Welcome to Fractal Forums

One of the many problems with 3d fractals is the "whipped cream", example below shows a quaternion:
(http://www.subblue.com/assets/0000/4473/Quaternion-Julia-10_medium.jpg)

Whipped cream is bad because that means there are regions of unbounded distortion. If you zoom in on one of these regions, you lose detail in the direction of the stretch.

In the mandelbulb the mapping function has "polar pinch" singularities due to spherical coordinates. However, I believe quaternions are not distorted since the rules of quaternion arithmetic are "natural" like complex numbers and singularity-free.

Instead, the distortion comes from the projection of 4d into 3d. Imagine you looking at a cube. You will see a projection of the 3d cube into 2d. All the faces are perfect squares but faces that are seen near edge-on look distorted into thin rectangles or parallelograms. I think a similar thing is happening with the quaternions, in which the 4d to 3d projection is causing the distortion.

What are your thoughts?

Unfortunately quaternion mandelbrots/julias are stretched. The q*q operation is non-conformal, meaning it introduces stretch... and since the operation happens an infinite number of times... whipped cream.
There are only 5 operations that don't introduce stretch in 3d (or higher dimensions)- translation, scale, rotations, reflections and sphere inversion. Or any combination of those, which is always reducible to just one of each operation. Liouville's theorum.

Mad thoughts - prompted by lack of formal math beyond "A" level:

In Euclidian space is the restricted conformality of Rn (n>2) directly related to the lack of true algebraic fields in Rn (n>2) ?

I'm guessing yes ;)

Also I would suggest that a true secondary "imaginary" vector/constant/function (j) is required to solve this issue at least for R3 or R4, probably such that j acts in two dimensions instead of one.....Hmmmm

  I tend to think that the "stretch" is caused by variable "interference" with one another (for the Mandelbulb, at the very least).

  You've got 3 components- 1 of which is influenced by the other 2.  In certain areas, the chaotic effects of the one variable cancel out the chaotic effects of the other 2- so you get a boring stretchy section. 

  I don't know if it's the same for quaternions.  I tend to see them as... lacking magnitude management- more on that if I look at the math later.  Just had an idea to pursue.  :D

I should add that it is Liouville's proved theorem. So, unless an operator uses the 5 mentioned transforms, it will have stretch.
It cannot be unproved by using a new squaring operator on 3d or 4d vectors, nor by a different management of magnitudes, or by a inventing new 'hypercomplex' numbers; it is just a fundamental fact of 3d (or greater) geometry.

  I'm not finding any reference to a proof of  "stretch" anywhere???  In fact, all I can find is the proof of the conformal mapping constraints.

  Anyways...  The additional variables interact to cancel out chaos in certain areas... unless set up properly.  :dink: :dink:

Yes, its a shame that the encyclopaedias don't explain very clearly... but conformal means that tiny spheres transformed will still look like spheres. Anti-conformal means that a tiny letter p transformed will look like a letter q, reflections are anti-conformal.
If it isn't conformal or anti-conformal then tiny spheres transformed will become ellipsoids, they will be stretched, or squashed depending on which word you want to use.

Nothing of value to add but I have to say this is a fascinating discussion.

I've been thinking about the definition of conformality lately and wrote a small program in Octave (Matlab clone) to compute the "stretchiness" for an arbitrary function.
Here's another way to think about it:
- To be conformal, the Jacobian of the function should be a rotation matrix times a scalar. A Jacobian is a NxN matrix (where N=the dimension). Its matrix entries are the derivatives of the x,y,z functions by x,y, and z.
- Another way to think about the Jacobian of a conformal function: every derivative = infinitesimal vector must be orthogonal to every other (dx . dy = 0, dy . dz = 0, dx . dz = 0) and every derivative's magnitude must be 1 (dx . dx = 1, dy . dy = 1, dz . dz = 1) times a scalar for all entries.
- The limits of conformal functions in N>2 must have to do with the fact that having the Jacobian = a rotation is akin to solving a series of differential equations, but there's too many equations and not enough flexibility for the function.

example, the Mandelbrot function,

fx = x*x - y*y;
fy = 2*x*y;

J = (2x -2y)
      (2y  2x)

by multiplication,

J = 2*sqrt(x*x+y*y) * (cos -sin)
                                   sin  cos)

J = scalar * rotation matrix

I haven't seen it mentioned yet, so I'll mention it. The true definition of a conformal transformation, valid in any dimension, is a transformation that preserves angles. If you have any two smooth curves intersecting at a point, there is a well defined notion of angle between the two curves, namely the angle between their tangents at this point. A conformal transformation will map these curves to different ones, but their angle of intersection will be preserved.

