

hobold
Fractal Bachius
Posts: 573


« Reply #2 on: May 01, 2012, 11:56:14 PM » 

Calling these things "Fractals" is a bit ... misleading, in my humble opinion. The dimension of this wrinkled torus surface is still two (an integral dimension rather than a fractional one), and its total surface area is still finite. It is not actually "selfsimilar" in the established meaning of the word.
It is a remarkable find nevertheless. A class of things that share some, but not all, qualities with fractals. That is in and of itself interesting, because it is an intermediate step between two classes of objects that were thought to be completely alien to each other.



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kram1032


« Reply #3 on: May 02, 2012, 12:10:19 AM » 

Well, the Mandelbrot set also has a fractal dimension of 2, so that's not really a criterion... Though in the limit, the normals wont any longer be defined properly. You can actually do the very same thing with the mandelbrot set, figuring out the gradient in every point. Any finite depth will give you proper normals but in the limit, only the inside of the set will have smooth transitions and the closer you get to the boundary, the messier it will be. Right at the boundary, the definition of normals fails and anyhting outside the set will have escaped to infinity. I see quite a bit of self similarity in there. Not quite of the toric kind and most certainly not 3D. but the wave"fronts" are selfsimilar to each resolution, only differing by rotation and scale. I wonder how well this thing would work as a gear... Probably (, in the limit), friction would be too high to make it useful but it certainly looks like it could work in a frictionless environment...


« Last Edit: May 02, 2012, 12:18:42 AM by kram1032 »

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Tglad
Fractal Molossus
Posts: 703


« Reply #4 on: May 02, 2012, 11:54:49 AM » 

I think hobold is right, yes it is everywhere C1 instead of everywhere C0 like a fractal curve or everywhere Cinfinity like a smooth curve. In fact I think you could say that it is the integral of a fractal surface, and hence its normals follow a fractal, so it has a well defined normal vector but not a well defined curvature at any point.
I find the final shape doesn't make sense to me... the firstlevel horizontal corrugations don't seem to help, since the inner ring of the torus is still much shorter than the outer ring.



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David Makin


« Reply #5 on: May 02, 2012, 11:58:07 PM » 

To be honest I think it is fractal  reread this bit: "The process continues indefinitely and, in the limit, builds an isometric embedding of the square flat torus. Of course, the program can only perform a finite number of tasks. We stop it at the fourth step. " The key phrases are "in the limit:" and "We stop at the 4th step"  if you stop at the 4th iteration of a Mandelbulb or even a Mandelbox then the resulting rendered surface is not strictly speaking "a fractal" This also reminds me to repeat something I've mentioned before  if you aren't *literally* rendering to the limit surface (which of course you never are unless you're doing something "real" and analog to plank length accuracy) then any surface you render to (whether based on DE value or iteration depth) is *finitely* defined and hence strictly not a fractal  in fact if rendering to a given iteration depth then at least for I(f(z)+c) where f(z)+c is a polynomial then In(f(z)+c) is itself also simply a polynomial though of course of degree p^n where p is the degree of f(z) and n is the number of iterations used.



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kram1032


« Reply #6 on: May 03, 2012, 09:27:25 AM » 

Tglad, well, the first step isn't yet a very good approximation. Also, it's quite difficult to see lengths of such wavy patterns. If you actually messure them, you'll probably find the approximation to be much better than you'd guess by eye.



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hobold
Fractal Bachius
Posts: 573


« Reply #7 on: May 03, 2012, 01:52:28 PM » 

The key phrases are "in the limit:" and "We stop at the 4th step"  if you stop at the 4th iteration of a Mandelbulb or even a Mandelbox then the resulting rendered surface is not strictly speaking "a fractal" It is a bit more complicated than that. The corrugations, that they apply iteratively, have an influence that diminishes with each iteration. Unlike, say, a Koch snowflake, where each iteration extends the length of the border by 4/3, the corrugations extend the lengths by a smaller and smaller factor each iteration. This is why the new "AlmostFractal" approaches finite length/surface area in the limit, while true fractals approach infinity. This is also the reason why four iterations are enough for this wrinkled torus, but not nearly enough for most fractals.



