haven't checked seahorse valley yet), in addition to other interesting fractal patterns. 

  Setting the angles and rotations as in the Mandelbulb B formula results in less interesting zooms in the elephant valley section of the fractal, but gives us a more interesting top/bottom section to explore.  It also gives us different bulbs to explore at higher z^n (z^4++).

I don't think we need to bother with standardisation. Leave that for the following generation of programmers and explorers and artists. Now is the time to be inspired and inspirational. I wondered if you had thought about Lie groups and how what you are doing might relate the mandelbrot to those and supersymmetry, and string theory. If you get time give it some thought.

I'd like to, but am not familiar with mathematical Lie groups.  For now, however, I have been inspired by your and others comments, and in general by the whole forum.  

  So here is another formula, one that rotates x axis vs the yz plane, y axis vs. the xz plane, and z axis vs. the xy plane.

  It is interesting, but starts out pretty plain at the z^2 area, as I've not got up into the 200th iteration (when the seahorse valley starts getting interesting).  I suppose this is a good place to put a teaser image to demonstrate the potential variety contained in the new formula.  This is a z^8 (8th order, as I call them) 10 iteration 3d fractal (zoomed pretty deep (especially for 10 iterations) in on some islands in this one quadrant (this is a set of tiny nubs off of a nub of a nub of a nub of a tiny island):

(http://www.fractalforums.com/gallery/1/1170_08_02_10_9_26_18.jpeg)

  This is based off of the kaleidoscope formula from earlier in this thread.  I was thinking that there had to be a way to create a fractal by rotating the components around the other 2 components planar magnitude, so went back to the old formula and tweaked it.  As the old formula was named "type D", I've gotten very original and named this one "type D2".  I suppose I should make a formula "type R"... so that I can modify it.
  I don't know why I didn't do this fractal thing earlier: I think I just didn't bother to zoom in enough (or set bailout correctly).  

MAKE SURE YOU SET BAILOUT HIGH, I set mine to 10^30, or you get a tiny little thing

  Pixel and variable assignment follow:
pixelr is the x component, as usual
pixeli is the y
pixelj is the z

pixelw is the yz planar component (I should name it plane_yz, but.. I didn't rewrite the formula yet)
  anyways:  pixelw= sqrt(pixeli^2+pixelj^2);
pixelv is the xz planar component:  pixelv=sqrt(pixelr^2+pixelj^2);
pixelu is the xy planar component: pixelu=sqrt(pixelr^2+pixeli^2);

I set all variables to their initial pixel components before beginning the iteration.  If you want to include a seed value, you can do so instead of setting the variables to the corresponding pixel and planar components.
sx=pixelr;    sw=pixelw;  // sw rotates with sx
sy=pixeli;    sv=pixelv;   // sv rotates with sy
sz=pixelj;    su=pixelz;   // su rotates with sz

  Here is the formula:
  One way to do it, checked it out, I don't think this method is "better" or even that much faster.  Not sure what the problem was (probably was running something in the background when I tried the other method).
r= (sx^2+sy^2+sz^2)^(n);    // I don't know why, but you basically get the same fractal if you split the components
r2=(sw^2+sv^2+su^2)^(n);    // like this, but for some reason it calculates a lot faster (I think it's because the
                                         // fractal is bigger)   
r=sqrt(r2+r)*.5;                   // I multiplied the magnitude by .5 to increase the size of the fractal...

You can just do this, but for some reason it's a lot slower and the details I've seen don't look that different:
r= (sx^2+sy^2+sz^2+sw^2+sv^2+su^2)^(n/2);

  A tangent (skip if you like):
  I also use an if statement to assign r=sqrt(sx^2+sy^2+sz^2) so I can make those cool kaleidoscopes I posted a bunch of earlier in this thread (it is the same code except for the magnitude).  There is one more mode I call "weirdmode" in which I set r=xyz and r2=wvu values and assign them to opposite variables: it's weird because you get these images that escape after about 2-3 iterations, totally solid, no change for higher iterations: but the images are interesting and different for different angles.  It's like completely solid mathematical objects or something (and I forgot the exact settings... bleh).... back to the formula:

phi=atan2(sw+flip(sx));       //  
tango=atan2(sv+flip (sy));
theta=atan2(su+flip(sz));
         
nx=r*sin(v*phi) ;
ny=r*sin(v*tango);     // you can just use whatever multiplier variable names you like
nz=r*sin(v*theta);     //  I used v for the xyz and mag for the wvu...  you can use only one, unless you like to mess
nw=r*cos(mag*phi) ;    // around... but there is enough variety in the stock fractals that.. well maybe there is
nv=r*cos(mag*tango) ;   // maybe we need another dimension
nu=r*cos(mag*theta) ;     // the 4d extension of this concept is pretty simple: just do 3d planes instead of 2d...
  
         
if (juliaMode) {
sx=nx+cr;
sy=ny+ci;     //  you gotta add in 3 more variables for julias... I haven't bothered yet, as you can see
sz=nz+cj;
   
} else {
sx=nx+pixelr;
sy=ny+pixeli;
sz=nz+pixelj;
sw=nw+pixelw;
sv=nv+pixelv;
su=nu+pixelu;
}

I just set bail= |sx+sy+sz|  because of the high cutoff, but... doesn't help much.  I haven't figure out why the bailout is sooooo high for this one.

  Or you can play with various bail formulas (I just have bail2=1 for the above), like a - bail1 and + bail2.. or whatever:

bail=abs(sw)^bail1*abs(sx)^bail2+abs(sv)^bail1*abs(sy)^bail2+abs(su)^bail1*abs(sz)^bail2;

  Now for some images of the new formula.  I messed around in the 2nd order one for a while, because it's just neat to find the patterns I found in a z^2 3d fractal.

  This is a series of zooms, from some location:
(http://www.fractalforums.com/gallery/1/1170_08_02_10_8_07_27_3.jpeg) (http://www.fractalforums.com/gallery/1/1170_08_02_10_8_07_27_4.jpeg) (http://www.fractalforums.com/gallery/1/1170_08_02_10_8_08_43_0.jpeg)

(http://www.fractalforums.com/gallery/1/1170_08_02_10_8_08_44_1.jpeg) (http://www.fractalforums.com/gallery/1/1170_08_02_10_8_08_44_2.jpeg) (http://www.fractalforums.com/gallery/1/1170_08_02_10_8_08_44_3.jpeg)

(http://www.fractalforums.com/gallery/1/1170_08_02_10_8_08_44_4.jpeg) (http://www.fractalforums.com/gallery/1/1170_08_02_10_8_09_43_0.jpeg) (http://www.fractalforums.com/gallery/1/1170_08_02_10_8_09_43_1.jpeg)

(http://www.fractalforums.com/gallery/1/1170_08_02_10_8_09_43_2.jpeg) (http://www.fractalforums.com/gallery/1/1170_08_02_10_8_09_43_3.jpeg)

  Here are a few random images, a 23, 36, then 30 iteration z^2:
(http://www.fractalforums.com/gallery/1/1170_08_02_10_8_07_27_0.jpeg) (http://www.fractalforums.com/gallery/1/1170_08_02_10_8_07_27_1.jpeg) (http://www.fractalforums.com/gallery/1/1170_08_02_10_8_07_27_2.jpeg)

Here is another 30 iteration z^2 (done with the slower formula: what a pain):
(http://www.fractalforums.com/gallery/1/1170_08_02_10_8_24_01.jpeg)

  All right,

  I checked regular 2d Mandelbrot sets.  You get a lot more details in a z^8 at 50 iterations than you do in a z^2, which explains the lack of super duper details in the above images (and the fact that I am now getting high amount of details in z^13s of the "new" type D2 fractal at low iterations (8 and a 9 iteration of the same location, for the 9 iteration in the same location I had to exponentially increase z resolution, which makes for a slooooowwwwwww fractal calculation: i suppose the cure for this time of calculation problem is to zoom way in when you increase iterations).  Anyways, I'll include those 2 images later.

  For now, here are two 2d Mandelbrots, one z^2 the other z^8, both at 50 iterations.  The z^8 is zoomed in a lot more, and just a lot more detailed.  Even seahorse valley is a little boring at low iterations for the z^2.
z^2, 50 iterations:
(http://www.fractalforums.com/gallery/1/1170_08_02_10_9_39_27_0.jpeg)
z^8, 50 iterations:
(http://www.fractalforums.com/gallery/1/1170_08_02_10_9_39_27_1.jpeg)

  I'm not even going to bother to post the 20 iteration images: they have far more details in the z^8 than z^2 version.

  Here are a couple of 13th order images, first one is 8 iterations:
(http://www.fractalforums.com/gallery/1/1170_08_02_10_9_39_27_2.jpeg)
  This one is 9 iterations, had to increase z resolution a lot, still got clipping issues because I wasn't going to wait 4-8 hours to render it at higher resolution/checking/other time consuming variables:
(http://www.fractalforums.com/gallery/1/1170_08_02_10_9_39_27_3.jpeg)

  Here is a 2nd order image, 18 iterations of more or less a whole quadrant (1/8th of a sphere, basically... maybe a bit less) of the fractal:
(http://www.fractalforums.com/gallery/1/1170_08_02_10_9_39_27_4.jpeg)

  You can see that 18 iterations is a lot less detailed than.. well, how about I post a 8th order lower iteration of the same spot?  I'll edit it in when it's done.  Doing an 8 iteration (higher resolution though... details require it) 8th order version of the same location.

  Here is the 8th order, 8 iteration quadrant of the fractal:
(http://www.fractalforums.com/gallery/1/1170_08_02_10_10_22_25.jpeg)

  It's a little bit clipped, but you can see the massive increase in details for the lower iteration count 8th order fractal.  Maybe even cubic instead of quadratic...

I introduced a "bubblebrot" mode by taking the magnitude r=(sx^2+....+su^2)^(n/3).  Makes lots of little bubbles that you can zoom into.  Also, weirdmode is done by setting r=(sw^2+sv^2+su^2)^(n/2), a lot like mandelamode uses r=(sx^2+sy^2+sz^2).  Weirdmode produces some pretty neat images, especially if you set the planar values negative (pixelw= 0 - pixelw...).

  You can increase the size of your base fractal by dividing the magnitude (r) by a number (or multiplying it by a number smaller than 1... darhhh).  It doesn't seem to distort the z^8 that I'm looking at now, although it seems to increase the time to calculate for some images (but maybe not?), and for some reason now its taking less than 1/2 the time (I divided the value by 4, zoomed into the exact same location of the fractal, maybe off by a few pixels).  Of course, multiplication by a small number that is easily converted to base 2 may be advantageous as well...