As was mentioned, in dimension greater than 2, the conformal group is finite dimensional, generated by translations, rotations, scale transformations and inversion. In two dimensions, at least if we are interested in conformal transformations defined locally, the conformal group is infinite dimensional. Seeing the plane as the complex plane, any holomorphic change of coordinate is conformal. Holomorphic meaning that the new coordinate w is a function of the old coordinate z but not of its complex conjugate \bar{z}. For instance the function iterated to produce the Mandelbrot set z -> z^2 + p is conformal, because there is only explicit dependence on z on the right hand side. From this example, we see that dimension 2 is very special, because it allows conformal transformations that are not gobally one to one. This fact is the root of many visual features of the Mandelbrot set. That's why, no matter how interesting it is by itself, the Mandelbulb cannot be a true 3d analogue of the Mandelbrot set. And why I believe there is no such thing. But from the richness of these new 3d fractals, it's well worth looking for it... :)

And I agree that whipped cream effects are a direct consequence of the presence of a non-conformal transformation.

 Highlighted an important statement.... and on to an explanation of what I thinks been going on (with the word "conformal"):

  I think here, at least if I remember correctly, Tglad and maybe others were using conformal to mean that a transformation of 3 points (a triangle) with the equation z^2 + c would result in an angle preserved triangle in a new location (perhaps within the Mandelbrot set, or outside the set specifically??).  I never bothered to check the math, so can't say that the M-set transform is conformal or not, and I haven't read a proof either- I've just seen it said numerous times.  I'm actually interested in a link.... but I suppose a simple triangle test (both inside and outside the set, as I don't recall whether it's supposed to be one or the other) would work as well:

A =(0,0)  B= (0, .1i ) C= (-.1,.1i)       right triangle
A'= (0,0)  B'= (-.01,.1i) C'=(-.1,-.08i)   not a right triangle... angles ain't preserved.  NC

outside the set:

A=(1,0)   B= (2,0)  C= (2,1i)    right...
A'=(2,0)  B'= (6,0)  C'= (5,5i)   not right...  NC  

    So I don't really know why.... or how the idea came about that the M-set transform is conformal?

    Any links to the original proof or statement about it?   There is this one... (http://www.fractalforums.com/index.php?topic=2133.msg10358#msg10358)

Matt,

 For the M-set, the angles are preserved in the infinitesimal limit, x = x + delta, y=y + delta, which is all that's required for conformality.
 My previous post in this thread has the simple proof. ;) (J = Jacobian)

-mike

Just to add to that, if it helps, here are some equivalent definitions for conformal:
-Angle between any two curves is preserved
-The Jacobian of the derivative of the transform only contains rotation and scale.
-Small triangles map to small triangles with the same internal angles
-Small spheres map to small spheres

If you are testing with triangles then the key word here is 'small' which is used in a very restricted sense, it means infinitely small. Or to write it in long hand- As you decrease the size of a triangle to 0, the transformed triangle's internal angles converge towards those of the original triangle.

