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hi all,

you know what ? i want to derive an algebra with 3 components :D

ok,  nothing new so far

but, i have the following line of thought:

- remember the derivation of the complex numbers out of the problem occured when square rooting -1 ? i am still thinking
about it, and think the line of thought could be defined as:
 - multiplication with a negative real number affects both axises, interpreting it as a rotation by 180 degree around the z axis,
   ending up with the polar multiplication expression, where 1i basically is 1/2 ( one halve ) of the 180 degrees defined by the negative
   scalar real multiplication

so, so far so good, and then i think of scalar 3d vector multiplication, multiplication of a negative scalar with a vector leads to a mirrored
result, e.g. -1*(1,1,1) = (-1,-1,-1)

so, BUT: the triplex math definition so far using the 2 angles to rotate around do not fullfil that, multiplication of the scalar
-1 using triplex math brings us to: -1 *(1,1,1) = (-1,-1,1)  <- and thus can in my eyes not be the wanted result, in my
opinion a 3component vector multiplication has to fulfill: -1*(1,1,1) = (-1,-1,-1) just as a scalar ... , or as triplex (-1,0,0)*(1,1,1)=(-1,-1,-1)


so, by using this as base, the derivation of (0,1,0) is just as simple as the complex multiplication, rotation by 90 degres (1 halve of the 180 degrees) around
the center, which leaves open the question how to derive the (0,0,1) multiplication ? should it be another halving, like 45 degrees around the center?

what do you think ?

Forgive me I'm both new here and fractal ignorant.  First thought is: hairy ball theorem.  Quaternions are pretty much as simple as 3D transforms get.  Unless you're thinking of some other domain space...like a torus.  Why 3 components?

To produce a 3D mandelbrot-analogue of course. :)

My own thought is that we want some mapping between 3D vectors and conformal, orientation-preserving linear maps of 3D space to define "multiplication". An obvious approach is to decompose vectors r and s into |r|r' and |s|s' where the primed versions are unit length and oriented the same, then define rs = |r||s|(r' x s') for some operation x defined on the 2-sphere. Options for x that would be associative and commutative that occur to me include:

• Interpret the 2-sphere as the Riemann sphere, with the sphere intersection with the positive X-axis being 0, the intersection with the negative X-axis being infinity, and the four other axis intercepts being 1, i, -1, and -i. Then use complex addition or complex multiplication on the Riemann sphere as x.
• Normal complex multiplication can be defined analogously in 2D, with operation x being an addition modulo 2pi on the unit circle. The most obvious analogy on the 2-sphere would distinguish a point as the zero on the sphere (the sphere intersection with the positive X-axis being a likely candidate, again) and parallel-transport the vector 0s' on this manifold along the arc to r'. Not commutative, and when r', 0, the real zero, and s' are coplanar, amounts to addition modulo 2pi on the circle that is that plane's intersection with the sphere. Upside: should reproduce the 2d Mandelbrot set when sliced on the xy plane, etc. Downside: will probably just be a surface of revolution like the quaternions produce. So we'd probably want to define "addition" differently.

Of the above, Riemann sphere complex multiplication might be the most promising, as even when 0, 0, r', and s' are coplanar it won't necessarily keep the point in that plane -- if not in the xy plane, points will get rotated about the x-axis when squared, for instance, as well as moving towards 1 or -1 if not in the yz plane.

Another notion just takes the conformal maps of R³ and directly tries to map them to vectors. The orientation-preserving conformal linear maps combine a scaling with a rotation, and have four degrees of freedom: magnitude of scaling, amount of rotation, and orientation of axis of rotation. The vectors have only three degrees of freedom. One simple map that occurred to me would be to define rs to scale r by |s| and rotate it about the axis parallel to s by an amount that also scaled with |s|, which I call "corkscrew multiplication" as the transformed r will revolve around s as well as lengthen as s lengthens.