Just a thought about the speed of these programmes: Does any one use  flags in the register for each iterated point to indicate if it is part of a sequence that  escapes or if it orbits? This would mean when the programme steps to that point it can skip the calculation calls as it already has determined this point is in the set 0r not. I know you add a flag look up call,but maybe overall this might save time?

If not using distance estimation then that might help on Julia Sets.
It's actually a better optimisation for 2D Julias - if your Julia formula is particularly complicated/slow and you only want to render the entire Julia to a given resolution (say up to around 4096*4096) then you can use a snap-to-grid method with per-pixel flags and as soon as any orbit hits a pixel that's already been calculated then you can stop iterating for that point as you've already got the result for the rest of its orbit.
Doesn't help for Mandelbrots though :(

 Here are a few more of the new formula z^2 (2nd order) images.  I think it really depends on where you look in the fractal: there is fractally stuff all over, but the most interesting variety (at these low iterations) seems to be constrained to certain locations (like the 2d set).  
  These first three (sorry about the clipping, I set my checking point at 5% back to increase speed, which chops off sections of the fractal) are simple iteration increments:

(http://www.fractalforums.com/gallery/1/1170_09_02_10_7_55_49_0.jpeg) (http://www.fractalforums.com/gallery/1/1170_09_02_10_7_55_49_1.jpeg) (http://www.fractalforums.com/gallery/1/1170_09_02_10_7_55_49_2.jpeg)
This one is a blow up/zoom in of the above, with a higher iteration:
(http://www.fractalforums.com/gallery/1/1170_09_02_10_7_55_49_3.jpeg)
This is a zoom in on a section of the above, with higher z resolution, lower total resolution and slightly higher checking point percentage (a little bit less clipping):
(http://www.fractalforums.com/gallery/1/1170_09_02_10_7_55_49_4.png)

  The last one, I am going to explore the twisty section on the right.  It's only 30 iterations, and already beginning to show signs of more interesting forms, which is good for a 30 iteration z^2 fractal (or so I assume).  

I think some people - rather than using DE - use an orbit trap at the center of the Mandelbulb, because this indicates a periodic cycle. Like David Makin said, this doesn't work for Mandelbrot Sets well, only for Julia Sets. This can be used to help points bail out faster if the point goes in any point outside the Antibuddhabrot/bulb. I don't know if these more complicated bailout mechanisms actually are worth the potential speed ups, but it can be relatively easy in some places. For example, if the 2D Mandelbrot, if Re(z)<-1.4 and abs(imag(z))>0.06 then you know it will bailout. Here's some code to plug in and see the difference: "bailout: (|z| < 4)&&(Real(z)>-1.4||abs(Imag(z))<0.06)".

Thanks guys. I think this may be because of scaling issues in the mandelbrot. It seems each mandelbrot is at a different scale and so smaller and smaller mandelbrots are being sculpted, but they are magnified up to look the same apart from the new detail sculpted out.

Re: True 3D mandlebrot type fractal
« Reply #38 on: November 19, 2008, 04:19:12 PM »   

Oh, my deeply sleeping thread!
I revive Thee from the dead... 


Ok, seriously, after staying away from FractalForums for a while because of RealLife(TM), I came back and (obviously) found this thread. I had a little exchange of mails with twinbee about the formula involved and he has asked me to post the results here. So, here it comes:


twinbee defined (a "few" posts back)

double r    = sqrt(x*x + y*y + z*z );   
double yang = atan2(sqrt(x*x + y*y) , z  ) // that would be theta in std polar coordinates
double zang = atan2(y , x);                      // that would be phi  in std polar coordinates

so I would suppose he has implicitly

x = r*sin(yang)*cos(zang)
y = r*sin(yang)*sin(zang)
z = r*cos(yang)

and I would have expected (doubling the angles!)

newx = (r*r) * sin(yang*2)*cos(zang*2)
newy = (r*r) * sin(yang*2)*sin(zang*2)
newz = (r*r) * cos(yang*2)

but he defines

newx = (r*r) * sin( yang*2 + 0.5*pi ) * cos(zang*2 +pi);
newy = (r*r) * sin( yang*2 + 0.5*pi ) * sin(zang*2 +pi);
newz = (r*r) * cos( yang*2 + 0.5*pi );

which can be simplified by taking into account the symmetries of sin() and cos() to

newx = - (r*r) * cos(yang*2) * cos(zang*2)
newy = - (r*r) * cos(yang*2) * sin(zang*2)
newz = - (r*r) * sin(yang*2)

which is not exactly equal to doubling the angles.
... but it leads to interesting pictures.
One DOES NOT need atan2(), sin() and cos() to implement these formulae,
because they can be simplified a lot by using the following identities for any
angle a :
cos(a)*cos(a)+sin(a)*sin(a)=1  (well known, I suppose)
cos(2*a) = cos(a)*cos(a)-sin(a)*sin(a)   (less well known, it seems )
sin(2*a) = 2*cos(a)*sin(a)

I'll spare you the details, but you end on:

newx = ( x*x + y*y - z*z )*( x*x - y*y) / ( x*x + y*y )
newy = 2 * ( x*x + y*y - z*z )*x*y / ( x*x + y*y )
newz = - 2 * z * sqrt( x*x + y*y )

no trigonometric functions at all, just additions, multiplications, divisions and a squareroot.
There is NO pole on the z-axis, BUT there might be numerical problems because of the
denominator, solvable e.g. by taking

if( abs(y) < really_small_value )
newx = x*x-z*z
newy = 0
newz = -2*z*sqrt(x*x)
else (view above)

(end of simplification)

Interesting side effect: when I first read about "doubling both angles" I wanted
to try that for myself. When doing geometry instead of math, I prefer measuring
the angle theta not against the z-axis, but against the x-y-plane, so in that case

phi = atan2( y, x ) // just like before
theta = atan2( z, sqrt(x*x + y*y) ) // exchange of arguments, angle is positive above, negative below x-y-Plane

then
x = r*cos(theta)*cos(phi)
y = r*cos(theta)*sin(phi)
z = r*sin(theta)

and simply doubling the angles

newx = r*r*cos(2*theta)*cos(2*phi)
newy = r*r*cos(2*theta)*sin(2*phi)
newz = r*r*sin(2*theta)

simplifying this analogous to above gives

newx = ( x*x + y*y - z*z )*( x*x - y*y) / ( x*x + y*y )
newy = 2 * ( x*x + y*y - z*z )*x*y / ( x*x + y*y )
newz = 2 * z * sqrt( x*x + y*y )

IDENTICAL to twinbee's stuff except for the sign in z !

Since I'm sitting at an OLD Mac (350MHz) and try to use POV-Ray to produce pictures I can't show any yet; that takes TIME! But the first little "thumbnails" I produced show that this change of sign dramatically changes the resulting set!

Forgive my ranting - I hope somebody might find these "simplifications" useful - twinbee thought so, at least!

Happy iterating...
Karl

So adding to Karls original analysis:
Power-Reducing/Half Angle Formulas

(http://www.sosmath.com/trig/Trig5/trig5/img8.gif)


Product-to-Sum Formulas

(http://www.sosmath.com/trig/Trig5/trig5/img10.gif)

For the mandelbrot in 2d

z= x+ iy=rcosø+irsinø

z²=r²(cos²ø-sin²ø+2icosøsinø)

                  =r²cos2ø+ir²sin2ø


For the 3d mandelbrot


z= x+ iy+jz=rsinΩcosø+irsinΩsinø+jrcosΩ

z²=r²sin²Ωcos²ø-r²sin²Ωsin²ø-r²cos²Ω+i2r²sin²Ωsinøcosø+j2r²sinΩcosΩcosø+ijr²sinΩcosΩsinø+jir²sinΩcosΩsinø
                  =r²(+i+j+ij+ji)

So doubling the angles is only part of the transformation if this analysis is right. Please check and point out my mistakes as it is easy to rectify.

   All right.  Removing the formulas for now.  Found out that the (old) radius formula might be incorrect, have to update it and really busy right now.

  If you mess with it:

r= sqrt( (sx^2+sy^2+sz^2)^n + (sw^2+sv^2+su^2)^n);    /*  the planar and linear magnitudes are separate: better math, better fractal  */
  Although the old formula makes cool images as well.  Must experiment.

From analysis of the trig methods i wonder  three things:
Hairy ball effect does it depend on stepping ? That is to say the finer the steps the less hairy the renders look. If so the hairy effect i due to the great cicle nature of the transformation and sculpting;
Fuzziness does this depend on what accuracy π is set to? In which case double or quadruple covers may mean pixel identification is not "spot on":
Whipped cream is this due to the fact that only only 2 degrees of freedom are used for rotations when in fact there is no such limitation in geometrical space, unlike flatland 2d world. If an axis is determined and a plane to which this axis is a norm then rotation about that axis in that plane can be defined. A norm to a plane is not necessary as a spin axis but it makes the math easier to describe and les relativistic: more vector math less tensor math.

Although i have not explored them yet clifford algebras and rotors are claimed to have  this property.

For the time being i am thinking of freeing up the degrees of freedom by adding a third angle of rotation to spherical coords so

is extended to where rotates around a normal to the radius which is in the plane of the great circle that the radius is in as determined by . This means that at certain times this nomal will coincide with the established orthogonal axes, principally when and are multiples of π/2.

  Your comment on degrees of freedom reminds me of something I just read on wikipedia, either yesterday or the day before.

  Gimbal lock (http://en.wikipedia.org/wiki/Gimbal_lock). 

  I tend to think that the "hairyness" is due to the fact that the objects are 3 dimensional.  There are spaces behind the front pixel that are out of the set, and if the z-axis resolution is not set fine enough, we can miss the pixel in front and hit one behind.

  So we end up with little shaded pixels (set into the object) because we missed the actual pixel (even though we checked (bisected) between the first pixel hit and last pixel not hit, when we ran our check, we hit towards the first pixel hit FIRST, although if we had done 2 iterations backwards we would have hit a pixel towards the last pixel not hit). 

  2d demo:
   1,2,3,4.....C,D,E = pixel in set  * = check point not in set     ( )= not in set  <-- second one is a space...   