In these conformal transforms of a clock, you can see that the large clocks can have weird shapes, but at the tiniest scales the clocks are all circular, so 'small' shapes are preserved-
http://en.wikipedia.org/wiki/Conformal_pictures (http://en.wikipedia.org/wiki/Conformal_pictures)

Another question about Louiville's conformality theorem - I hope this makes sense:

The theorem states that the conformal mappings in Rn (n>2) are limited to those as described in this thread and here:

http://en.wikipedia.org/wiki/Liouville%27s_theorem_%28conformal_mappings%29 (http://en.wikipedia.org/wiki/Liouville%27s_theorem_%28conformal_mappings%29)

My question is, is it possible that other transfoms in Euclidian space of Rn (n>2) will be conformal if we only consider a lower dimensional slice of that space, e.g. a slice in Rm (m<n).
I'm pretty sure that the answer is yes ?
If so then I still think it is possible to achieve the elusive "perfect" 3D Mandelbrot equivalent but only as a 3D slice of a Mandelbrot in Rn (n>3).
This would mean that there's no point really persuing anything in plain R3, but IMHO it's still possible that there may be an R4 (or higher) algebra such that a 3D slice gives the holy grail.

If you mean performing a transformation in only say two of the dimensions then the answer is no...
For example, a transform that did Z = Z^2 on the r and i axes, and J=J on the j axis is not conformal.
If you mean literally only transforming a 2d plane that happens to be in 3d space (e.g. only when j=0), then yes the transformation is conformal, it is an exception because Liouville's theorem applies to smooth transforms, and that wouldn't be smooth or continuous.
It does not follow that 4d space can contain a 3d non-mobius conformal transform, regardless of what algebra you use, the theorem is a property of euclidean geometry, not any particular algebra.

   Thanks Mike (and Tglad).  I should have looked closer... to see the infinitesimals.  

I realise it does not *necessarily* follow, but it seems to me that Louiville's theorem assumes that all dimensions are considered both before and after the transform/s.

For example say we have an R4 transformation but we only actually consider R3 space before and after the transformation, I don't think that Louiville's proof is complete enough to state that a non-mobius 4D transform couldn't exist that is not conformal with respect to the R3 view of the R4 space - I realise that this couldn't be described as treating things as strictly Euclidian though, but to render an object that appears like the "holy grail" 3D Mandelbrot the number of dimensions actually involved in the calculation is not really relevant.

To put it another way, is it not possible that a non-Mobius transform in Rn is such that all failure of conformality in Rn involves only Rx where x>=1 and x<n-2 ?

David: Good idea. The way I think of it is that you have hidden dimensions that absorb the bad stuff. But, df/dx, df/dy, and df/dz would remain mutually orthogonal.
Likely, in the 3D projection, you would have lots of discontinuities as density moves in and out of the hidden dimensions. So, connectedness would have to be sacrificed.

Here's a sort of equivalent to help me picture it:
All the angles in a triangle add to 180 degrees, that is proven in euclidean geometry, and it doesn't matter how many dimensions.
But, if you imagine a curved sheet in 3d, like the surface of a sphere, then a triangle on this sheet will have internal angles add up to > 180 degrees.
This doesn't break the proof, since it isn't a triangle in 3d, what we're actually talking about is a triangle in 2d in a non-euclidean space.

And that is what I think David is describing... any 3d volume that is bent in 4d space is the same as just operating in a non-euclidean 3d space. Liouville's theorem doesn't apply because it is non-smooth in the 4d space.
(If David is not suggesting the volume is bent in 4d, then no, Liouville's theorem still applies).

So... the question is, what sort of conformal transforms can be done in a non-euclidean 3d space?
Answer: I don't know...   but...

A bit of googling suggests that Liouville's theorem is also valid for any 'Hilbert space' (http://eom.springer.de/q/q076430.htm (http://eom.springer.de/q/q076430.htm) "Liouville's theorem is valid both in the case of Hilbert space [36] and..")
So I looked it up and Hilbert space is a big generalisation of euclidean space, which basically covers all spaces where calculus can be used (wikipedia- "Hilbert spaces are required to be complete, a property that stipulates the existence of enough limits in the space to allow the techniques of calculus to be used").

I guess the same issue can be applied to being "complete", the statement regarding Hilbert spaces I assume also assumes that we're considering all n dimensions of an Rn rather than just x dimensions of Rn where x<n ?