Corkscrew multiplication is non-associative, though, and squaring only squares length. The latter leads to the notion that the axis needs to not simply be s. Making it be halfway between a fixed direction (say, the positive x-axis) and s is a possibility, or making it be the direction of the vector one gets by changing s's x-coordinate to 0 or something. The cross-product of s and a fixed vector (e.g. (1, 0, 0)) is another option. These also suggest other options for the rotation's magnitude, including s's dot-product with the same or another fixed vector or with r.

hmm, you seem to be far further than me :) lol, i wonder if it would be possible to define it via reductionism, or circling in the real thing,
what do we know, or demand from such an algebra?

- true extension of complex numbers and real numbers ( they shall be contained )
- multiplication of (-1,0,0) or negative one real has to switch signs of all components

so, what else does it have to fulfill ? :(

so, i have had some thoughts about it, keeping complex multiplication alive,

what do you think about formulating:

(1,0,0) = rotation by 0° around Z AND Y PLANE
(-1,0,0) = here it becomes tricky for the first time, to incorporate the negation of the third component as well ( when multipliying with 1,1,1)
we do the following: rotate by 180° around z ( just as usual ) AND rotate around X by 90°

next is the second component unit vector, or the imaginary part, leaving complex multiplication intact ? (not sure about it, but at least for 1i it works ;) )

(0,1,0) = rotate by 90° on 2 axis, but in the order: X and then Y ... which brings us to -1 as well, but clearly it is affecting both axises

(0,-1,0) = so, what to do with this friend, we know from the complex multiplication that we have to land on -1 as well, so, formulating it then with 2 axis rotation i would come up with rotate by 270° around X and then ...   rotate by  270  around the Y axis

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little break here

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so, i hope everybody got my point, i want that each unit vector takes influence on all 3 components of the vector, thus i defined all operations with 2 axisses, although we where lying on the plane, the examples above demonstrated for the unit multiplication how this could be achieved,
the inbetween angles could be created by just interpolating ( similar to what complex multiplication in polar form is doing )

or, even better, just the same way as it is done for polar representation

---

so, which brings us to the (0,0,1) unit vector ....

using the above line of thought i have 2 ideas:

- the first would be to implement it in the same way as the imaginary multiplication described above, but with just switched axis orders, or generally changed axises ...

- the second thought would be to make use of the sequence 180->90->45 degrees, so, generally speaking behave the same like the imaginary
component but with even further reduced angle ...

what do you think about it ?

argh, where is the next brute force raymarcher ???

argh, oje, so, this just seems to be not it, but perhaps someone can help me out with something....

... reflections can be defined through rotations ? or not ? how to express a point mirroring through the center,
the above described method of 180° z and then 90° x is just working for halve of the points :(

i am fiddling around with the right ordering, and amounts, i think 2 axis should be enough, but it might
be possible to achieve a 3 dimensional through 3 axises, but somehow i can not find an example for it :(


so, or isnt that possible :( ???

I don't see how these ideas could evade the fact that there are no many to one conformal map in dimension 3... The idea in the post above the one quoted here will produce a non-conformal transformation (it will be conformal in each plane, but not in 3d).

On the other hand, if you're only using the conformal maps of R^3, you have 1 to 1 maps, and you won't get a fractal by iterating them.

Sam

The short, inexact and anal answer is: no you cannot form reflection out of rotations, but you can formulate rotation from reflections.  But that doesn't matter.  If you can represent translation and can represent and compose scaled rotations, then you can simply define exactly one reflection with a plane through the origin and use translation and rotation to generalize reflections.  If by "mirrored through the center" you mean point inversion, then negation.

I was actually clever enough to deduce that (go me)...I'll start a new thread at some point to ask "why".

so, how is point inversion just using rotation achieved in 3d ? ??? ??? ??? ???

That was an example of me getting distracted.  What I intended was: do you mean point inversion?  So if you have directly represented point P then you want -P?  You could formulate point inversion as a rotation, but it seems unlikely that you'd want to.