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kram1032


« Reply #8 on: May 03, 2012, 02:13:30 PM » 

That's also the reason why they call it a C^{1} fractal. I wonder what actually happens if you look at the first derivative of the surface, e.g. at the normals. In the limit, they should jump just as much as typical fractals would, right?
Furthermore, I wonder if we could come up with similar C^{1} fractals... It should be not too hard to come up with some iteration sets where the surface doesn't grow indefinitely.
If I got it right, they basically used some kinds of harmonics to do the job. E.g. they added sinewaves in the normal direction, updating that direction per iteration step. I wonder how in particular they chose their harmonics. Since every single step increases the surface, the series should assymptotically approximate the correct linelengths from below, rather than above. To come up with that (or with a way to define it so a computer can do it for you), is probably the hardest part about that.
But if you don't pose an actual geometric problem to be optimized, just basically doodling with this kind of process (as we did for various fractal formulas), it should be relatively easy to do?



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jehovajah


« Reply #9 on: May 13, 2012, 10:02:04 AM » 

Our program generates pictures of the isometric embedding of the square flat torus that reveal some kind of selfsimilarity in the infinite succession of corrugations. This strongly suggests a fractal structure. This seems even more surprising because the fractal nature is incompatible with the presence of tangent planes. This seeming paradox is resolved when we look more specifically at the behavior of the corrugations at different scales. At each stage, the amplitude of oscillations decreases too quickly to ensure a perfect selfsimilarity. As a consequence, the limit surface is not as rough as a fractal. Since the limit surface is C¹ regular, we call it a C¹ fractal . The pictures below show the differences of roughness between a fractal curve and a C¹ fractal. While it is useful to make distinctions, i do not think arguing over the term Fractal is of much use nowadays. The term has grown beyond its strict mathematical sense, and now is a cultural paradigm, or near to become one for most people that encounter it. Thus it is useful to distinguish that the term covers various different types of behaviour which result from iteration or recursion, and then go on to specify which behaviour is relevant, but it is not useful to attempt to establish a purist meaning to the term. Fractal iitself has become so meaningful it deserves to stand as such as a monument to Benoit, who acknowledged that it had entered into the popular arena in a way he never imagined it would. We would of course have to redact Benoit's original work to keep pace with this development, but this would accord him a status i think he well deserves.


« Last Edit: May 13, 2012, 10:44:02 AM by jehovajah, Reason: quote added »

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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!



jehovajah


« Reply #10 on: May 13, 2012, 10:51:14 AM » 

That's also the reason why they call it a C^{1} fractal. I wonder what actually happens if you look at the first derivative of the surface, e.g. at the normals. In the limit, they should jump just as much as typical fractals would, right?
Furthermore, I wonder if we could come up with similar C^{1} fractals... It should be not too hard to come up with some iteration sets where the surface doesn't grow indefinitely.
If I got it right, they basically used some kinds of harmonics to do the job. E.g. they added sinewaves in the normal direction, updating that direction per iteration step. I wonder how in particular they chose their harmonics. Since every single step increases the surface, the series should assymptotically approximate the correct linelengths from below, rather than above. To come up with that (or with a way to define it so a computer can do it for you), is probably the hardest part about that.
But if you don't pose an actual geometric problem to be optimized, just basically doodling with this kind of process (as we did for various fractal formulas), it should be relatively easy to do?
This makes my point. We re inspired by the term "fractal" to explore new ways of creating art and solutions. To solve this issue would mean that at least for artists, they could render good approximations more quickly?



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May a trochoid of ¥h¶h iteratively entrain your Logos Response transforming into iridescent fractals of orgasmic delight and joy, with kindness, peace and gratitude at all scales within your experience. I beg of you to enrich others as you have been enriched, in vorticose pulsations of extravagance!



