Why there isn't 3D 1:2 conformal transformation?

[Alef]

Is there any mathematical reason why there are no 1:2 conformal transformation in 3D?
I don't believe if there is no proof of that;)
All 2 dimension graphs can be mesured, 2 alsou are even number. 3D structures are much complex in this respect and 3 is odd number. Even 4 dimensions are much more algebraicly easy, jet there are no such 4D space and 4D creations aren't visualy appealing exept rotating tesseract.
Probably numbers eventualy control everything, say even numbered atomic numbers are more stable than odd numbered ones and there alsou are magical number (2, 8, 20, 28, 50, 82, 126) atoms representing exceptional stability. (Scientists will kill me for this explanation.)

It could be just nonsence, but is this becouse 1/3 = 0.333333 is irrational number, thus grailish 1:2 3D transformation should be irrational and 1: 2/3 in 2D mapping??? And in 2D cutout [strike]Holly[/strike] Holy Grail should be something like 2/3 mandelbrot probably with herman rings as infinetely small stalks would have infinetely small volume and thus non-visible but 2D rings could represent some larger 3D structure.
In z=0 cutout this could only corespond to 2 variable equations or powers larger or equal than 3 divided by something of z.
p.s.
I can imagine that by mathematical standarts what I said is pure trash.

[Roquen]

I find this notion counter-intuitive as well.  As an example it seems like the quaternion equation:

f(p) = (pr^*)² r + c

with 'r' a unit bivector should be a 2:1 mapping everywhere in the domain except the line through the origin in direction 'r'.

[Tglad]

Is there any mathematical reason why there are no 1:2 conformal transformation in 3D?

Yes, in 3D (or higher) the only conformal transformations are mobius transformations (http://en.wikipedia.org/wiki/Liouville%27s_theorem_(conformal_mappings)). Note that this is proven.

That means you can only make transformations out of combinations of translation, rotation, scale and inversions... and any combination of these (e.g. 100 of each all performed in some random order) is still just composed of at most a single translation, rotation, scale and inversion.

So... since all of these four transforms have single cover, that is your mathematical reason.

You don't necessarily need double cover for fractals, Kleinian limit sets in 3D are conformal because they simply branch each iteration and use a conformal transform (as above) on each branch... or randomly pick a branch (which gives the same result at its limit).
If you allow anti-conformal transformations (reflections) then you can make 'nearly conformal' 1:2 transformations by folding 3D space, so it is conformal/anti-conformal everywhere apart from the infinitely thin fold surface. This is what KIFS do.

Roquen, quaternion equations such as q^2 are not conformal.

[Roquen]

Let me babble through my train of thought and perhaps you can point out where I'm going astray.  Starting from Liouville's and only including (uniform) scaling, (proper) rotations and translations as the set of transforms under consideration, then indeed any arbitrary application of these three can be composed into a single set and are indeed 1:1.  This this is also true in 2 spatial dimensions.  All of the following is making the implied assumption that the transforms are being applied globally.

Consider the m-set: f(p) = p²+c.  Which can be validly described locally as a uniform scaling (magnitude square), proper rotation (torque minimal angle is doubled WRT the x-axis) and a translation.

So if Liouville's is the proof, then there must be some corollary about composition of local transforms when n>2 of which I'm unaware.

WRT: q² vs. (pr^*)² r.  There are some difference.  The former is locally an improper rotation + scaling of R⁴ and the latter a proper rotation + scaling of R³.

[Roquen]

The former is locally an improper rotation + scaling of R⁴ ...

Actually I retract the "improper rotation" part...I didn't think that through enough.

[arki]

Is there any mathematical reason why there are no 1:2 conformal transformation in 3D?

I would like to understand your question. What do you mean by "1:2" in this context? Can you elaborate a little bit?

[Syntopia]

Consider the m-set: f(p) = p²+c.  Which can be validly described locally as a uniform scaling (magnitude square), proper rotation (torque minimal angle is doubled WRT the x-axis) and a translation.

Squaring the magnitude is not a uniform scaling, and doubling the angle is not a rotation (since the angle varies). A rotation is a rigid body movement, and as such is a 1-1 mapping.

I would like to understand your question. What do you mean by "1:2" in this context? Can you elaborate a little bit?

I think the reasoning is that complex squaring is a (conformal) 2:1 transformation (every non-zero complex number has two square roots), and since all conformal transformations in 3D are 1:1, no obvious generalization is possible.

[Roquen]

Let me try this another way.

For a point:  p = m(cos(t), sin(t))

f(p) = m²(cos(2t), sin(2t))

The point 'p' and the set of all point infinitesimally to close to 'p' are being uniformly scaled by 'm' and being rotated about the origin by 't'.

[Syntopia]

Let me try this another way.

For a point:  p = m(cos(t), sin(t))

f(p) = m²(cos(2t), sin(2t))

The point 'p' and the set of all point infinitesimally to close to 'p' are being uniformly scaled by 'm' and being rotated about the origin by 't'.

Every point must be transformed by the same rotation. You apply different scaling factors and rotation angles depending on the input point.