WRT: conformal maps I have a question.  Does that include 3 dimensions embedded in a higher dimension space?  And is even conformal a requirement?

as far as i known so far 3 dimensional point inversion can not be expressed through just rotations :(
at one hand i want this to use for visualising the 3dimensional mandelbox rendering, and on the
other hand i would like to formulate a 3component algebra from it ;) so, right now my visualisation
of a multiplication with a negative real number comes down to rotate by 180 degree around z
and then plane mirror it on X/Y plane ... :( has anyone a solution for expressing it just with rotations ?
because this would make the visualisation far more attractive!!!

I don't grok the system being described.  The only reason to do point inversion via rotation would be if it were easier to perform the equivalent of finding an axis orthogonal to the vector to the point and rotating about it by pi.

Cite? Well, technically there isn't in dimension 2 either, for instance complex squaring isn't conformal at the origin, though it is everywhere else. We'd be wanting a map that was conformal everywhere but some set of measure zero, and was many to one.

We already have non-orientation-preserving ones, for that matter -- the folds used in the Mandelbox calculations are two-to-one and conformal away from the folding plane, but non-orientation-preserving. Is there a proof that there is no orientation-preserving map on R³ that is conformal except on some set of zero measure?

And to answer my own question, what about this?

First, consider the rotations of the 3-sphere (adjoin a "point at infinity" to R³, or consider the unit vectors in R⁴). It's possible to rotate it without any fixed points analogous to a "north pole" or "south pole" -- pick a plane through the origin in 4-space and rotate in that plane, and pick a second plane in 4-space that intersects the first in only the origin and rotate in that plane, and compose these two rotations -- only the 4-space origin is fixed, and it's not part of the 3-sphere.

Rigid rotations of the 3-sphere should be conformal.

Now, consider not a simple rotation but doubling the angle from some fixed reference direction, in one or both planes. This should produce a 2-to-1 (or even 4-to-1) map of the 3-sphere onto itself, and since angle doubling in the complex plane is conformal everywhere but the origin, it seems this should be conformal in 4-space everywhere but the origin -- and everywhere on the 3-sphere.

That gives us what seems to be a many-to-one family of conformal maps of a 3-dimensional manifold that is conformal and orientation-preserving everywhere. Regarding the 3-sphere as R³ with a point at infinity adjoined, the maps' restrictions to R³ should have the same properties everywhere but at the preimages of infinity, which are "poles" of the map analogous to zeros of denominators of meromorphic functions on C.

Those maps, in turn, should give some sort of potentially interesting Julia sets, particularly when combined with one-to-one conformal maps such as translations (analogous to the "+ c" part of the regular Mandelbrot formula). The four-to-one map from doubling angles in two linearly-independent planes of R⁴, in particular, seems possibly promising as an analogue of "squaring". (And, of course, it could also be used to generate 4D fractals for slicing or projecting.)

That's actually an interesting point. You are correct that squaring is not conformal either everywhere in 2 dimensions.

However, I think the reason why there is no analogue of squaring in higher is the following. By studying locally what it means for a map to be conformal, you can figure out what kind of freedom you have in deforming the map (locally), while keeping it conformal. In dimension 2, you have an infinite number of deformations, corresponding to the fact that the space of local conformal transformations (holomorphic functions) is infinite dimensional. In dimension 3 or higher, there are new constraints, which were trivially satisfied in 2 dimension, and which imply that there is only a finite number of local deformations, corresponding to the global conformal transformations. Now if there were analogues of squaring in higher dimension, they should be seen as possible local deformations of a conformal transformation. I feel this kills the idea that there are partial conformal transformation beyond the global ones and mirrors, but I agree it's not a full proof. At least it shows that such hypothetic transformations are different from the squaring in two dimension in the sense that they necessarily look like global transformations locally.

The place where I've seen the computations mentioned above is in the book on conformal field theory by Di Francesco, Mathieu and Senechal, chapter 4:
http://books.google.com/books/about/Conformal_Field_Theory.html?id=keUrdME5rhIC
There should be better ressources however.