  Checking in this direction ---->        *   123  45      678   9A  BCDE

   So we have something like this, miss the first group of pixels and hit pixel 9.  We do 1/2 back to previous no hit checkpoint:

  Checking in this direction ---->        *   123  45 *    678   9A  BCDE

  We hit nothing, move 1/2 towards 9:

  Checking in this direction ---->        *   123  45 *    6*8   9A  BCDE

  We hit 7, now we check ~1/2 towards previous miss:

  Checking in this direction ---->        *   123  45 *   *6*8   9A  BCDE

  hit nothing.  Now we check 1/2 towards previous hit on 7:

  Checking in this direction ---->        *   123  45 *   ***8   9A  BCDE

  And hit 6.  We've totally missed the groups of pixels in the front due to our bisection algorithm and end up with an indented pixel.  Perhaps if we get a value deeper than the 9 pixels around the pixel, we should just bring it to the level of the pixels around, or at least have it as an option (to do so).  Otherwise, we have fuzz, some of which I'm running into now with these Julia's, as I prefer the higher details of one more iteration, but don't have the patience to set z axis resolution high enough to hit every pixel (at least for now, until I find a set of amazing images that I want to make intensely detailed and correctly rendered):

  1) 9th order (z^9)       c= -.25, .1, -.4
  2) 15th order (z^15)    c= -.29, -.29, -.29
  3) 15th order (z^15)    c= -.29, -.29, -.29    (rotated)
  4) 15th order (z^15)    c= -.31, -.31, -.31
  5) 15th order (z^15)    c= -.37, -.23, -.31

(http://www.fractalforums.com/gallery/1/1170_11_02_10_6_43_02_0.jpeg) (http://www.fractalforums.com/gallery/1/1170_11_02_10_6_48_27_0.jpeg)(http://www.fractalforums.com/gallery/1/1170_11_02_10_6_48_28_1.jpeg)(http://www.fractalforums.com/gallery/1/1170_11_02_10_6_48_28_2.jpeg)(http://www.fractalforums.com/gallery/1/1170_11_02_10_6_48_28_3.jpeg)

Thanks for that insight. I have no clue of the rendering algorithms necessary to visualise these escapes and orbits. I have only recently realised that the visualisations are every bit as computational intensive as the iterations if what is happening in the sculpting is going to make sense.

i am essentially talking about missed points and planes due to the calculation schemes referencing points in the 'tracks' ie orbits or escapes in different ways. The trig formulae will sculpt in the planes of great circles thus leaving a noticeable combing effect which i think is the basis of hairy ball. If so this effect should diminish with finer and finer stepping. But now add on top the rendering difficulties again in a great circle plane and you end up with a multiple effect.

Your explanation give insight into the development of spirals in some of the images that twinbee posted, where sharp images suddenly appear to exude cream before becoming sharp again .

Gimbal lock. Thanks  ;)

Here is yet another z^2 fractal of the new type (although this one has the magnitude set by the new rules).

  I've heard that coastlines are fractals, but did you ever see fractals that are the edge of an ocean?

(http://www.fractalforums.com/gallery/1/1170_12_02_10_11_37_39.jpeg)

  This is just a zoom in of a quadrant of the fractal.

  There is plenty of variety within it, but I followed the quadrant line, and zoomed into the lakes that exist in the z^2 version (there are lakes in the others as well, but I thought we needed some true 3d z^2 fractals, so...).

So here are a few different  mandys at different iterations.

Now a mandy found in nature/yh~w.

This is a few sculpts from acorn mandy and lathed mandy. These are not slices per se but true lathed or "rotated" sets. These were produced by terry gintz mac version 1.02 of Quasz.

these two images show variations of the lathed mandy which indicate that the lathed effect can be controlled sufficiently to reveal the details of the mandy.

Although not as intricate as the trig formulae it is still a good sketch of the structure at power 8

Dragons Teeth! The rise of Godzilla mandy!

The wizards staff! A julia of arcane power. He who wields it has power to subdue dragons!

These sculptures arise out of an exploration of the sin and cos functions and the principle range.

The road to the holy grail is long and winding. Why not fly or go by boat?

Lets stop for breakfast and fry an egg! Wait what appears in the distance......?

[x=real(z)
y=imag(z)
s=imaj(z)
a=real(c)
b=imag(c)
d=imaj(c)
r2=1*(|c|+d*d)+0*(|z|+s*s)
t0=1*sqrt(|c|)+0*sqrt(|z|)
|z|<10
|c|<10
s2t=sin(2*t0)
c2t=cos(2*t0)
z=1*r2*(s2t*cos(2*b)+s2t*sin(b)*i+c2t*j)+0*r2*(s2t*cos(2*y)+s2t*sin(y)*i+c2t*j)+0*c+0*z

Have fun with the formula. Maybe you might have an interesting adventure on the way to your holy grail! Yom Tov!

Grail mandy settles down to lay a mandy egg. But she becomes egg bound right at the crucial dropping point! Argh!!!

First some variations based on the following grailmandy with the approximations to theta and phi incorrectly applied in the tan and cos versions.

y=sin(imag(z))
s=imaj(z)
b=sin(imag(c))
d=imaj(c)
r2=1*(|c|+d*d)+0*(|z|+s*s)
t0=1*sin(sqrt(|c|))+0*sin(sqrt(|z|))
|z|<10
|c|<10
s2t=sin(2*t0)
t2c=cos(2*t0)
z=1*r2*(s2t*cos(2*b)+s2t*sin(2*b)*i+t2c*j)+0*r2*(s2t*cos(2*y)+s2t*sin(2*y)*i+t2c*j)+0*c+0*z

wherey,b,t0 are approximations to theta and phi.

ctd

And now the intended versions for cos and tan.

I deliberately altered the colour palette in the last one. Not so boring n'est ce pas?

y=sin(z)
s=imaj(z)
b=sin(c)
d=imaj(c)
r2=1*(|c|+d*d)+0*(|z|+s*s)
t0=1*sin(sqrt(|c|))+0*sin(sqrt(|z|))
|z|<10
|c|<10
s2t=sin(2*t0)
t2c=cos(2*t0)
z=1*r2*(s2t*cos(2*b)+s2t*sin(2*b)*i+t2c*j)+0*r2*(s2t*cos(2*y)+s2t*sin(2*y)*i+t2c*j)+0*c+0*z

A set based on this variation of grailmandy now follows.

a cos and a closeup of a sinz. Remember these functions are approximating theta and phi, so other functions that have a link to theta and phi may produce interesting results: for example e^iz.

Thought that it's relevant to this thread too, so here's a cross-link for anyone interested:

http://www.fractalforums.com/theory/summary-of-3d-mandelbrot-set-formulas/msg14463/#msg14463 (http://www.fractalforums.com/theory/summary-of-3d-mandelbrot-set-formulas/msg14463/#msg14463)

Just to finish this part of the exploration, using cosz as an approximation to theta and arranging the formula for a traditional mandy thusly:
 y=cos(z)
s=imaj(z)
b=cos(c)
d=imaj(c)
r2=0*(|c|+d*d)+1*(|z|+s*s)
t0=0*cos(sqrt(|c|))+1*cos(sqrt(|z|))
|z|<10
|c|<10
s2t=sin(2*t0)
t2c=cos(2*t0)
z=0*r2*(s2t*cos(2*b)+s2t*sin(2*b)*i+t2c*j)+1*r2*(s2t*cos(2*y)+s2t*sin(2*y)*i+t2c*j)+1*c+0*z

to finish with grailmandy i have set the formula to reflect mandlebulb formulae, and used sin and tan as approximations for theta, phi.

Just had a really crazy idea and tried it, I didn't expect it would actually "work" and it didn't - you just get a sort of slightly plumped out 2D Mandelbrot, but I like the Julias:

UF code:

            magn = sqrt((m=|zri|)+(s=sqr(zj)))
            if magn>0.0
              zri = sqr(zri)
              magn = sqr(magn)/sqrt(sqr(m)+s)
              zri = magn*zri + cri
              zj = magn*zj + cj
            else
              zri = cri
              zj = cj
            endif

zri/cri are complex, magn/zj/m/s are real and in UF |zri| means the magnitude of zri squared.

David, am I right in thinking that this isn't the whole formula? Is it done after regular Mandelbulb angle multiplying/radius squaring? Pics please! (Mandel and Julia!)  ^-^

Nope - that *is* the whole formula :)

I suddenly wondered what happens if you apply the 2D Mandelbrot but scale the result of that plus the third dimension such that the new magnitude of the triplex is the square of the magnitude of the original triplex, hence:

Get the triplex magnitude, if not zero square the x and y component as if complex then scale the new triplex by dividing by its magnitude and multiplyiing by the magnitude of the original triplex squared. Finally add the constant.

Pics as requested:

"Electric Mandelbrot"

(http://www.fractalforums.com/gallery/2/141_04_04_10_12_40_43.jpeg)

If I was in the market for a custom-made electric guitar.....

And a test Julia showing I hope why I rather like the Julias for this one :)

(http://www.fractalforums.com/gallery/2/141_04_04_10_12_48_32.jpeg)

Wow, cool pics, they're pretty close to a real grail... cool formula David! The Julias really do sounds exciting, if they can hold all the variety of the regular 2 Mset, preserved well. ...a Douady rabbit (sp?) with long "fur"!  ;D

 :thumbsup1:

 :dink: love the palm leaf julia dave.

Hmm, I didn't hear, how those pics sound (Timeroot ;P) but they indeed look really nice :D
The Julia resembles certain kinds of leaves :D
Are there also other leaf-structures in those sets?

I don't know, I've only rendered the Mandy and 3 Julias. The formula was just a sudden idea.

I haven't investigated it further as atmo I'm adding some pre-render code for the Mandelbox that automagically finds the correct scale factor for the DE and the extent of the cuboid- I'm doing that after giving up trying to work it out analytically i.e. for all values for xfold,yfold,zfold,spherical fold and scale. So far the result is very promising - the adjustments to the DE it gives agree with my manually discovered values to 4 significant figures, I'm just trying to get it to around 6 and also to have the algorithm calculate what the scale factor should be for the DE value to convert it to a step value i.e. to work out the maximum error in the DE and calculate a scale to apply to convert the DE to a step that allows for the maximum error in the DE.

POWER 8 David?, POWER 8!!!!  O0

OK, OK !!!

Using:

            magn = sqrt(|zri|+sqr(zj))
            if magn>0.0
              zri = zri^@mpwr
              magn = (magn^@mpwr)/sqrt(|zri|+sqr(zj))
              zri = magn*zri + cri
              zj = magn*zj + cj
            else
              zri = cri
              zj = cj
            endif

With seed (-0.05,0.8,-0.20397)

Here's a Jellyfish:

(http://www.fractalforums.com/gallery/2/141_06_04_10_7_58_34.jpeg)

yay, nice :D
Not overly fractal though^^

 :angry:

OK, not too  many fractal details, but nice in its own way...

Thanks David!

( Not going to ask you to zoom in... ;) )

The search continues. A signal 2 mandy followed by a signal 8.

