Fractals And Chaotic Mixing

[Da Blob]

In chaos theory and fluid dynamics, chaotic mixing is a process by which flow tracers develop into complex fractals under the action of a fluid flow. The flow is characterized by an exponential growth of fluid filaments.[1] Even very simple flows, such as the blinking vortex, or finitely resolved wind fields can generate exceptionally complex patterns from initially simple tracer fields.[2]
The phenomenon is still not well understood and is the subject of much current research.
http://en.wikipedia.org/wiki/Chaotic_mixing

http://www.lactamme.polytechnique.fr/descripteurs/Galerie_GeneralitiesVisualization.FV.html

IMO, when this is combined with State-Of-The-Art Applied Signal Theory (wavelet transforms), a more realistic view of the universe will result... The phrase 'dynamical dimension' may even catch on...

Dynamical characterization of mixed fractal structures
Luiz Bevilacqua and Marcelo M. Barros
Vol. 6 (2011), No. 1-4, 51–69
DOI: 10.2140/jomms.2011.6.51
Abstract
We present a new technique to determine the fractal or self-similarity dimension of a sequence of curves. The geometric characterization of the sequence is obtained from the mechanical properties of harmonic oscillators with the same shape of the terms composing the given sequence of curves. The definition of “dynamical dimension” is briefly introduced with the help of simple examples. The theory is proved to be valid for a particular type of curves as those of the Koch family. The method is applied to more complex plane curves obtained by superposing two generators of the Koch family with different fractal dimensions. It is shown that this structure is composed by two series of objects one of which is fractal and the other which is not rigorously a fractal sequence but approaches asymptotically a fractal object. The notion of quasifractal structures is introduced. The results are shown to provide good information about the structure formation. It is shown that the dynamical dimension can identify randomness for certain fractal curves.

 

[PhoenixRising]

wow, some very fascinating concepts here. I'm actually a chaos theory enthusiast, so I've done some studying of fractals, fluid dynamics, complex systems, and the fractal formation of things in nature, including thoughts. One of the most interesting concepts I've come across so far is that fractals are shapes with infinite lines. If you've ever played with a fractal generator, you'll notice that with the evolution of the system the fractal continuously expands and grows. The notion that the universe is infinite does fit nicely with the mathematical realities of fractals. I have long thought that the overall structure of the universe is probably fractal, especially since the forms we observe are all fractal in nature. The principle of self similarity would point to the universe having a fractal form as well.

The gallery reminds me a lot of the strange attractors I studied in my research. Phase space is a fascinating concept, thinking in terms of 4+ dimensions really tends to stretch the mind. I would like to hear more about your theory on chaotic mixing and Applied Signal Theory. Quantum computing does seem to lend itself to chaos, being that our brains are basically quantum computers.

 

[Da Blob]

Amongst various metaphysical speculations is the idea that the universe itself is alive and humans themselves are just part of its nervous system. Following this hypothesis is rather interesting as it relates to signal theory/Fourier transforms.

In the human body, signals are constantly being sent and received and the result is usually a transformation of one kind of another.


Much has been made of the Butterfly Effect, but I wonder if anyone has considered the butterfly as a signal for the hurricane?

I wonder about the driving force behind transformation/emergence. Could the signal itself be a vehicle containing power, not just information?

Again, the multidimensional POV provided by Fourier transforms, may not just be access to information, but access to power as well...

 

[Velo]

wow, some very fascinating concepts here. I'm actually a chaos theory enthusiast, so I've done some studying of fractals, fluid dynamics, complex systems, and the fractal formation of things in nature, including thoughts.

The gallery reminds me a lot of the strange attractors I studied in my research. Phase space is a fascinating concept, thinking in terms of 4+ dimensions really tends to stretch the mind.

Ah, you're bringing back some memories from grad school 20+ years ago. Lots of time spent studying strange attractors in phase space, both computationally and experimentally. It wasn't the main part of my research, but it was definitely the most fun. Finding out that a fractal could have an infinite perimeter and a fractional dimension somewhere between 1 and 2 at the same time was mind-boggling to me.

Ever try to explain to someone that the length of a coastline can't be determined exactly because of its fractal qualities? The deer-in-the-headlights look is priceless.

