Whipped cream look of quaternians an artifact of projection?

[M Benesi]

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

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

[David Makin]

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.

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 ?

[msltoe]

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.

[Tglad]

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 "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").

[David Makin]

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).

[Tglad]

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.

[David Makin]

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 ?

[msltoe]

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).

[M Benesi]

  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...). 

[Tglad]

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 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.

[M Benesi]

  lol.   

[s31415]

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

[msltoe]

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.

[kjknohw]

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.