Before considering "the" true 3d M-set, ask what kind of dimensions you're extending to.  Like dimensions are Euclidean, I think.  Ttwo in complex are algebraically related but unlike.  They do, under multiplication, mutually contribute to each other. More later, badly outta time.

After extensive experimentation this is the leading candidates using Nylander like fomulae and QuaSZ (http://www.MysticFractal.com/QuaSZ.html) by Terry Gintz.

To continue my 4-16 comment, it seems many renderings seen here are manifestations of "designer" dimensional systems.  It so much depends on what you might want in dimensional extension.  I already expressed doubts about any single "true" extension beyond the complex, but one good candidate might be one that contains those dimensions along with square roots of i expressed in 2 others, totalling 4.  sqrt(i) would be represented by j, and the other dimension is ij as a multiplicative consequence.  I've worked out the math for all 4 operations (div is complicated, but not necessary here) but it needs to be programmed as a generator on its own, say, in C++.

But i has a square root in C,

Yes, it can be expressed like that, but again, that's not necessary either. We'd only be plotting coefficents of j and ij just as we do for Re and Im, so that f(v)->v^2+g, v=x+yi+zj+wij, g=a+bi+cj+dij.  We get a set of 8 rotating powers of j, just as we have 4 of i.  It's the relationship between j and i and real that counts.  Note that when j and ij dims are zero, we have the standard M-set in two dims.  Preserving that, and going to a dimensional extension that is meaningful to the function is what appealed to me.  Has anyone done this yet, only I did not recognize it?

(came back to fix above, may as well add:)
j^2=i, j^3=ij, j^4=ii=-1, j^5=-j, j^6=-i, j^7=-ij, j^8=j^0=-ii=1, so that Re=j^8-j^4 values, Im=j^2-j^6 values, sqrt(i)dim=j^1-j^5 values, and i*sqrt(i)dim=j^3-j^7 values.  That's how the coefficients are processed.  Any nonzero value for c or d above will contribute to x+y+z+w under iteration, though of course we are only interested in eventual fates of the coordinates for now, and can only look at (at most) 3-d slices.

Hope I put it down right this time.

Contrary to  predicted it is the corrected expansion that is  less interesting or possibly more! The image sculpted here is thin slice in the yz  plane seen at an offset angle. This may mean that the x(ij) resultant of the yz  plane rotation under vx is in fact a 3d operator of sorts, like the two planar right triangles in the twinbee nylander formulae. The important thing here and there is the orthogonal or otherwise linking of the two planar operators so as to span geometrical space. When i used the i23 linking, dave produced a solid object and so did i. (see the second image.) This seems to be the result of the mistakes.

By the way the formulae do not need to be fully expanded to draw them. i think i can use the building blocks and their definition to   draw them i the latest quasz by terry.

Continuing 181, if anyone's interested, the consequent formulation is:

x part: x^2-y^2-2zw+a+
y part: (2xy+z^2-w^2+b)i+
z part: (2xz-2yw+c)j+
w part: (2xw+2yz+d)ij

These 3 formulae produce linked but different shapes. There is no plane as guessed but a curious linking surface, which apparently changes orientation with the order of the BC product.

@fracmonk

using this:
wx= x#*x#-y#*y#-2*z2*w
wy=(2*x#*y#+z1-w^2)*i
wz=(2*x#*z2-2*y#*w)*(1+i)/1.142
ww=(2*x#*w+2*y#*z2)*i*(1+i)/1.142

i get these. Do not know if it is what you are after. A julia and 2 mandy's.
z1=z^2 and z2=z
However have had to use quad c to get this as your extension of the polynomial numerals in i cannot be rounded off in 3d. You just get a "sausage".

I'm not familiar with the code, and can't be sure if it executes as intended.  If you wrote your own from scratch, you'd keep the four parts separate (as you did) but treat as real computationally, only *assuming* the r,i,j,+ij after each component, and all values would go where they belong and change in each step appropriately.  You got interesting results, but I'd guess they're just artifacts.  I'll print it, take it with me, & try to decipher.  Best I can do, for now.

jehovajah-  Familiar with FractInt (http://www.Nahee.com/spanky/www/fractint/fractint.html)?  It's what I use.  If you know it, it automates complex math so you can use a single variable which it treats as complex.  Stayed up late "fooling" it into treating my variables as 1 dimensional real.  Then followed my own advice left to you in last post and did 2-d slices of some of the 6 possible combinations:  a by b with c+d as set constants, etc.   It's good, and if you get your code right, you will have the glory (and it is glorious!) of doing its 6(?)  3-d index sets first.  Good luck.  Later.

(http://www.sosmath.com/trig/Trig5/trig5/img8.gif)
(http://www.sosmath.com/trig/Trig5/trig5/img10.gif)
                 =r²(+i+j+ij+ji)
                 =r²(+i+j+ij+ji)

I thought i would derive the alternative cosine formula.

z= x+ iy+jz=rcos¥cosø+ircos¥sinø+jrsin¥

z²=r²cos²¥cos²ø-r²cos²¥sin²ø-r²sin²¥+i2r²cos²¥sinøcosø+j2r²cos¥sin¥cosø+ijr²sin¥cos¥sinø+jir²sin¥cosYsinø
                  =r²(+i+j+ij+ji)

With some initial sculpures of the sinΩ version.
And one i like by just squaring

z=rsinΩcosø+irsinΩsinø+jrcosΩ.

Hiya fracmonk.

I am using QuaSZ (http://www.MysticFractal.com/QuaSZ.html) by Terry Gintz which does not completely match with FractInt (http://www.Nahee.com/spanky/www/fractint/fractint.html) but is close.

The description you gave of the extension of the polynomial numerals in i i combined with the post that explained the square root of i as a polynomial numeral.  That is where the revision of your formula came from.

Now I am reminded of Kujonai and his mod 3 sign formulation, but what I cannot follow at preset is how you algebraically link these systems to the Cartesian or polar coordinate system.  Your advice and explanation do not at present help me to do that.  When you have a thought like this sometimes you are on your own until you can get others to play with the ideas the way you do.  I had a go because your formulation was suggestive of a way to realize it.  I assumed in one attempt that the formulation was attached to the quaternion math structure but with the coefficients being modified by the square root of i.  So by mistake I added quad c and got the sculptures posted.  I then corrected c to reflect the polynomial basis you outlined and got the "sausages" which I did not post.  On analysis my assumption would produce sausage results because your definition of the basis is in terms of i and that represents only 2 dimensions in quaternion math.

Hope I have explained clearly where I am up to with your polynomial numeral extension.

my second assumption would be to use your formulation as coefficients for r,i,j,k with the sqrt(i) function and see how that plays out.  What do you think?  :dink:  

hey, jehovajah!  O.k., first to correct last msg- 4  3-d index sets, assuming unused dim param=0.  If u remember early diy M-set programming advice, it was given in real math- newx=x*x-y*y+a, new y=2*x*y+b, and the important part is that there is no "i" to be seen anywhere.  So, your generator's core iterator should *actually* be made to carry out the following for each iteration:

x=y=z=w=o (to start)  2 or 3 parts of your constant are auto, 2 or 1 are fixed, initially

newx=(x*x)-(y*y)-(2*z*w)+a
newy=(2*x*y)+(z*z)-(w*w)+b
newz=(2*x*z)-(2*y*w)+c
neww=(2*x*w)+(2*y*z)+d

I made the bailout test high, and just added them all together for it. Crude, but effective.
in 2-d:
For any given a+b, c by d exhibits origin symmetry.
There is x-axis sym for b by d for any a when c=0
   "     "     "        "     "  b by c  ''     "    "    "    d=0
   "     "     "        "     "  a by d  "     "    b    "    c=0
   "     "     "        "     "  a by c  "     "     "    "    d=0

Time!  All for now. Sorry.
 (time passes...)
Let me explain my situation:
I get all internet access from a public library computer.  Very limited time, purposely disabled drives, and lots of competition for equipment.  Very frustrating!  Got limits?  Always have to rush, and don't mean to be terse. Please forgive that. Thanx                               

Jehovajah!  Should have mentioned earlier this system is not quat, or any other similar scheme.  If i've got it right, it's a field.  Why give up field properties if you don't have to?  Pretty sure it's associative, commutative, distributive, etc.  Early on, stigomaster pointed out that sqrts of i are in C, but more than that, they're on the unit circle @ 45 deg angles to 1,-1,i, +-i.
C is the smallest closed subfield of a set of dimensions 2^n.  It is n=1, and the environment I propose here (n=2) is in turn a subfield of an 8-D system n=3, which would pare down the unit circle further into 22.5 deg arcs, etc.  You are very good at polar, I suspect, + I hope that offers insight.  I suck at it, + look at it from p.o.v. of algebra.  I mentioned that it's the relationships betw the dims that count.  Let's square a number: (distributively)

(i+j+ij)^2=-1+ij-j+ij+i-1-j-1-i=-3-2j+2ij    Notice in this case that real was zero before this particular number was squared, but now there's a real part in the product, and i terms canceled.  If still lost, refer back to the earlier table of 8 powers of j.  I'm deadly sure you can figure it out.

Also, I made a small mistake indicating symmetries: where it said "xaxis", that only referred to my 2-d rendering.  They have a "butterfly's symmetry" like in M.  No other plane in right angle orientation to C has simple connectedness. M on C is the only one.  Accept no substitutes.  I'll try to stay in touch but have pressing matters in personal life.  Later.

Thanks for that. So i did assumption three which was my first thought that the coefficients are in fact the extensions for the four orientations i⁰₀ i¹₀ j₀ ij₀ where these are totally the same basis as i₀ v₀ j¹₀ k₀ as described in http://www.fractalforums.com/complex-numbers/polynomial-rotations/ (http://www.fractalforums.com/complex-numbers/polynomial-rotations/).

Quasz as Terry has updated it makes this a simple thing to do using the rfun..rend bracket.

I am touched by your situation and wish you feelings of gratitude along with opportunity to pursue your dreams  at every available instance resulting in joy to you and all around you.