The Rn space could be "complete" so that when we're calcuating the DE for instance we use all n dimensions, but when we visualise say R3 (were n>3) or consider the conformality of that R3 mathematically it may be that the R3 sub-space is conformal under the Rn transform.
The problem is finding the correct transform and the necessary value of "n" in Rn to give a conformal "z^p" when viewed in some R3 subspace of Rn under z^p.
Only problem with that idea is that the Rn DE value could sometimes be very, very small even in apparent empty space because there's "solid" very close by but along one of the dimensions not included in the R3 we're rendering/viewing (a similar issue applies to using DE to render complex Julibrots again due to the DE being of different dimensionality to the viewed/rendered object).

No, as I was saying, a bent 3d volume in 4d (or higher dimensions) can be considered simply a 3d space in a non-euclidean geometry. So the transform happens in 3d in a Hilbert space.

In short, the only conformal transforms in >2d are the Mobius ones.

OK.

But (probably just my ignorance)....another point comes to mind, we're dealing with fractals any type of which considered as the entire, complete entity (to the theoretical limit) is essentially a single transcendental transform.
Do Louiville/Hilbert theories cover transcendental transforms as well as finite ones ?

Another question for Tom: is the Mandelbox transform reducible to a single Mobius transformation, or is it an apparent violation of Liouville's theorem? I say "apparent" because Liouville's theorem may not cover C0-only functions (those without continuous first derivatives).

  I'd like to ask a question:  why do we need conformal transforms?

  I recently introduced an even "more" non-conformal z^2 formula over in the theory thread (it has conditional sign assignment for new x components on the positive x-axis that depend on the initial values of x, y, and z).  The formula produces many unique, interesting quasi-fractal patterns which vary with the location explored, and we've only just started exploration.  Even in the stretched areas, zoom in enough and you'll start to get emergent patterns (maybe not awesome ones), although it requires a large number of iterations and deep zoom (which takes time on older computers- and you can just go for a nice low  <20 iteration area...). 

(https://lh6.googleusercontent.com/_gbC_B2NkUEo/TWsD7O1SvXI/AAAAAAAAA-0/HDRxA84XJjo/s144/z%20squares%20is%20pretty%20big.jpg) (https://lh6.googleusercontent.com/_gbC_B2NkUEo/TWsD7O1SvXI/AAAAAAAAA-0/HDRxA84XJjo/z%20squares%20is%20pretty%20big.jpg) (https://lh4.googleusercontent.com/_gbC_B2NkUEo/TWtGPNMw86I/AAAAAAAAA_Q/b0GGr30XZ8E/s144/side%20a%20mine%20270%20degrees.jpg) (https://lh4.googleusercontent.com/_gbC_B2NkUEo/TWtGPNMw86I/AAAAAAAAA_Q/b0GGr30XZ8E/side%20a%20mine%20270%20degrees.jpg) (https://lh6.googleusercontent.com/_gbC_B2NkUEo/TWtdYa65e7I/AAAAAAAAA_8/o8cSalyufBg/s144/radio%20tower.jpg) (https://lh6.googleusercontent.com/_gbC_B2NkUEo/TWtdYa65e7I/AAAAAAAAA_8/o8cSalyufBg/radio%20tower.jpg) (https://lh5.googleusercontent.com/_gbC_B2NkUEo/TWtiK8GeHHI/AAAAAAAABAc/YzVG-36FE8Q/s144/melted%20pile.jpg) (https://lh5.googleusercontent.com/_gbC_B2NkUEo/TWtiK8GeHHI/AAAAAAAABAc/YzVG-36FE8Q/melted%20pile.jpg) (https://lh3.googleusercontent.com/_gbC_B2NkUEo/TWwFMpD8ojI/AAAAAAAABA0/TVS22BBas3s/s144/cookie%20cutter%20with%20filaments%20correct%20formula.jpg) (https://lh3.googleusercontent.com/_gbC_B2NkUEo/TWwFMpD8ojI/AAAAAAAABA0/TVS22BBas3s/cookie%20cutter%20with%20filaments%20correct%20formula.jpg) (https://lh3.googleusercontent.com/_gbC_B2NkUEo/TWygzfaVdPI/AAAAAAAABBQ/lVHQQjRi9oI/s144/loopy%20area%20done%20more.jpg) (https://lh3.googleusercontent.com/_gbC_B2NkUEo/TWygzfaVdPI/AAAAAAAABBQ/lVHQQjRi9oI/loopy%20area%20done%20more.jpg)

