Fractals & Nature

[Power 8]

I am often surprised at how often a fractal image will look like something out of nature.  Vegetal, mineral or animal shapes frequently come to mind. This is even more true with 3D fractals like mandelbulbs and such. Does any one have thoughts, ideas or theories about this?

[Power 8]

Here are a few images that illustrate this:
Some I made and some  found on this forum
Animal:



by mjk1093:


Vegetal:



by msltoe:


Mineral:


Vegetal & Mineral!:by Tglad:

[Sockratease]

I am often surprised at how often a fractal image will look like something out of nature.  Vegetal, mineral or animal shapes frequently come to mind. This is even more true with 3D fractals like mandelbulbs and such. Does any one have thoughts, ideas or theories about this?

Well, we do have an entire section devoted to the concept - so there must be something to it!

http://www.fractalforums.com/fractals-in-nature/

[Power 8]

Oops yes indeed! Sorry, hadn't seen that section yet...

[aluminumstudios]

I am often surprised at how often a fractal image will look like something out of nature.  Vegetal, mineral or animal shapes frequently come to mind. This is even more true with 3D fractals like mandelbulbs and such. Does any one have thoughts, ideas or theories about this?

I kind of don't like this line of thought for two reasons.  One is that the human brain is a pattern matching machine.  Our brains, automatically will try to match shapes to known things, no matter similar, or dissimilar things are.  For example, our brains process a colon, dash, closing parenthesis as a smiley face :-)    If you really think about it, ) and : are so simplistic they don't resemble a face in the slightest.  Our brain just insists on matching patterns.  It's probably an evolutionary device that suited us well in the tens of thousands of years that we were here before civilization.  Now it just makes typing more fun :-)   :-P   (^_^;)

I guess the second reason I like to avoid deep contemplation of fractals is that fractals are essentially numeric models of chaotic systems.  Many things in nature are chaotic systems, so naturally there are going to be a lot of parallels between the two.  But that doesn't imply any deep connection between the two or underlying principles that connect things like our beloved Z=Z^2+C to the branches of a tree any more than Newton's equations cause (or are caused by) the orbits of planets and behavior of gravity.

I love fractals and think they are truly aesthetically and scientifically fascinating, but I don't think that they hold any more secrets than the mathematical equations that have helped us model (the key word is model, not be responsible for in some way) other things in the universe.

This is just my opinion on things.  I mean no disrespect to your observations.

[Power 8]

No disrespect taken Aluminumstudios, on the contrary thanks for your views, which are insightful and exactly the kind of thing I was expecting when I made this post. Still I remain fascinated by fractals and get a pleasant feeling every time one reminds me of a natural shape... Why that is I don't know... Of course the laws of gravity existed long before Newton, and  Romanesco broccolis existed long before Mandelbrot but I find the whole idea that man can find equations and formulas that describe natural geometrical or physical phenomenons to be intellectually stimulating.

The opposite works for me too:

detail of Romanesco broccoli

[Sockratease]

Well, I believe very strongly in a connection between fractal patterns and those of the real world!

Aren't Lorenz Fractals just his equations (which happen to work reasonably well!) for describing the weather?

I am Part-Scientist and Part -Mystic-Crazy-Hippy-Artist-Bastard!  I have no problem believing that DNA acts on an iterative mathematical equation, and Only an iterative mathematical equation, and that Fractals will turn up as the driving force behind the evolution of that equation.  

But then again, I like to believe that Qetzacoatl is returning in 2012 to EAT this Universe and poop out a new one.

I have no Scientific Basis for this (or many!) of my beliefs...  But I do enjoy believing them, and don't care if I'm wrong, so it's just what I believe.

[Timeroot]

AluminumStudios, please check out the work of Max Planck. There are several systems that are "universal" - that is, the same effects can be observed in all of them, even if the equations or the large scale effects are completely different. For instance, the Mandelbrot version of the Newton fractal produces the exact same shape as the regular Mandelbrot set, because the two are "locally similar". The Mandelbrot set uses the same equation that appears in population dynamics, dripping water, convection currents, etc... in fact, there are some observations of "complex temepratures" (Yang-Lee zeroes), leaving the Mandelbrot set quite close to the real world.  

[aluminumstudios]

No disrespect taken Aluminumstudios, on the contrary thanks for your views, which are insightful and exactly the kind of thing I was expecting when I made this post. Still I remain fascinated by fractals and get a pleasant feeling every time one reminds me of a natural shape... Why that is I don't know... Of course the laws of gravity existed long before Newton, and  Romanesco broccolis existed long before Mandelbrot but I find the whole idea that man can find equations and formulas that describe natural geometrical or physical phenomenons to be intellectually stimulating.

Maybe we share more of the same viewpoint than I thought.  I find it satisfying and elegant that mathematics can model irregular things.  I just shy away from the idea that some people put forth that there is some kind of deep, almost mystical or unknown connection between fractals and the universe.  I remember reading on some web site a while back that fractals can heal us because of the fractal patterns of body tissues or something like that.  I don't believe in those such things at all.  But I certainly do acknowledge the elegant mathematical relationships in the world.  For example (one that I heard and was fascinated by), is that for electrons to have stable orbits around atomic nuclei they must have certain energy levels.  The energy of photon's that electrons emit and absorb corresponds exactly to the energy electrons must shed or gain to be in a stable nuclear orbit.