I am attempting to post to the gallery and cut down on attachments as this uses up too much of Trifox's precious and appreciated space, but i think it is appropriate to attach these images for you to look at now,and if they are what you are finding to gallery them later. A julia and 2 mandelbulbs.
pixelsculpt mandy
(http://www.fractalforums.com/gallery/2/410_01_05_10_2_56_10_1.png)
xyz view of ordinary mandy
(http://www.fractalforums.com/gallery/2/410_01_05_10_2_56_10_0.png)

Jehovajah- went to link in your last msg to print it & take it away for study. At a very quick glance, I have doubts about this being the same animal (for instance, nothing about rotation in these), but will read, just the same.  I could not leave without adding a couple thoughts (& there are always more):
I've done a survey of 2-d "sliced mandelbread" studies, in which one param was zero, another fixed for each picture, and two became screen coordinates.  The issue I was trying to bring to bear in this discussion is about what is the most meaningful and appropriate extension of C not just for M, but anything complex.  I think this might be it.  It might be helpful to do a 3-d visualization that includes a+b recognizable cross-section in one of the butterfly symmetricals I listed.  Look for:

1. Any 3-d visualizations for a non-escaping value of the unseen component should be in one piece.
2. Is condition 1 necessary?
3. Does anything in 4-d extend beyond |2|?  It shouldn't, & my studies don't suggest it.

There IS a sound and provable way to do division in this system, and though it's not relevant to M-set generation in it, I think the question of field properties hinges on it, and that will go later to appropriateness arguments.

Thanx for the pix. I will look for coincidence between things in them and my own.

Later!

I recognised something like the complexified quaterinions in these images and refer you to the follwing,copied from Terry gintz Quasz manual for mac.

9.2 Tutorial
  An Introduction To CQuat Fractals By Terry W. Gintz
In the process of exploring all possible extensions to a fractal generator of this type, I
considered using discrete modifications of the standard quaternion algebra to discover new
and exciting images.  The author of Fractal Ecstasy [6] produced variations of the
Mandelbrot set by altering the discrete complex algebra of z2+c.  The extension of this to
quad algebra was intriguing.  There was also the possibility of different forms of quad algebra
besides quaternion or hypercomplex types.
Having modeled 3D fractals with complexified octonion algebra, as described in Charles
Muses' non-distributive algebra [7], it was natural to speculate on what shapes a
"complexified" quaternion algebra would produce.  Would it be something that was between
the images produced with hypercomplex and quaternion algebra?  Quaternion shapes tend to
be composed of mainly rounded lines, and hypercomplex shapes are mainly square (see
Figures 1 and 2.)
  i   j  k
i -1 k -j
j k -1 -i
k -j -i 1
Table 1 Hypercomplex variable multiplication rules
   i  j   k
i -1 k -j
j -k -1 i
k j -i -1
Table 2 Quaternion variable multiplication rules
In both quaternion and hypercomplex algebra, i2=-1.  The hypercomplex rules provide for one
real variable, two complex variables, (i and j) and one variable that Charles Muses refers to as
countercomplex (k), since k*k = 1.  It would appear from this that k = 1, but the rules in Table
1 show that k has complex characteristics.  In quaternion algebra there is one real variable
and three complex variables.  In hypercomplex algebra, unlike quaternion algebra, the
commutative law holds; that is, reversing the order of multiplication doesn't change the
product.  The basics of quaternion and hypercomplex algebra are covered in Appendix B of
Fractal Creations [8].  One other concept important to non-distributive algebra is the idea of
a "ring".  There is one ring in quaternion and hypercomplex algebra (i,j,k). (There are seven
rings in octonion algebra.)  If you start anywhere in this ring and proceed to multiply three
variables in a loop, backwards or forwards, you get the same number, 1 for hypercomplex,
and 1 or -1 for quaternion, depending on the direction you follow on the ring.  The latter
emphasizes the non-commutative nature of quaternions.  E.g. : using quaternion rules, i*j*k =
k*k = -1, but k*j*i = -i*i = 1.
For "complexified" quaternion algebra, the following rules were conceived:
  i    j   k
i -1 -k  -j
j -k   1   i
k -j   i   1
Table 3 CQuat variable multiplication rules
Note that there are two countercomplex variables here, (j and k).  The commutative law holds
like in hypercomplex algebra, and the "ring" equals -1 in either direction.  Multiplying two
identical quad numbers together, (x+yi+zj+wk)(x+yi+zj+wk) according to the rules of the

complexified multiplication table, combining terms and adding the complex constant, the
following iterative formula was derived for the "complexified" quaternion set, q2+c:
x -> x*x - y*y + z*z + w*w + cx
y  -> 2.0*x*y + 2.0*w*z + cy
z -> 2.0*x*z - 2.0*w*y + cz
w -> 2.0*x*w - 2.0*y*z + cw

I do think Terry would be best able to direct you on how to use quasz to get what you want.

By the way the link to polynomial rotations is to an unfinished peice of work.

Hope these views help
xy plane view
(http://www.fractalforums.com/gallery/2/410_01_05_10_2_56_10_2.png)
xz plane view
(http://www.fractalforums.com/gallery/2/410_01_05_10_2_56_10_3.png)
zy plane view
(http://www.fractalforums.com/gallery/2/410_01_05_10_2_56_10_4.png)

As is clear from the formulae your complexified quat is different to the one in quasz and more like a hypernion based math, but again unique to your choice of extension.

Kujonai might be possible to do now that i am more familiar with quasz and the similarity of your descriptions.

jehovajah-  I was lucky enough to unexpectedly get on a private machine for this.  Read last messages, and here's the thing: Quasz (sp?) is, if i'm correct, built from the ground up to do quat.  You may have the chops to fool it into doing this, and I'm no judge of that, but that seems WAY, WAY more complicated than it needs to be!  I "fooled" fractint into doing it by using imag(z), imag(p1), etc. to convert the i component value to real so that there are 4 real channels of computation and wrote an equivalent of the black box you saw in a previous msg. that keeps the proper relationships between components.  Absolutely true results that way, but I don't have time to do 3-d there and do not have time to get familiar w. that for a while.  But that is my BEST advice, since it's easy, straightforward, and as simple as it gets without a dedicated generator.  Saw latest pix of yours and recognized some M features.  There are surface details in the main body that don't SEEM to correspond, but might.  I can't tell so far which components are used and which is unseen, and to know what was intended may be a help.

Also, clarified point (1.) in last msg. end of previous page of this thread. These points may help demonstrate the usefulness of this system.

Actually had some leisure to look around elsewhere and found there are other extensions that preserve more properties than quaternions.  Bicomplex is one, (194 formula?) and this is not that, either.  Latest pix look look like "bicomplex".  This one seems so simple to me, I can't believe I'd be first to stumble onto it.

Still can't find it elsewhere, however.
 
Is there a way for Terry G. to review our dialogue and comment?

later, and thanx again.

Anyone planning on using fractint in the way I described and is unfamiliar with that program should know this caveat regarding the blackbox: the z variable is used for bailout purposes by the program, and in normal use it is complex, so another variable set, say x1,y1,z1,w1, should be used to indicate old v without causing confusion (error) to the program for each iterative step.  An index set can be initialized with z=0, and then making the above v components = real(z).  Choose screen components from a,b,c,d, (ie. a=real(pixel), b=imag(pixel) ), and fix the others (ie. c=real(p1), d=imag(p1) ), for 2-d views, and then use the black box I wrote, only substituting z1 for z, y1 for y, etc.  The bailout test lines should be something like z=x1+y1+z1+w1  and   |z| <  127.9 (How I did it, unless you have a better one).  That's all you need to write the formula file, after review of program syntax requirements. All you need, I think.

Really gotta go! Be good.

Jehovajah et al- Been away in the wilderness for most of the week, and thought i'd check in.  Very surprised to see no new posts.  Anyway, I'll be doing a lot of that, since the work is seasonal.  It'll have to get cold again before I'll be able to have a look in here on any regular basis again.  I do promise to stay in touch, but don't count on quick answers if you've got any questions for me.  Be well!

Check this out for a c. Pickover mandy.

(http://www.fractalforums.com/gallery/2/410_08_05_10_1_11_11.png)

@ fracmonk

Good luck and good fortune with your work, who knows you may be able to get a cheap computer out of it this year.

I noticed you are using the pixel version of the mandelbrot formula, This always adds more detail to the image if it is there. Quasz really does make this kind of thing easier once you have figured out what you want to do. Although you can not do internal slices the external detail is accurate to the chsen level.

I cannot render all the possible views that may help you to check your results as that would take too long,so i have just done top side and front view but front view has z axis going horizontal and y axis vertical.

I have tested your bail out conditions and they make no significant difference except to slow the render down.

The formula for Quasz is

[rfun
y=y#*y#
x=x#*x#
z2=imaj(z)
z1=z2*z2
w=imak(z)
wx= x#*x#-y#*y#-2*z2*w
wy=2*x#*y#+z1-w^2
wz=2*x#*z2-2*y#*w
ww=2*x#*w+2*y#*z2
rend
((z=wx+wy+wz+ww) && (|z|<130))
z=wx+wy*i+wz*j+ww*k+c
xy
(http://www.fractalforums.com/gallery/2/410_08_05_10_2_52_57_0.png)
xz
(http://www.fractalforums.com/gallery/2/410_08_05_10_2_52_58_1.png)
zy
(http://www.fractalforums.com/gallery/2/410_08_05_10_2_52_58_2.png)

Jehovajah- Thanx once again for pix, you may be getting closer, but it doesn't look quite there yet.  Is QuasZ limited to Macs?  Is that what you use?  Fractint was written for DOS, and I don't know if there's a good emulator.  I avoid a home hookup by choice, and work on a standalone (old one, but fast enuf).  The way it is, i can't upload, and that would make a big difference.  I started w. an admiration for Mandelbulb here, and shared what seemed to be others' frustrations w the inadequacies of quaternions.  That's why I worked out what I did, finally, to share it.  Looked at T.G.'s site, at cquat, (which you mentioned) and that's not it, either, but again, getting close. The formula is explicit.  It is exactly that or not.  My pix indicate symmetries I listed, and for two of them, the M-set is in the middle, and left-hand, right hand volumes bulge out and hang down in the minus y direction on both sides.  (Pretty sure it's minus)  If that helps.  I'm by accident still around for now.  See you...

Jehovajah- I've been thinking (watch out!) maybe I've got the Rosetta Stone that puts us on the same page.  After visiting the QuasZ site, saw his mult. tables. If you're fond of them, try this:

     i   j   ij

i  -1  ij   -j

j   ij   i   -1

ij  -j  -1  -i

When you compare them to those offered by QuasZ, notice the block of 4 in the lower right in particular.  It is why I have so little faith in the idea of bending QuasZ to do this formula for this environment.

This experience pushes me further into the suspicion that quaternions have been a delusional curse on mathematics since 1843.
W.R. Hamilton, not A. Hamilton, by the way.  I thought that absolutely hilarious from here in the states.