 

[PhoenixRising]

Ah, you're bringing back some memories from grad school 20+ years ago. Lots of time spent studying strange attractors in phase space, both computationally and experimentally. It wasn't the main part of my research, but it was definitely the most fun. Finding out that a fractal could have an infinite perimeter and a fractional dimension somewhere between 1 and 2 at the same time was mind-boggling to me.

Ever try to explain to someone that the length of a coastline can't be determined exactly because of its fractal qualities? The deer-in-the-headlights look is priceless.

I agree, infinite perimeters are really a profound topic. That concept really ties into the infinite possibilities of form in nature, I think. A while ago I read a book that described how the uncertainty principle could cause "random" genetic mutations. This, in turn, would effect the system of the mutated creature to generate a new form. I have had the concept for a while that if a genome could be translated into a set of base equations, then we could simulate the development of the specific traits associated with that genome. Kind of a long shot, but perhaps if it were done with attention to detail, the process would work.

It is a shame that chaos theory isn't taught in schools. Or at least the concept of fractals in detail. We are taught to think in such limited terms, categorizations and expectations. The universe as it really is, is filled with complex mechanisms that can not be described by standard mathematics or linear reasoning as we know it.

 

[Velo]

It is a shame that chaos theory isn't taught in schools. Or at least the concept of fractals in detail. We are taught to think in such limited terms, categorizations and expectations. The universe as it really is, is filled with complex mechanisms that can not be described by standard mathematics or linear reasoning as we know it.

The dirty secret of undergraduate physics education is that it focuses on presenting problems that have tidy mathematical solutions. An apt analogy is that students are being led down a well-trodden narrow path and missing all of the interesting features hidden in the wilderness only 2 steps off of the trail.

I do disagree with your comment that some complex mechanisms cannot be described by standard mathematics. In many cases, the mathematics exist but are not presented to most students and therefore remain hidden. The interesting cases are often found in the last chapters of textbooks but are never covered for two reasons. First, too much time is spent on the historical topics to leave room for anything new. Second, some of these cases require simple computational exercises which aren't quite as easy to cover in the standard lecture format.

For anyone who wants a good introduction to chaos theory and its history, start with the book "Chaos" by James Gleick (1987). It does a fine job of showing how the field started to appear in the 1960's but remained outside of the mainstream until the 1980's.

 

[Da Blob]

Just a note, I remembered how utterly dismayed I was to discover that often even college-level classes were nothing but 'meaningless' history, without any reference at all to fascinating modern developments or ongoing research.

The problem is, IMO, that a historical perspective has become the status quo of academia and I truly wonder how effective a historical perspective is in the solving of modern and current problems (?)

I almost believe, that those who lack a historical perspective are the most likely to solve problems. This makes a certain amount of sense when I thinks about it...

If one sees a problem as unique, new and Now, instead of an 'updated' old problem to which an updated old solution can be applied, one could tend to consider solutions that are also unique, new and Now...

 

[PhoenixRising]

Yes, I have heard about that circumstance in undergrad physics, and other sciences. I almost went into cosmology instead of commercial art, but my astronomy professor warned me that I probably wouldn't be able to research "off the beaten path" topics.

You are right that there are many complex systems in nature that can be described by standard mathematics. I suppose I was thinking about things like organic forms and quantum mechanics. It is a shame that history is emphasized so much in school. However, I can see the reasoning behind that reality, without knowing what has already been discovered, how can we know our discoveries fit in with the rest of science? In a perfect world, there would be more of a balance between history and new theory and research.

I actually did read Gleick's Chaos this last year. Very fascinating and informative read. I have a stack of books of that nature I have been slowly getting through.

 

[PhoenixRising]

@Da Blob

I agree that in a lot of cases individuals who have learned about a topic by logically analyzing it rather than just studying other people's analysis, are more likely to identify small details, and therefore formulate efficient theories and solutions.

Perhaps someday we will be able to interface in a way where all the history we need to know can be downloaded into our consciousness instantly. Kind of like the Matrix maybe.. Then we could focus on new research without worrying about conflicting with the laws of nature as science understands them.