Mandelbrot to Mandelbulb - 3D Infinity

These following clips are from an audio book entitled, Complexity & Chaos. The segments below give a history of fractals and highlight the importance of fractals in nature, and their place in current scientific theory.

Arthur Clarke: Fractals - The Colors Of Infinity

Considering that everything in existence is a manifestation of the interaction of four (known) basic forces, it's little wonder why our universe shows such an affinity for repeating fractal patterns. Self-similarity is a key concept in fractals, and Murray Gell-Mann has spoken on this aspect at several venues where he relates the astonishing elegance in physics.

Fractals are a simplistic and effective visualization of a broader phenomenon called Chaos. Like all chaotic systems, fractals rely on a number of key traits such as self-similarity, sensitivity to initial conditions, a system of feedback (iteration). So the two are closely related, but they are not synonymous terms. A point of note: Chaotic systems are inherently deterministic, however this doesn't mean they are predictable.



Chaos Theory should further not be confused with Complexity Theory, though may often sound like similar concepts - and they are closely related. A key difference between complex systems and chaotic systems is that chaotic systems are linear, whereas complex systems are non-linear. Complex systems are not nearly as sensitive to initial conditions, often being able to adapt and compensate for fluctuation in bottom level conditions. However, this isn't always viable - leading to complex systems being simultaneously highly resistant to change as well as extremely fragile. Our climate and biosphere are complex adaptive systems.

Emergence is the sum of a non-linear equation, the result of a system which is more than just the sum of it's parts due to the interaction of those parts. Ultimately, complex systems are governed by deterministic forces - but their adaptability and lessened sensitivity to initial conditions make their reaction to any given interaction unpredictable.

Consciousness, is emergent.




Fractals are powerful insights into the way the universe functions, but they are not the be-all, end-all explanation. Yes, I suppose you could accurately describe our universe as a "holographic fractal" of sorts... but the Mandelbrot (or Mandelbulb) sets do not do it justice. Not all fractals are ideal or infinite, and fractal expression (self-similarity) in nature is generally only exhibited within ranges on differing scales of resolution.

An example of the problem, and proposed correction, to the tendency to over-apply fractal geometry to real-world application.

PubMed: Correcting for finite spatial scales of self-similarity when calculating fractal dimensions of real-world structures

Fractal geometry is a potentially valuable tool for quantitatively characterizing complex structures. The fractal dimension (D) can be used as a simple, single index for summarizing properties of real and abstract structures in space and time. Applications in the fields of biology and ecology range from neurobiology to plant architecture, landscape structure, taxonomy and species diversity. However, methods to estimate the D have often been applied in an uncritical manner, violating assumptions about the nature of fractal structures. The most common error involves ignoring the fact that ideal, i.e. infinitely nested, fractal structures exhibit self-similarity over any range of scales. Unlike ideal fractals, real-world structures exhibit self-similarity only over a finite range of scales.

reply to post by Alxandro

How can something so mathematically and scientifically complex still have such a beautiful element of DESIGN?

It's not complex. It's actually very simple.

zn+1 = zn2 + c

The equation is a little borked because of the format, and I don't care to properly display it now, but that's the gist. It gives you an idea of how simplistic it is. The "infinite design" is merely the deterministic result of the execution of that equation.

As for why it's beautiful... I'd venture to say it's likely a result of evolution. Beauty is not an abstract concept independent of the human mind. Our concept of beauty and our appreciation for it is wired by your brain, and is largely variable. However, there is a strong underlying desire to correlate beauty with symmetry. We are bi-laterally symmetrical creatures (our left sides resemble our right sides), as are all of our ancestors down to the emergence of chordate. Beforehand, really, but let's stick with creatures that have spinal cords. All vertebrates are bilaterally symmetrical.

In terms of reproductive fitness, typically those members of a population with the greatest level of symmetry are likely to be the most fit and healthy. Those with a greater frequency of divergence between their left and right sides are likely to unfit. Chimpanzees are a bit more attuned to this, showing greater preference for symmetry than humans who preferred a "slight" variance... but overall, there is that strong tendency to rate Kobe Bryant more "beautiful" than Sloth from the Goonies.



Now, of course, this isn't all there is to the human concept of beauty. However, it's a strong enough trend to spill over into other facets of observation... and has become a staple of human designs from fabric patterns to dinnerware to web space layouts. That fractals are self-similar and as such often display a great deal of symmetry is (IMO) a significant factor in why we are so attracted to fractals and consider them beautiful.

It is important to note that much of what we consider mysterious quirks of the mind is merely normal operation misapplied. For instance, we often see faces pop out from random background noise. This wasn't expressly an evolutionary advantageous trait to develop, nor is it indicative of some mystical spiritual realm peering back to us through the mundane. It's merely your brain's facial recognition processes constantly and persistently looking for facial patterns regardless of the source. This is one of the first processes in visual perception, and without other visual cues - the pattern can take several milliseconds longer to process, causing facial recognition to overlap with other visual processes - such as spatial recognition and association. In effect, you see both the objects for what they are - AND - the face simultaneously.



Beauty in fractals is based, partially, in sexual selection.