Anyway, if you still think you can make that program do the above, good luck.  Started looking at 2-d slices of Julias.  In M, sets are simply connected non-escaping pts or dusts w none.  Any set of 4 param coordinates not escaping should yield a Julia connected over 4 dimensions, even if not shown to be in 2 or 3.  Any view of such a set with non-escaping pts should be connected at least in that way.

Apologies to all-  Before quat fans hunt me down and stone me, I will admit usefulness of H in certain space-time concerns, but M here is more of a number-theoretical object.  But what I've been trying to say all along here is:

more field properties=more validity

The "curse" has really been against number theory more specifically.

I've drafted a paper detailing this system and would like to engineer an upload here, for that and half a dozen small views.  Can anyone help me w this site's procedures?

J-sets for non-escaping coordinates involving nonzero values for c and/or d appear never to be connected in any plane parallel to, or @ 90 degrees to, the complex plane.  The question is whether such sets are connected in 3 dimensions or whether they require 4 to be connected.  If 0,0,0,0 is non-escaping, the set should be (simply) connected.  Any a+b from M (not escaping) when c=d=0 will yield identical J-sets to those on C on the z by w plane with rotation of (guess what?) 45 degrees.  2-d views of J-sets of y by z when x=w=0 have a 4-way symmetry.  All 5 other combinations of 2 where the remaining 2 dims=0 have origin symmetry.

This sis a polynomial rotation mandy at last.

(http://www.fractalforums.com/gallery/2/410_20_05_10_12_44_08.png)

jehovajah- I was wondering if you gave up with this.  Been away myself, but had occasion to look at 100's of 2d x-secs.  May have been wrong about simple connectedness.  For that, one component being zero may be nec.  Looked at one spot where there seems to be something torus-like, from all angles, but it could be an illusion.  V. strange territory.  All speculation.  A couple 3-d views would be nice, but-

Do we have an implementation problem?  Has anyone succeeded in doing the x-secs w FractInt on their own? 

I still need directions on upload procedures here.  Might help.

To elaborate on last msg, no matter  what formula I might examine, I tend to gravitate toward interior pts arbitrarily near an edge.  In M, this includes minis. Structures in such places are generally more intricate.
In higher dims, say 3 for now, the equivalent is viewing pts near the surface.  In the environment I've been lately pitching here, the usual bulbs and minis of M are usually stretched and drawn, usually beyond recognition.  It's harder to say w certainty exactly WHAT features one's looking at.  Maybe bubbles and pass-thrus ARE in fact possible.  In that case, 2-d x-secs are not adequate for full overview, but valuable for looking at internal structure, if any.  It would be superb to have verifiable views in 3-d of this formula, yes?

K- Just checked back in this thread, + now I see where it was hiding. Have to pkg. it to meet size requirements, and will get .frm's on first, if I can. Thanx again!


Time has passed and the .frm file below has been a bit improved upon since, and that version can be found in the next page (15) of this thread.  Also at reply #222 is an updated version of the paper below as well.

A still better and hopefully final version of this paper is posted at the very end of page 16 of this thread.  (Go for THAT one!)  NO, DON'T!  See the BAD CODE NOTICE: posts 251, 252

this thread is going to be closed, or at least needs to be splitted up, please open up another thread for off topics

K- Think I managed it. Never uploaded before, and am using Vista while used to XP.  Trifox- hope I'm not doing wrong here.  Sorry if I did!

OK, I removed my "upload-help" posting, if that was what you meant Trifox...

Trifox-  Not clear whether the problem is size or subject, so if the problem is subject, does that mean this thread remains open, and we should just stay more on topic?

Jehovajah- Still lurking about by any chance?  You haven't weighed in for awhile in this thread, and I wonder if all is o.k. w you.  Posted some of my 4-d (in 2d) in "implementation" section, and if categorization is followed strictly, I should consult w those in "programming" thread at some point, if no other progress can be made.  Due to unexpected (+ otherwise unhappy) events in real life, had fairly unrestricted access to net lately, so I got the chance to put out what I promised. Look around for it, alright?  Silver lining...

See you later.

Yes thanks fracmonk. Just been busy elsewhere and tidying up other lines of thought. For me the search for the true mandelbrot is nearly over,as i have seen many wonderful exhibits of it and my real interest was to explore why polynomial numerals of the form a+ib+jc apparently were not able to produce good mandelbulbs. I now know that this is not true but it requires serendipity to find them and a willingness to be creative abut the squaring formula.

I would like to see one of your slices but i do not have the inclination to figure out how to do it with my current technology, as i am much more concerned with the exploration of the most fundamental conceptions underlyng thinking and algebraic thinking specifically as it derives from general language forms,structures, procedures and syntax,parsing etc.

I will keep looking here though especially if i find a new mandelbulb formula, and who knows what you are explaining may suddenly become clear to me how to represent it. I am still "figuring out" how to implement kujonai's mod 3 unary operator system, although he calls it signed.

Just seen trifox warning. Think he means that a new thread will make this topic more accessible to new searchers. It is very weighty now.
Fracmonk catch up with you in any new thread you start.

Jehovajah- Ha! That may be programming...got a raw idea about color-coding for depth.  Length, width, depth, 3-d...instead of the traditional use of black for nonescaping areas and color according to speed of escape, thought of escapees all being colored black (as a night sky) and then coloring the parts of the surface of a 3-d mass of non-escapers according to distance from the computer screen in the foreground, so to speak. Thought of a pretty efficient timesaver scheme for that approach.  Needs programming (like everything else). Follow your nose... 

J-  Thought I might add that if you get fractint (free) and put the pix and the .frm file together w it in the same subdirectory (folder), and call up fractint, u can view the pix, you can get info about them, zoom into them, save the palette, etc.  Not hard to use at all. The supplied palettes could be better in most cases...these pix use my "standard" one, though there is little hint of its range at the magnification shown.  I'd like to supply a collection of interesting coordinates for anyone interested, since in future I once again will not be able to send pix unless I get an opportunity on a "normal" machine again, and that's an unpredictable thing.  Only wish I had more time for this instead of other things I just must do.

Some coordinates for those who might want to use 2d to evaluate worthiness of conjuring up a 3d generator for full-field-property 4-d M-set:

I suggest, to know where you are, the famous "San Marco" J-set, (first line below):
        
a   (real)                             b     (i)                            c      (j)                         d     (ij)
-1                                      0                                    0                                  0              , and next to try:
0                                        0                                    0                                  .583679392
0                                      +.999                             -.047194378                  +.220344925 , ("zebra" J-set views)
-.048401254                       -.803391959                    -.810697561                      +.31742613

and try one where values for all (a,b,c,d) are: -.333361438

Enjoy!     (This is, of course, for hardcore enthusiasts only...no really, anyone...)

Thought i'd spend some time describing some things i'd found so far, without too much rehash:  I've noticed that many features found are very elongated, and if you've ever mixed paint or saw handmade papers, you'll notice that these cross-section pix have that feel to them.  If I did successive pix w small incremental changes, I get mostly the suggestion of some very complicated detail structurally, likely exceeding the ordinary 2-d M-set.  One set of coordinates i'm looking at now may have a bubble of escaping points fully enclosed within 3 dimensions, with an outlet, however, when one of those 3 is exchanged for the previously unused 4th.  That would mean that simple connectedness would require all 4 dimensions, in such cases.  The requirement of the origin point being non-escaping for the J-set to be connected (even in 4 dims) seems to still apply.  Trying to fully grasp the meaning of these concepts extended to 4-d is strange territory for me.  Don't want to sound like a broken record, (or a skipping cd), but this really needs a 3-d generator.  Terry Gintz, where are you?!!

Is everyone else on vacation?  I have yet to see see any fractal fans read the paper or try those formula files in FractInt and either concur with (confirm) or refute my results.  The grail belongs to anyone courageous and pure of heart AND knows where to look for it...in other words, anyone who wants it...

Gaston Julia would have been supremely interested, and then asked: "What's a computer?"

While working in totally unrelated biz, I've kept the machine busy w hi-res assignments.  They're a bit more time-consuming, but I've found another peculiar set of coordinates close to the surface of the 4-d object (but still within it).  In this one, there are vortices of non-escaping points surrounding eyes of escaping points.  Non-escapees get thinner and closer together as they spin in.  The structure is too similar in character to Hawaiian earrings, which is bad news for MLC conjecture (unless I misunderstand COMPLETELY).  There are in many j-sets some pretty incomprehensible structures, but the likelihood of simple connectedness in 3-d (as opposed to needing 4 for that) seems stronger generally the closer I study these things (in those cases when the point at the origin of a j-set does not escape, of course).  It would be really nice...and it doesn't escape me that the real in the M-set has an UNBROKEN AND CONTINUOUS set of non-escaping points -2<a<.25, so that if ever the capability shows itself, an animation of a 3-d, continuously shape-changing object;   b by c by d   , for microscopic incremental changes in:   a   , would make a very outstanding animation.  Dream fractal dreams...

To continue last reply: coordinates referred to are: -.790323258-.193903776i+.25j+0ij, and were easy (by cj+dij in a by bi) to come by.  2-d "mandelbread" slice studies suggest that connectedness (to standard a by b M-set) is robust, substantial, and persistent in 3-d. Tenuous (filament) connections in the standard M-set can be either MORE or LESS substantial in higher dimensional extension, depending on views chosen.  Good news for the MLC conjecture, unless I misunderstand it completely (see last msg) OK, I admit it was a provocation of timid academic lurkers, if present...I had this daydream about some Big U. math dept. prof. whacking me on the head w. a rolled-up math journal, shouting: "BAD topologist, BAAAAD topologist!!"

In the past, I had run into filled-in Julia-like features in 2-d index sets of other complex functions, sometimes neatly symmetrical and apparently simply connected.  When they weren't, I had assumed, in most cases rightly, that I did not get the critical point, and there was only local coincidence.  But this offers some insight into why j-sets take their characteristic shapes.

I have been able to include a zoom series of the aforementioned coords. for a by b in mags. 1 thru 100k in this & last post.  They should have 4 to 3 aspect ratio, but don't.  They look really good in 2048x1536 and proper proportion.  (Hint, hint...)

And now for something completely different:

This is as good a time as any to thank David Gilmour for an ancient (pre-fractal) favor, related to inspiration and empathy.  We're even now.  Stay connected.  Breathe...

One more thing: There's no reason why fractal artists wouldn't want to include kaleidoscopic effects from trigonometric manipulation of quaternion-like schemes in their repertoire.  "Hall of mirrors" comes to my mind inevitably when considering the search here, which is of a mathematically scientific nature more than an artful one, however.  Everything has its place.  

As a result of some dialogue in the "Programming" ("FPP") section of this site, there Schlega came up with a slight improvement (about 5% in overall running time, on average) for the .frm file.  I updated it, available below.