I'm pretty sure this map is not conformal. It is clear that you are stretching in which ever tangent direction you're doubling the angle. In 2d, angle doubling is obviously not conformal. Only when you combine it with a rescaling of the radius to obtain the squaring operation does it become conformal. It is not possible to compensate in the same way in higher dimension...

Seems it should work in all even-dimensional spaces, at minimum. Double angles in mutually-wholly-perpendicular planes, plus square radius.

For that matter, in three dimensions doubling angle in the xy plane, squaring radius in the xy plane, and doubling z should look locally like a uniform scale by two and rotation -- this is just the complex-squaring map "extruded" into 3 dimensions and forced to stay conformal. I doubt that it will give a very interesting fractal, since the treatment of z is somewhat trivial, but it's there, and it's a two-to-one map that looks to be conformal everywhere but the z-axis.

This obviously generalizes: partition an orthogonal basis for Rⁿ into sets of sizes 1 and 2, with each size-2 set choose an angle doubling and radius squaring of the corresponding coordinates, and with each size-1 set choose a scaling of that coordinate by 2, and compose all these transformations, and the result ought to be conformal.

Back to the 3-sphere, the 4D map with partition into two sets of 2 should fix the embedded 3-sphere of unit radius, and deleting a point can't magically make it non-conformal, so it should give a 4-to-1 conformal map of R³...unless someone can show good cause why it somehow doesn't.

I don't believe the maps you describe are conformal. There are at least a few flaws I can see in your reasoning. For instance, it is not true that if you have a conformal map on X and a conformal map on Y, you get a conformal map on X x Y (also locally). The scale factor of the squaring map in 2 dimensions is not uniformly 2, it is |z|. The only map with a scale factor uniformly equal to 2 is the dilatation by 2. You seem to be mistaking the multiplication by 2 of the angle with the conformal factor. The conformal factor can be detected only by analysing how to nearby points are mapped. Moreover, unless I misunderstood the argument, the "angles" you're speaking of are really 2d angles. It is not clear to me how you can double them globally.

But I'll be happy if you prove me wrong: if it is really conformal 2 to 1 everywhere but on an axis, it is bound to yield very interesting fractal patterns. You should try to implement it. And in case it's not conformal you'll immediately notice it by getting "whipped cream" patterns.

Sam

I'm fairly sure you're wrong. Consider C² with the map that pointwise complex-squares the two components of a vector. Regard it now as R⁴ with the usual inner product, with the two copies of C being the xy and zw planes.

Away from these two planes, point neighborhoods experience uniform dilation if their xy projection and their zw projection have the same distance from the origin. So restrict to the 3-manifold of all points satisfying this property. The map fixes this manifold, since if |xy| = |zw| then |xy|^2 = |zw|^2.

No nonuniform dilation = no whipped cream.

Conformally mapping from this 3-manifold into R³ (except for a set of zero measure) or else implementing a raytracer that works in that manifold instead is left as an exercise for the reader. :)

To be more precise, we're talking about the map:

• x -> x² - y²
• y -> 2xy
• z -> z² - w²
• w -> 2zw

on the 3-manifold defined by the single constraint x² + y² = z² + w². (It has a cusp of sorts at the origin and its intersections with axially-aligned 3-subspaces are 45-degree double-cones.)

Well I take that as a sign that you don't have much more confidence in your constructions than I do. :P

But if you are confident, I don't think you want to leave the glory of having found a 3d M-set to one of your readers.

I have plenty of confidence, but lack the specific knowledge needed to implement the raytracer. OTOH, it seems to me that simply projecting into the xyz cube should almost suffice. That map is two-to-one, so one would render either the positive-w or the negative-w half with this naive approach (and they're likely to be symmetrical in many cases anyway, like the top and bottom halves of a normal Julia set in C are 180-degree rotations of one another).