(https://lh5.googleusercontent.com/_gbC_B2NkUEo/TWyg2ORYOVI/AAAAAAAABBY/YA5ZaQBkRng/s144/oops%20my%20fractal%20got%20twisted.jpg) (https://lh5.googleusercontent.com/_gbC_B2NkUEo/TWyg2ORYOVI/AAAAAAAABBY/YA5ZaQBkRng/oops%20my%20fractal%20got%20twisted.jpg) (https://lh6.googleusercontent.com/_gbC_B2NkUEo/TWyg2GQx4fI/AAAAAAAABBc/ffhqCshaBS8/s144/stringy%20near%20loop%20area.jpg) (https://lh6.googleusercontent.com/_gbC_B2NkUEo/TWyg2GQx4fI/AAAAAAAABBc/ffhqCshaBS8/stringy%20near%20loop%20area.jpg) (https://lh3.googleusercontent.com/_gbC_B2NkUEo/TW8EK-B4RfI/AAAAAAAABCI/NZ1REOW6ppc/s144/the%20tendrilized%20outcropping.jpg) (https://lh3.googleusercontent.com/_gbC_B2NkUEo/TW8EK-B4RfI/AAAAAAAABCI/NZ1REOW6ppc/the%20tendrilized%20outcropping.jpg) (https://lh4.googleusercontent.com/_gbC_B2NkUEo/TW8EIJok0BI/AAAAAAAABB8/wvYOYSXFS2w/s144/stalky.jpg) (https://lh4.googleusercontent.com/_gbC_B2NkUEo/TW8EIJok0BI/AAAAAAAABB8/wvYOYSXFS2w/stalky.jpg)

Phew :)
We don't need conformal transforms, but pretty much every fractal you can think of uses conformal or anti-conformal transforms, e.g. http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension (http://en.wikipedia.org/wiki/List_of_fractals_by_Hausdorff_dimension) or indra's pearls or gaskets or any well known IFS. Depends on what you mean by fractal, but if it has any stretch in one iteration then in infinite iterations some areas will have infinite stretch, which probably prevents that direction from having a fractal dimension (since it is smooth). Does it matter when we are making interesting shapes anyway? Probably not.

"is the Mandelbox transform reducible to a single Mobius transformation, or is it an apparent violation of Liouville's theorem?"
No, it is several Mobius transforms connected by surfaces. Liouville's theorem applies to the volumes (where the transform is smooth) and doesn't apply to the border surfaces (where the transform has no derivative).

@David- well I don't know quite how you classify such an infinite transform, what does the mandelbrot transform 0.1+0.2i to? Generally Liouville's theorem covers all smooth transforms, whether they require infinite recursions or not. If a transform has rough areas then the theorem still applies to the smooth areas.

  lol.   :dink:

You can perfectly draw fractals with non-conformal transformations, there are tons of them in the Ultra Fractal formula database. But the point is that non-conformality is directly responsible for the "whipped cream" effect that was the primary subject of this post. Somehow this effect is unaesthetic, because the fractal look of the pattern is lost in one or more directions. If we want to avoid this we need conformal transformations, or some very mild breaking of the conformal symmetry.

The "Kaleidoscopic IFS" in 3d and Duck-like fractals in 2d are examples of mild breaking of the conformal symmetry. These formulas use "foldings", ie mirror transformations replacing one half-space by the mirror of the other. These are 2 to 1 transformations that are conformal (or anti-conformal) everywhere, except on the plane (in 3d) or on the line (in 2d) separating the two half-spaces. You can read more about the 2d case here:
http://www.algorithmic-worlds.net/blog/blog.php?Post=20110227

Tglad: Thanks. It's another case of don't shoot the messenger. Liouville's theorem has got be the most depressing theorem in all of mathematics. Who'd thought theorems could have emotional value :). I'm sure there's still some fun with fractals of the form you've developed.

I think projection hurts things, but this thread convinced me that non-conformality is also at fault. The only hope is to cancel these non-conformalities or preventing them from building up.