A further result of that discussion prompted two updates of my paper on this subject, also below. notquat9.doc has the latest, most tragic news:
The division scheme long discussed most likely has a FATAL FLAW, described within.  The rest is fine, however.

The best and hopefully final version of the paper is posted at the end of page 16 of this this thread.  The aforementioned flaw is subject to your point of view, and the situation is explained fully there.

BAD CODE NOTICE: See posts 251, 252

Here are views of the J-set for the coordinates given in reply 221.  They have the origin at center and take slices at various axis combinations.

And here is the rest of that set.  Notice the similarity between xy and zw.  Notice also that it is very difficult to describe this space without lots of pictures...

As nearly as I can tell, the poor behavior of division in this space would make it a commutative ring.  The 2d .frm file for cross-section views I made available only hints at the complexity of indexes and the expanded collection of Julias that exist in this space. A lifetime would not be long enough to do even a cursory survey of these.  One day it could become a favorite vacation destination of mathematical minds, and as such, never get too crowded.  But without input from others, it is little more than a message in a bottle.

These things are there, but do not become a part of our reality until they are discovered and revealed.  This job is far from done, and since this is more important than commercial considerations, I've provided what I could manage without restriction.  As an avowed enemy of orthodoxy, I've made some mistakes along the way and said some foolish things, but I feel this will emerge and flower on its own, if it gets enough water and sunlight.  To be clearer, the grail cannot be possessed with the intention of hoarding it selfishly.  The search is not over.  This is only one way of projecting the M-set outside the complex plane, even if it turns out to be the best-justified one.  Every picture NEEDS a story.  What have YOU got?

One result of (let's call it) hypercomplex analysis is:  (j+ij)/sqrt(2)=i.  ...which means that the M-set, a little compressed sideways, can be seen on the a by c=d parameter plane (45-degree angle).  By NOT plotting bi dimension, you get a simply-connected version of M besides the one we're so familiar with.  It IS a single 3-d object that it comes from, but will always require 4-d calculation to render.

See it below, along with a location zoomed into, along with a Julia view for the same angle on the complex plane.

The plane shown in the 1st pic above cuts the volume of the a by c by d object exactly in half, because of its own symmetric properties.

Coordinates for the 2nd & 3rd pix there are a=-.744381711, b=0i, c=.148284347j, d=.148284347ij.

The 1st pic below is of a by c for constant d. From the view of that plane, it would seem unlikely that a zoom into the picture's ctr. would yield a view indistinguishable from that seen in the second pic above, but it does.  2nd pic below shows the entire origin slice of the Julia set in x by z orientation for the same coords.  Obviously not connected, but the 3rd pic shows its ctr. magnified 1000x.  Close to the origin, connected, locally. Away, falls to pieces.  This is only a 2d effect, as it remains connected, as previously discussed, in 3d.  The location was very randomly chosen.

Here are a couple more views of the julia ctr detail.  Stretch and spin in these are inevitable because of the orientation of the first view shown.  Interestingly, xy looks like zw there, xw looks like yz, and xz looks like yw.  See the pattern in that?

I'm really not a monologuist, you know.  I feel a little like I'm wandering city streets after a nuclear blast.  Or maybe it's just MK...

Comments?

Has anyone else tried to image the Julia set for 1+i-sqrt(2)j, or similar coordinates?  It seems that though this point translates to zero, it escapes rapidly, because of its multiple components, analogous to something like 20+20-40=0 (if it was pertaining to a Euclidean situation).  It is sheer divergence, cloaking the ordinary complex behavior with another entirely different one, which strikes me in a way as entirely natural, while the complete conversion is quasi-euclidean, in some respects.  There is no reason to think that extension won't reveal unexpected behaviors.  A complete conversion from 4d to 2d is easy, and elucidates it nicely, but I found that an idea to do a sort of conversion which keeps 3 of 4 components in each iterative step, using the (j+ij)/sqrt(2)=i identity will preserve the 4d results you have already seen.  It is also time-consuming, however, and changes nothing.  Still pretty nice to see it in both incarnations.  Any takers?  Other thoughts?

Reflecting on things, I realize I probably could have been clearer in saying that there is no advantage to conversion to pure complex except understanding of a few algebraic questions.  As in a Euclidean plot, the dimensions must be kept separate to get any meaningful results.  In that way, 20 minus 20 is a position, not a sum, or problem to be solved, by an answer such as zero, as in the case following.  I must confess, for awhile I was having doubts about the veracity of my results because of the canceling effect of, for example, 1+i=sqrt(2)j.  This only applies to the purposeful collapse of 4-d to 2-d before iteration begins, when you have a position 1+i-sqrt(2)j.  If you want 4d, you just don't collapse it!  Some days, I just feel stupid, but I do come around again...

Whenever I can, I'm working on the "final" version of my paper, of which early drafts were posted here, which will have more elaborated info about M in this space.  Busy w. other things too, so it takes awhile.  I really need to get it right, including all the things recently found, in some coherent form.  Asimov's Foundation trilogy, if I recall correctly, ended with 2 chapters entitled: The Answer that Satisfied, and The Answer that was True.  Before I submit this for your consideration, I have to research some applications speculated about for M in the past.  A bit haunted by the notion that M is an electrostatic model as well.  Extra dimensions that behave the same should be of more than passing interest to those who study electrodynamics, even QM, where explanations could always make a little more sense!  Not expert on such things, that, for me, is really reaching a bit...  Sorry for the delay.

Thought I'd show you a pretty picture while you're waiting.  Funny, I always used the same palette and never even rotated it for everything I ever put on this site...

And how about some spelunking down some of the odder-shaped canyons of the 4-d M-set?  I was doing a coordinate on the a by c=d plane, and looked at other views parallel to, or at 90 degrees to, the complex param. plane.  Too many times, I would see something interesting, but never have the time to go there.  It's a zoom exploration here in *b by c*, starting at mag. 100.  *Oops!  Pix are mislabeled, sorry!*

(to be continued...)

Never know what you'll find in a cave, and in lo res, you can miss good stuff...but you can follow yer nose wherever it goes...

There seems to be a problem with connex today- tried to upload remaining pix in the series here.  I'll try again...it seems to only take text right now, if it does even that...

So now (under "modify"), I try again.  V. stubborn guy...

And finally, the last of them.  Nothing at all special about the location.  The journey offers good sightseeing, as these usually do.  FractInt only allows a 9-digit decimal assignment in input variables, so that limits the depth of Julia views, which always correspond w. index views.  So this last pic has a big black blob at its center, to all but insure that the coordinates are of a connected Julia set.  This is a problem when one does not see the trademark mini-M.  Its edges have that repetitious quality, however.  The series was done with 100,000 iterations.  Infinity is another story entirely.  I should find an interesting set of Julia views to go w. this series.

Also, on radio, heard an entertaining short story called "Chivalry", by Neil Gaiman, and na       by Jane Curtin in a show called "Selected Shorts" which had at its center the Holy Grail.  Absolutely nothing to do w. this, but v. funny.  If curious, maybe you can search it out.

Afraid posting this pic is also prohibited as w. last post, so once more, we'll have to wait for better conditions.

The line up above contains "and na       by", which refuses to be fixed.  The word is "na      ".

NA      !

N
A
 
 
 
 
 
 
!

Here is the last in the series, and some z by w Julia views, which are really equivalent of x by y ones, turned 45 degrees.  I've noticed that one does not have to go as deep into J-sets to get the roughly corresponding features in an index set, as in the relationships between the standard M-set & its Julias.  Anyone have an explanation for that?

Tried to fix last msg.  The word is DETARRAN spelled backwards.  This site doesn't like that word in some uncanny way.

The more I think of msg. 234 I laugh harder- it's like something out of Monty Python (& the Holy Grail?)
"We are the Knights of NA       !  We want your fractal pictures.
Yes, indeed, all of them will do.  Thank you very much.  Thank you very, very, very..."

From the left side of the first picture in this last zoom series are a couple more detail pix.  They continue to show how the complexity of M takes on a wholly different character in 4-d.  Currently working on hi-res, hi-iteration pix that take a long time and contain too much data to show here, according to the rules as I understand them.  Am proofreading when I can, too...

Well.... what a long thread! -  keep up the good work. Although it'd take me a while to fully understand everything written in this thread (my math is by no means top notch), the search is still ongoing, and for all I know you're getting closer.

To toss ideas into the mix, I'm not sure if it's been mentioned before, but one might try alternation of rotation and scaling with the original Mandelbulb formula, rather than performing both rotations (phi & theta), and then one master scaling afterwards. My preliminary attempts weren't successful, though I haven't tried further alternating with the vector additions as of yet (splitting up x, y & z).

twinbee-  For awhile I thought I was being shunned or something!  I recently looked at mandelbulb for the first time, and can't see how the M-set there is not a flower pot from which trig projections are made to grow.  That's my first impression. Where can I find understandable info on what its geometry represents mathematically?

  Anyway, a couple things I'd planned for today may help you here:

Back to basics- The pix below are for the familiar 0+i+0j+0ij Julia set, various 2-d views.  I'd love to see a 3-d rendering of this from anyone with such a capable generator.  I still work only in 2-d for now, and don't have one yet.  Even w. max iterations, it would be fast, since its nature is not to pinpoint non-escaping points, but for escaping points to shadow them closely.  (Pleeeeez!)

Also, I've taken my paper about as far as I could, keeping it as simple as it can be, I hope.  It aims for those like myself who would not take my word THAT this space is a good, maybe best environment for extension of M, but need to convince themselves of HOW and WHY it works.  (Just wanted you to know-) You are not alone.

PAPER MAY NEED FURTHER CORREX: see posts 251,252

Fracmonk nobody is shunning you, i hope. The interest in this space is waned because we all have a surfeit of the mandlebulb, any of which are a lifetime study! Or rather a playing field.

I am enjoying your work but from afar and am busy on my own interests, so i am sorry if you have been feeling left out. Sometimes pioneering work means a lonely trail.

You say you have updated your paper so repost the link and i will have a look.

Msg. 239 this thread, end of last page (16).  I feel that this space is a universe unto itself as well.  Maybe each of us will just wind up w. their own.  Hopefully not IN their own, though.  No prospect for mathematical explanations of Mandelbulb outside of "Oh, look what happens when you do this!"?

In the very beginning of my comments on this subject, I mentioned "designer dimensions".  Did a bit of research on the loathsome subject of math terminology, and found that my approach uses "binary arithmetic operations" w. a 2-arity consistent with how the original M-set is generated.  M-bulb & many others use a 1-arity unary math atop it somehow.  (Correct me if I'm wrong about that!)  I don't see consistency in such a dimensional mix. Don't get me wrong- M-bulb is a tremendous mathematical work of art, but is it really a proper extension of M?