To whatever extend fractals play a role, nature is awesome :-)

[reesej2]

On the contrary! I find it extremely likely that there IS some sort of underlying connection, but it's more simple than you'd think. When you think about it, fractal geometry is an ideal design for life--instead of having to specify the entire structure of a tree in its genetic code, all that's needed is a rule for generating the tree, like an L-system. It's no mystery why various forms in nature bear such a close resemblance to the fractals we work with. It's just efficient.

Though I have to agree with aluminumstudios--the notion that the connection is somehow mystical or supernatural is pretty strange. Fractals are, at their most basic level, purely mathematical constructs, and therefore any connection they may have with the physical world must also be purely mathematical.

Incidentally, Timeroot: it's always strange when we encounter complex numbers in physical quantities. I'd heard of an application of imaginary numbers in electrical engineering, but never of these Yang-Lee zeroes. Interesting.

[Tglad]

I guess my feeling why fractals are seen in nature is that the rules that drive them are very simple, even though the results can look complicated. Moreover, euclidean geometry like straight lines or circles are really just special cases of fractals. What is a straight line if it isn't a recursive structure made out of self-similar smaller lines? There are just more ways of generating geometry which is rough (fractal) than the few ways that turn out smooth (like a perfect circle). If the rules operate about the same at different scales and in different directions and at different locations then that roughness should be fractal roughness.

I don't think all of nature is fractal, but I do think fractal geometry is one step better at describing nature than euclidean geometry (cylinders, boxes etc) since it generalises it.

And just as some natural objects are surprisingly euclidean (like pearls), some objects seem to be surprisingly fractal (like oysters http://www.fractalforums.com/index.php?action=gallery;sa=view;id=1977).. but most things aren't well described by either in my opinion.

[reesej2]

I agree. The human body, for example, shows both fractal and Euclidean characteristics--our circulatory system is distinctly fractal, the interior of our lungs even more so, but our overarching structure (arms, legs, et cetera) is decidedly Euclidean. And, of course, the analogy to fractals breaks down at a certain scale, because everything has nonzero size--a living thing is built up of cells, a cloud of water molecules, etc. But I'd say that some sort of hybridization between Euclidean and fractal geometry ought to be sufficient to describe everything.

[Power 8]

Thanks to all for your thoughts and insights.
Euclidian + fractal geometry can describe a lot of things but I guess there's still more to discover concerning the geometry of nature. Since Euclid lived in 300BC and fractal geometry was discovered in the second half of the 20th century, (Mandelbrot's publications are from the 70s) I hope it takes us less than 23 centuries to find the next step which is possible now we have the internet to share and bounce back ideas to an extent that Euclid never dreamt of. (or maybe he did dream of such things, he was a pretty fly guy for his time.)
I also feel, though I'm no mathematician that the triplex algebra discovered here is a pretty interesting discovery...
Reading Tglad's post where he says

What is a straight line if it isn't a recursive structure made out of self-similar smaller lines?

makes me think maybe euclidian geometry is englobed by fractal geometry which itself might be part of something still to be discovered...
Tglad also says:

some natural objects are surprisingly euclidean (like pearls), some objects seem to be surprisingly fractal (like oysters

so for fun I made a fractal pearl using Mandelbulb3D's Real Power formula, and a pearl is a sphere which is just a power 1 mandelbulb


power 1.2 / zmul 1 / R stop 1.02

Finally about reesej2's post where he says

a living thing is built up of cells, a cloud of water molecules,

, well living things such as ourselves are also made out of water molecules as a cell is mostly water and 75% of our body mass is water but that is a different subject altogether that I also find fascinating, (the physics and proprieties of water ) . Water shows "fractality" too as for instance in these Google Earth pictures of rivers that look like trees:





...and trees are fractal shapes too and they're made mostly of water... are trees vertical rivers?
Hmm I think i should go to bed now...

[aluminumstudios]

Nice pictures and some interesting thoughts, but I don't think that we should get too hung up on what percentage of our bodies or other things are water.  Carl Sagan (a man who I deeply respect), once said "It's not the molecules that go into things, but the way they are put together."

I wonder how often fractal patterns creep into human affairs without people ever being aware of fractal geometry and the like.  This is Wat Arun (Temple of the Dawn), in Bangkok.  I shot this photo a year ago, interested in the pattern, and only now realized how fractal it looks.  (I know this isn't nature, but I see it as another example of the omnipresence of fractal patterns)

[reesej2]

Sort of reminds me of the Mandelbox... I think it's likely that humans have a built-in appreciation for the visual beauty of fractals, and that's why they crop up so often. I've seen decidedly Mandelbrot-like patterns in paisley, and some patterns that look a lot like other types of fractals in patterns on carpets. That'd also explain why most of us are here at all...