Rotations are isometries, they keep distances unchanged, and will preserve the global shape of the input (the way we intuitively expect rotations to behave). Doubling the angle, as in your example, will seriously distort the input, and change distances.

[Tglad]

For a point:  p = m(cos(t), sin(t))
f(p) = m^2(cos(2t), sin(2t))

Yes, this is equivalent to z^2 on complex numbers and it IS conformal... but it ISN'T 3D or higher. Liouville's theorem applies to 3D or higher.
In 2d there are all sorts of conformal transformations.... you can't do this in 3D or higher, that is the result of his theorem.

What do you mean by "1:2" in this context?

He means a double cover, where the transform covers the space twice, like z^2... if b = f(a) then every b comes from two separate a values.

Liouville's theorem seems to apply not just to flat euclidean space, but also to spherical and hyperbolic space.... however it is still possible to get 1:many transforms in some spaces, for example a toroidal space can quadrupal cover simply by scale by two. However, in this case I can't see how to make fractals out of it because there is no infinity point.

[Roquen]

I mental working with the following rough description of a conformal transform:

The map f(p) of a domain S in Rⁿ into Rⁿ is conformal at point p in S if (scaling consistent), (angle preservation) & (orientation preservation).  f is a conformal map of S if this holds for all p in S.

and I'm attempting to define a collection of infinitely small S's that fill Rⁿ and 'f' such that the conformal property holds when moving between connected subsets.

And what I'm hearing is that I'm misinterpretation Liouville's and that it states that no such 'f' is possible.  So my proprosed conformal 2:1 map above, if I bothered to check, would be either non-conformal everywhere or perhaps only in the region around the line of symmetry.

So a potentially interesting question would be can one form 'f' like I'm attempting to do above such that it's conformal everywhere except: along some line or probably more interesting everywhere except in the region of some fixed point(s).

[Alef]

About the mandelbulb pow2. There are some formula claiming that it is a perpendicular mandelbrot of pow 2 mandelbrot. And it just a mandelbrot with one mirror, so could be that there are one mirror in mandelbulb.
<a href="http://www.youtube.com/v/Aow0DoyFC3U&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/Aow0DoyFC3U&rel=1&fs=1&hd=1</a>

Here is slightly like 2D render of mandelbulb involving alsou escape time value fog. Its great, but aren't streched only in Z=0 and  its twice as big in y=0 plane 
http://krzysztofmarczak.deviantart.com/art/Real-shape-of-Mandelbulb-381908457


Actualy I were thinking that something like this could lead to something more grailish. In y=0 It streches but eventualy streches back, but then it have two variables.
http://www.fractalforums.com/new-theories-and-research/few-steps-behind-perfect-3d-mandelbrot/

[Tglad]

So a potentially interesting question would be can one form 'f' like I'm attempting to do above such that it's conformal everywhere except: along some line or probably more interesting everywhere except in the region of some fixed point(s).

If you allow anti-conformal transformations (reflections) then you can make 'nearly conformal' 1:2 transformations by folding 3D space, so it is conformal/anti-conformal everywhere apart from the infinitely thin fold surface. This is what KIFS do.
But you can't make something that is conformal everywhere (in 3d or higher) except just a point or except a line, because all you have is translation, rotation, scale and inverse for the volume that is conformal.

[Edit]- you can also get 'conformal everywhere except a surface' by just using a different mobius transform on each side of the surface... but this tends to give broken looking fractals because the transform is discontinuous at the surface... hence folding works better.

[Roquen]

I promise I'm not trying to be an obtuse nutcase.  And I'm not in any way disputing Liouville's.  By the nature of the legal transforms it's obvious (the most dangerous word in mathematics) that one cannot find any finite partitioning of space and apply different legal transforms in each and be conformal at the boundaries of the partitions.  Skimming a half a dozen proofs didn't help me either..they only deal with a single sub-set of space and I'm not clever enough to work it through myself.  An attempt at a more generalized search only lead me to papers that were so far above my head I couldn't even tell if they pertained to my question.

So what I believe I'm hearing is that my attempted generalized construction above cannot be conformal anywhere?

toroidal space can quadrupal cover simply by scale by two. However, in this case I can't see how to make fractals out of it because there is no infinity point.

Wild hair thought - what about via a clifford torus?

[Tglad]

I'm attempting to define a collection of infinitely small S's that fill Rn and 'f' such that the conformal property holds when moving between connected subsets

So what I believe I'm hearing is that my attempted generalized construction above cannot be conformal anywhere?

Your construction is valid in Rn if f() is a mobius transform, but these aren't 1:2. Any other attempt to bend f() in any direction with stop being conformal. An inverse (which scales v by 1/v^2, and then reflects it) is an example where the scale, translation and rotation change from place to place, and the whole thing remains conformal... this is the only one that 'changes', the other transforms that can compose a mobius transform are translation, rotation and scale.

Wild hair thought - what about via a clifford torus?

Had to look it up (nice shape!), it is a surface that happens to be in R4, so the space is 2D so you don't have Liouville's constraint, but never-the-less, I'm not sure you could do many 1:many conformals apart from scaling by 2 (or some integer).