Firing squad volunteers line up below.

Most everyone must be by now aware of the passing of our beloved Dr. M. on Thursday, as I heard it reported finally Sunday.  This was my first chance to get back to you.

Below is an account of my own, if you care to indulge me in my own grief.  They say it's better to share these things.

I've spent some time gathering my thoughts for this.  One of the reasons I got involved here was the perception that fractal art, with all its polish, was eclipsing fractal science.  Over the years, I saw the Spanky site fall victim to link rot as New Agers competed to dazzle you with, well, polish over substance.  Often I wonder whether I'm too late.  Looking around this site, I saw a comment that asked whether Dr. M could weigh in on Mandelbulb and give his thoughts on it, but everyone seemed afraid to ask.  I don't know if his health problems were what might have prevented him, but he apparently never did.  If you're one of few who downloaded my own account from the last post, you will see that I also have cause for extra grief for the same lost opportunity.  I would have been greatly honored to get his critique on my own work.  He was primarily a scientist, and I feel that I must be true to the same spirit that drove him.  I tell myself to be patient, and they will come around, as they did for him.

Because of the univerality of M, it shows up easily in odd places, and lends itself to many kinds of extensions.  The one I've been describing to you relies on the behavior of the numbers themselves in iteration, and is more consistent with the original complex dimensions that produce M than any others suggest.  That seems to have gotten yawns so far, and let me tell you, I JUST DON'T GET IT!  Is there something in the water?  I hope that eventually others will realize the significance of this.

Anyway, if I've missed some key thing here, have the decency to let me know.  Everyone has blind spots, and I can face up to mine.  I think I've made a fair case, though.  Again, correct me if I'm wrong.  I can take it.

Lovely read, thanks :-)

Amazing that you actually get to talk to the fella. He presented a lecture at the University I went to, annoyingly the year before I started.

Sad to see him go but what a legacy he's left us!

post #234 made me laugh too.
Narrated narrated narrated
NARRATED

hey I can say it

Tglad & miner49er-  As they say, that was then, this is now.  Narrated.  It seems to work...it may have had something to do w. pix being overposted then.  Why that symptom?  Still beats me!  Glad U got it, though.

Still need more serious and genuine feedback for the math, and plain english about how Mbulb works, since I REALLY want to deal w. the powerful feeling I have that somehow I'm missing something!

Also, colder weather will force me back to more limited internet access, where I will seldom get my hands on equipment that can do uploads and downloads, and I will be forced into verbal-only messages like this one again.

On public computers in some places, they impose this handicap for fear of (shudder) terrorists and hackers (and bears, oh my!)

The last line of my last post, for those too young or otherwise unfamiliar, is a takeoff of one of Dorothy's lines in the Wizard of Oz.  Another goes something like: "Toto, I don't think we're in Kansas anymore."  I really love that one...

For newcomers to my "notquat" formula, my comments in this thread begin at p.12.  The .frm file to use w. FractInt for 2-d slice images is in p.15, reply# 222, and the most concise version of the paper, again, is @ end of p.16.

6 most basic 2-d views are available in replies 21&22 in "Implementation: 3d Mandelbrot type fractal" thread.

Schlega did some 3-d views in July that can be found in reply 6 in the "Programming" section, under "A Great Need...
He's got some code there, too.  Not familiar w. ChaosPro, I don't know if it's right.  The pix may be, but I think they're under-iterated, and less detailed than optimum.  Can anyone: verify/provide orientation info/elaborate upon/improve, these "possibly historic" images?

That should gather up available info a bit tidier.

There is an observation I'm very fond of, I think from Zen, that says that everything you can see is in the way of something then that you can't.

We still have serious problems, that I earlier bemoaned, with how dimensions are defined, and whether they should be somehow better classified.  Do we speak of directions and distances, on one hand, or of qualities or properties, on the other?

In the former sense, I think most of us here are usually concerned, rather than the latter.  I've talked of 2d & 3d slices of a 4d object, one based on square roots of i.  I often wonder if most dimensions we experience in this world are unseen.

Would anyone be mildly interested in a new thread concerned with CUBE roots of i, (which HAS to be very different somehow, w. my best guess) so that if the M formula were used, one would get a 6d object to look at?  According to Hurwitz, thankfully, there's no division possible.  Please let me know...

people, this thread is growing too large, are you ok if i close it, with notice for next thread ?

Trifox- Glad you're here for this:    It's a serious NEED TO KNOW!

In the core code for the square root of i object I've done, I made a profoundly WRONG 2-character mistake that, copied into all view formulas, makes most of them give WRONG results most of the time.  I only discovered this at the 11th hour of the 11th day of the 11th month, and could not notify til now.  That means that MOST but not all of the pix I posted on this site are WRONG if pretty artifacts, and MANY but not most observations I made based on them are erroneous as well.

I'm posting the corrected version of the .frm with MY DEEPEST APOLOGIES below.  The wonder is that in the previous version, ANY of it turned out basically right.  Up to now, I have become inadvertently guilty of promoting the "designer dimensions" I was so busy complaining about.  There are the same symmetries, though the character of edges, and presumably surfaces in 3d will be different.  I'm also adding sample 2d pix done w the correct version.

To continue,
Oddly, most of the info in the paper I wrote on the subject is still accurate, but I have to review and correct that too.  I've done THOUSANDS of time-consuming pix w that bad code, so as bad as you may feel, being responsible for it makes it that much worse for me.  I will fix this, no matter what it takes.  For those few who have followed my work, I apologize once again.  They say you only hurt the ones you love...

For chaos theory, it seems that small errors can turn into systematic ones.  (see?)

Because of my access problems and other commitments, I won't be able to fix this mess as quickly as I would like.  I will use every moment I can, however, to do so.  I only wish anyone using it might have found it sooner.  The .frm is prefaced w a comment giving the specifics.  I hope no one interested becomes discouraged as a result.  I'm really trying not to myself!

Trifox, in this light, do you have any suggestions?

More details:

For those interested, it is truly remarkable that my paper will need very little amendment.  The problem is with the code and the pix it creates, not with the theory, concepts, or the math.  Locations cited in earlier posts naturally correspond to a wrong-shaped object and the Julias that are then consistent with IT, and the correct object is apparently completely M-like in its features, now that I've taken a better look at it, having Julias consistent with a correct formulation.

I cannot upload anything right now, and I will be very cautious in future- once burned, twice shy.  This is good advice for anyone.  Check and understand what the code is supposed to do, if you're as serious about it as I am.  If you're not, enjoy the pretty pix, (the OLD ones) even if they're not meaningful in the least.

The paper will be corrected and posted accordingly post haste.  (When it's absolutely right!)

Reflecting a bit:
Using the corrected code, you will find that ABR.gif (not shown in post 251) would be the standard M, that ACR & ADR are identical, & that BCR & BDR are in a left-hand, right-hand relationship w. each other.  So it may be the canonical extension of M that eluded us all this time after all.
Despite my being (up to only very lately) the most terrible ambassador for this object, I am sure it can stand on its own now.
I firmly believe that this thread, in all its tortured history, will be judged to have finally found its salvation before the end...

So Trifox, if it makes everything easier, I have no objections to your putting it out of its misery now!

You agree?

Probably better to start again fresh.  Very little will need repeating.

I understand your grief and your passion Fracmonk. Respect!

Hope to see you in another thread.

J-  thanx bunches, it's really appreciated.  I'm not worthy, of course...

Since this thread is still open, I guess it won't hurt to do a few parting shots.  The only reason I chanced on finding that code error was that I was taking the basic formula structure and editing it for a 6d version (based on CUBE roots of i) mentioned recently in an earlier post, and spotted it then.
6d really stretches FractInt's capabilities, maybe too far.  Not sure if it can do it accurately.  I don't know about opening a new thread for that.  We'll see.

Back to 4d, here are 4 pix, plotting cj by dij for a=-1.75, mag 5x, and a=-1.25, a=-.75, & a=.25, all 1x.  In each then, the center is just a, and bi=0.  Pinch points.  With the right code, even!

I downloaded ChaosPro, and if ever I can find the time, I'll see if I can learn how to get it to do this stuff.

I wouldn't mind if someone else opened a new thread based on squareroot(i) to continue this exploration, based on interest.

Proofreading hopefully final version of paper.  But of course, you've heard that "final" bit before...                    (-later!)

If all 2d slices of your 4d object looks like the last pictures, the 3d view must be very promising.

Can you post the corrected formula? 

trafassel-  The formula was ok all along, but the code had an error, as I explained.  Post 251 here has the corrected views that made the pix there (not v. visible, above them). 

I was also going to mention that a by c by d plots (while b=0) will yield a handsomely symmetrical 3d object, turned 45 degress on the real axis, which splits in the middle to reveal a PERFECTLY whole M-set, a by c=d.  This is, however, is narrowed widthwise in a 1:sqrt(2) proportion.  I can't wait to see a 3d rendering, which, of course, I don't know how to do yet since I'm not familiar enough with any 3d generator...

But you might be able, no prob., I dunno...best wishes!

I can try to render your fractal. I only need the formula

For my program input i have to converted it in a formula like this (8-Mandelbulb Cos):

    public override long InSet(double ar, double ai, double aj, double br, double bi, double bj, double bk, long zkl, bool invers) {
            double aar, aai, aaj;
            long tw;
            int n;
            int pow = 8; // Mandelbulb
            double gr = 10; // Bailout value
            double theta, phi;

            double r_n = 0;
            aar = ar * ar; aai = ai * ai; aaj = aj * aj;
            tw = 0L;
            double r = Math.Sqrt(aar + aai + aaj);

            for (n = 1; n < zkl; n++) {

                theta = Math.Atan2(Math.Sqrt(aar + aai), aj);
                phi = Math.Atan2(ai, ar);
                r_n = Math.Pow(r, pow);
                ar = r_n * Math.Sin(theta * pow) * Math.Cos(phi * pow);
                ai = r_n * Math.Sin(theta * pow) * Math.Sin(phi * pow);
                aj = r_n * Math.Cos(theta * pow);

                ar += br;
                ai += bi;
                aj += bj;

                aar = ar * ar; aai = ai * ai; aaj = aj * aj;
                r = Math.Sqrt(aar + aai + aaj);

                if (r > gr) { tw = n; break; }

            }
...
}

sry people this thread has to be closed, threads with more than 10 pages tend to become desinformative!
open up a part II thread please!