cKleinhuis
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« on: June 12, 2013, 02:55:06 PM » |
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hi all, you know what ? i want to derive an algebra with 3 components 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 ?
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divide and conquer - iterate and rule - chaos is No random!
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Roquen
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« Reply #1 on: June 12, 2013, 03:21:33 PM » |
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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?
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Pauldelbrot
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« Reply #2 on: June 12, 2013, 10:43:24 PM » |
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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.
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Pauldelbrot
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« Reply #3 on: June 12, 2013, 10:53:50 PM » |
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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.
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cKleinhuis
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« Reply #4 on: June 12, 2013, 11:25:40 PM » |
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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 ?
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divide and conquer - iterate and rule - chaos is No random!
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cKleinhuis
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« Reply #5 on: June 13, 2013, 12:44:59 AM » |
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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 ---- little break here ----- 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 ?
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« Last Edit: June 13, 2013, 12:50:15 AM by cKleinhuis »
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divide and conquer - iterate and rule - chaos is No random!
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cKleinhuis
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« Reply #6 on: June 13, 2013, 12:49:13 AM » |
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argh, where is the next brute force raymarcher
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divide and conquer - iterate and rule - chaos is No random!
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cKleinhuis
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« Reply #7 on: June 13, 2013, 04:52:06 AM » |
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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
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divide and conquer - iterate and rule - chaos is No random!
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s31415
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« Reply #8 on: June 13, 2013, 09:21:11 AM » |
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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.
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
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Roquen
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« Reply #9 on: June 13, 2013, 10:01:53 AM » |
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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. To produce a 3D mandelbrot-analogue of course. smiley
I was actually clever enough to deduce that (go me)...I'll start a new thread at some point to ask "why".
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cKleinhuis
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« Reply #10 on: June 13, 2013, 01:25:16 PM » |
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divide and conquer - iterate and rule - chaos is No random!
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Roquen
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« Reply #11 on: June 13, 2013, 03:25:01 PM » |
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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?
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cKleinhuis
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« Reply #12 on: June 13, 2013, 03:36:23 PM » |
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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!!!
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divide and conquer - iterate and rule - chaos is No random!
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Roquen
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Posts: 180
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« Reply #13 on: June 13, 2013, 04:49:21 PM » |
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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.
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Pauldelbrot
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« Reply #14 on: June 14, 2013, 10:55:25 AM » |
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I don't see how these ideas could evade the fact that there are no many to one conformal map in dimension 3... 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?
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