To add to the above: the angle between the surface and the w-axis would need to be considered too, and a stretching in some dimensions performed on the projection to make the map conformal. The surface normal at [x y z w] is given by the partial derivatives of x² + y² - z² - w², so is [2x 2y -2z -2w] and is parallel to [x y z w]. The plane containing the surface normal and the line segment from [x y z 0] to [x y z w] will intersect the 3-cube in a line (as a basis for the plane is the normal and [0 0 0 1], and the latter projects to 0) and that line is the projection of the normal, so is the line parallel to [x y -z] through [x y z]. A local scaling along this line by the w-coordinate of the unit-length vector parallel to the normal ought to modify the projection to make it conformal. Multiplying x, y, z, and w by a constant doesn't move us off the manifold, so we can restrict our attention to the intersection with the unit 3-sphere and then generalize later. The vector [x y -z -w] parallel to the normal is then unit, and we can simply use |w| as the scaling factor, so the scaling factor is in general |w|/|[x y z w]|. This gives us differential equations to solve: d(xform)/ds, where s is a coordinate parallel to the vector [x y -z], is equal to [x y z]*|w|/|[x y z w]|, and given the rest t, u of an orthogonal basis s, t, u, d(xform)/dt and d(xform)/du are both equal to 1. Composing the projection onto the [x y z] subspace with a solution to these differential equations should yield a map from one branch of the multicone manifold to R³ that is conformal.

Another way to do it: convert [x y] to polar coordinates [r t] and convert [z w] to more polar coordinates [s u]. The manifold then is defined very simply by r = s, so the coordinates [r t u] suffice to address the manifold (and in a one-to-one way this time). The map into [r s t u] is not conformal, but by analogy with the complex logarithm, the map into [(ln r) (ln s) t u] should be, given the same constraint r = s we already have. Then, since the surface is parallel to the [t u] plane everywhere and is at 45 degrees to the (ln r) axis in the [(ln r) (ln s)] plane, we need only project to [√2(ln r) t u] for a conformal map into a 3-manifold bounded by -pi..pi in two axes and 0...infinity along the third, so shaped like an "infinite brick" of sorts. This projection is simple (no differential equations to solve!) but has the downside of branch cuts -- the fractal will continue from the left side of the "brick" to the right, in wraparound fashion. One pair of branch cuts can be eliminated though, conformally, by sending [√2(ln r) t] to a vector of length re^√2 whose orientation is determined by the angle t, regardless of u, and scaling the u coordinate by √2re^√2/(2 ln r). Or the renderer could repeat the "brick" infinitely along the t and u axes.

It occurs to me, meanwhile, that the four-to-one squaring-of-4-space map used to implement v -> v² + c would probably just give us some variation on the theme of a Cartesian product of standard 2D Julia sets, as the xy and the zw coordinate pairs evolve independently. That's easily fixed, however, by replacing "+ c" with a 4-space rotation (six degrees of freedom) that doesn't in general fix those two planes, or any other family of one-to-one conformal maps that don't generally fix those planes. Or, a single such map can be used along with a + c. (Note that any + c can't quite be a straightforward translation in 4-space, as those don't fix the r = s manifold. A translation in the 3-space we conformally map that manifold into will work fine, however, to induce a conformal translation-like map on the manifold. A translation in the [√2(ln r) t u] space induces a 4-space rotation that preserves the xy and zw planes and a scaling, sort of like a complex multiplication by a constant.)

Since I'm fractal clueless I just spent about an hour peeking at various 3D fractals and one thing that struck me pretty fast is that all the quaternion based fractals looked pretty much like surfaces of revolution or twisted versions thereof.  Am I missing some?

this is exactly why we search for a 3component algebra :D, the closest so far you can find when googling for "mandelbulb" and another nice 3dfractal is google for "mandelbox"

I'm rather stunned that there aren't any interesting quaternion based fractals.  I find this excessively counter intuitive.  They have all the tools one needs to create something at least reasonable.  I was rather expecting that mandelbulb was defined as a quaternion function.  I may know nothing about fractal, but I can say with a reasonable certainty that all of quaternion based fractals I've seen are based on analytic functions over quaternions.  Regardless the complexity of a given analytic function, the result will be Complex (as in literally) and therefore bound to a plane (and that's a good thing).

I'll create a new thread.