Much like the Mandlebrot fractal, your mind... your consciousness... is similarly extremely simple in it's most basic expressions and interactions. The complexity and richness of what emerges is far greater than merely the sum of it's parts. It just doesn't require anything more THAN the sum of it's parts to exist.

"You don't need something more to get something more". ~ Murray Gell-Mann

I wonder what limitations there are on the types of shapes that can be found inside this structure. The ones shown are all reminiscent of biological structures. Plants, coral, some even remind me of a human spinal column. But perhaps these shapes are not as limited as they appear. It may contain structures that are more difficult to find, that resemble an even wider variety of familiar objects. Imagine finding a structure that looks exactly like an entire human skeleton, or a wristwatch, or a Ford Pinto! Can't rule it out.

Originally posted by Astyanax
reply to post by Evasius

 

SNIP
In the view of the universe as a giant fractal, is there room for quantum effects? Is there anything to explain how fractal spacetime manifests matter and energy in theories of quantum gravity? Is that process also fractal? And is there any evidence supporting such ideas, or just aesthetics and a sense of 'rightness'?
END SNIP

Hey Asty:

I read your questions and you appear to have a viable mind so I will attempt to give the mathematical perspective on fractals in geometry and how and maybe even why other avenues of academia such as philosophy, physics and even to some small extent, theology have embraced fractal geometric iterations to support various ideas.

This post will have at least 3,0119 @ characters after the @ symbol.

That is a retractile projection based upon the little box at the bottom left of the screen. That means after my typing ordeal ends, I can go back and change it. In effect I am altering the past, the present and the future, should I run out of words before this ends. A Fractal projection is different because it has a defined state in the past that cannot be altered, but must be iterated or repeated. Fractal projections occur in nature. Retractile projections do not. Well at least not until the internet you could argue. Hah Ha.

Since fractal projections are everywhere, a lot of people in the fields of chemistry, physics, biology, theology, and anything else that ends in either "ics" or "ogy" have tried to relate fractal spatial iterations to their study. Some quite well, others well, not so much. An ice-crystal is fractal, a snowflake is fractal, a coral reef is fractal, the earths plates are fractal, the sand on the beach is fractal as are the clouds in the sky and the waves in the oceans, molecules are fractal, fruits and vegatables are fractal, all living organisms are fractal, the universe is fractal, and so are the hairs on my backside. Big Hairy Deal!

There is room for Quantum theory as well as String theory in this but the compelling relationships are found in theology and philosophy, to my mind at any rate. There is no theory that relates quantum gravity, to the Hausdorff dimension, that I am aware of. In Theology and philosophy, there are primary arguments that are compelling and provocative that could engage many a brain in witty repartee but it ultimately boils down to this question...Is it the hand of God that holds the mouse? Who clicks on a particular area and alters spatial reality, each image succeeding the in the exact same size as the preceding image?

I am short by 1261...

Fractal Metal Watch to the 3/4 mark and put on your seat belt
www.youtube.com...

Fractal Metal Watch to the 3/4 mark and put on your seat belt
www.youtube.com...

Fractal Metal Watch to the 3/4 mark and put on your seat belt
www.youtube.com...

Fractal Metal Watch to the 3/4 mark and put on your seat belt
www.youtube.com...

Whoaa... the infamous quad post. I had a computer crash when I posted that and didn't think it went through. Guess its a fractal post yeah that's it

Great thanks for all the posters on this thread! Best source for wallpaper on ATS

Fascinating subject as always.

So to sum it all up?

All things all information, matter, antimatter space and time are on an infinitely subatomic to an infinitely multi universally multi dimensional scale as we are able to perceive it and it all exist because of consciousness?

Everything just IS ?
It IS what it IS ?
We just ARE ?
I am that I am ?

We long to experience all things that have been happening for infinity because of consciousness.

No beginning no end, just a paradox

reply to post by St-Patrick


That is amazing, I too saw the same patterns and images after eating 1/4 ounce of mushrooms. I remembered it for the longest time, then one day i saw this thread. I was simply astonished.

This is a fascinating subject, and one that will hurt the brain if thinking about it for to long.

I'm gonna bookmark this thread.

[edit on 8-4-2010 by TravisT]

Infinate aye? And to think the big bang caused all this?

I just want to say WOW! Thanks for posting all of the videos and beautiful pictures. I showed everyone in my house your thread and they were also impressed. Thanks for the great pictures, I made a slideshow for my desktop screensaver. These images are quite the conversation piece when students come to my office. Quite often, they forget what they came to see me about after viewing the slideshow.

Though I would share a TED link to Benoit Mendelbrot speaking. This is a real treat for me, hope you also enjoy.


TED.com

I just found this thread today. This is some kind of amazing!

To be completely honest, recently, when I observe certain types of fractal plants, it makes me wonder if this is indeed how the universe is. And yes, I'm so sure this has been discussed many times, at length.

I love those things. They make me feel very small, and yet connected to everything else.

I like how neurons also look like the universe. For all we know, our world and the universe could be part of something larger we can't even comprehend.
Maybe we are all very tiny.

I know when I see some nebulae and images of galaxies, some of them sort of remind me of cells dividing.

Everything we see at a microscopic level is happening on a grander scale out in space I think. As below, so above.