Fractal Geometry of Nature

[InfiniteNow]

I chanced on a PBS show last night entitled Colours of Infinity. Hosted by Arthur C. Clarke and featuring Mandelbrot himself, Steven Hawking, and Prof. Ian Stewart, it presented some breakthrough applications of fractals. ...The show will air again on February 4th;

:beer:

 

Sadly, my friend, it appears this was/is a local phenomenon. Upon reading your post, I was filled with a giddy excitement and launched your program schedule links. When I recognized they were specific to the pacific, of the northwest variety, I searched my local PBS schedule. But alas, it 'twas not to be. We have a wonderful Nature program on schedule, but my infinity will be colorless that day. :beer:

[Racoon]

I chanced on a PBS show last night entitled Colours of Infinity. Hosted by Arthur C. Clarke and featuring Mandelbrot himself, Steven Hawking, and Prof. Ian Stewart, it presented some breakthrough applications of fractals. Specifically, a new technique in image data compression that is phenominal! Basically, (as if!), any image is scanned or otherwise digitized and then a

a fractal equation is formulated on that image. Then taking that new fractal equation, it is applied to the zoomed in/pixelated view and the detail is 'filled' in! :)

The show will air again on February 4th; here's a couple of links:

 

Program Information | PBS 06:00:00&use_gmt=t

PBS | TV Schedules

 

Did you remember to tape it Turtle-San??

 

There was no way I could stay awake until 4 in the morning for the re-airing of the first one I missed...

[Turtle]

Did you remember to tape it Turtle-San??

 

There was no way I could stay awake until 4 in the morning for the re-airing of the first one I missed...

 

Alas:doh: , I saw Sunday, but the 4:00 am part didn't register. In other words, I was ready to go at 4:00 am MONDAY. :) Sorry I missed the mark.

I do have the homepage of Prof. Ian Stewart however and he has some cool fractal images as well as e-lectures on the subject. I think he's the one who developed the fractal compression scheme. Sorry again for my lameage.

Ian Stewart Home Page v 2.1

[Racoon]

Alas:doh: , I saw Sunday, but the 4:00 am part didn't register. In other words, I was ready to go at 4:00 am MONDAY. :) Sorry I missed the mark.

I do have the homepage of Prof. Ian Stewart however and he has some cool fractal images as well as e-lectures on the subject. I think he's the one who developed the fractal compression scheme. Sorry again for my lameage. :eek_big:

Ian Stewart Home Page v 2.1

 

 

I get the gist of it. Revolutionary video technology that can be used for many applications from Space exploration, to Hollywood, to FBI profiling

 

Thanks for trying ol' chap.

It'll get re-ran or made available to those who don't have access to Oregon Public Broadcasting at 4 am. :camera:

 

Fractals Rule!

[Racoon]

Heres a really sweet video of Fractals... B)

 

Blue Oyster Spiral Fractal Zoom - Google Video http://video.google.com/videoplay?docid=1619313842463920970&q=Fractals&hl=en

[Pyrotex]

Cool. 10^33 magnification.

 

If the original "Mandelblot" was the size of our entire Solar System (the orbit of, say, Neptune), then the details in the last frame would be far smaller than a Quark inside a Proton inside an Atom. :Waldo:

[InfiniteNow]

Cool video from TED.

 

Ron Eglash: African fractals, in buildings and braids.

 

TED | Talks | Ron Eglash: African fractals, in buildings and braids (video)

 

"I am a mathematician, and I would like to stand on your roof." This is how Ron Eglash greeted many African families while researching the intriguing fractal patterns he noticed in villages across the continent. He talks about his work exploring the rigorous fractal math underpinning African architecture, art and even hair braiding.

[Thunderbird]

This link is the best I have come across that ties together fractals, phi,Emergence, Maximum Complexity, Self-organised Criticality, Autopoiesis, and Chaos, into a cohesive whole.

 

 

Dynamical Symmetries: Autopoietic Architecture

The Areas of Mathematical Synthesis Between Complexity, (edge of) Chaos Theory , Fractal Geometry and the Golden Mean: leading to an argument for an Autocatalytic Architectural approach based on emergent Self-Organised Criticality

 

 

 

Phi-Lo-Tactics:(Phi)Recursion/Self Re-Entry-Heart of Self Organization?

 

Heraclitus of Ephesus, sixth century B.C.

 

Morphology is not only a study of material things and of the forms of material things, but has its dynamical aspect ... in terms of force, of the operations of energy. This is a great theme. Boltzmann, writing in 1886 on the second law of thermodynamics, declared that available energy was the main object at stake in the struggle for existence and the evolution of the world.

 

D'Arcy Thompson, On Growth and Form, 1917

 

...to (in)form buildings with thematic meaning, they must convey a gestalt, the whole must be more than the sum of the parts, and there must also be an ambiguity and paradox immanent within that gestalt, as a tension. (And quoting Heckscher on composition...) It is the taut composition which contains contrapuntal relationships, equal combinations, inflected fragments, and acknowledged duality's. It is the unity which maintains, but only just maintains, a control over the clashing elements which compose it. Chaos is very near, its nearness, but its avoidance, gives ...force.

 

Robert Venturi, Complexity and Contradiction in Architecture, 1966

 

 

The Universe is constantly seeking to lower it's energy (like a stream meandering down a valley, sometimes becoming rapids or cataracts), as it continually breaks it's initial higher (energy and dimensioned) symmetry. This symmetry-breaking can be pictured as bifurcating branches of probabilities, of chance and causality intertwined. The reason for this is that the higher energy/dimensions of the Universe's origin were also highly symmetrical, which paradoxically, means that they were also extremely unstable. Highly symmetrical objects are unstable ( imagine the beginning of the universe as being like you sitting on a perfect, highly polished sphere), and have a high probability for instability because any deviation by you from the north pole will have you rapidly slipping off. Here, gravitational instabilities quickly break your symmetrical position at the north pole and pull you to a lower gravitational potential energy state. So small initial deviations in unstable, far from equilibrium situations can lead to massive, even cosmological consequences. The Butterfly effect (also known as sensitive dependence on initial conditions), is literally, Universal.

Ludwig Boltzmann is known to us as the first to provide a probabilistic, statistical interpretation of entropy. This is simply the tendency of everything in the Universe to cool to a minimum energy or temperature --- known as thermal equilibrium. The route to this second law of thermodynamics is via increasing disorder in the organisation of energy and matter.

The current symmetry-breaking from the initial condition leads therefore, from a highly symmetrical, ordered and energetic state towards the opposite, an asymmetrical, disordered and lower energy one; from a low entropy Big Bang to a higher entropy present and future.

The great paradox of the second law then, is the evident, increasingly complex, emergent and hierarchical order we see all about us. How is this ordered, structured information (expressed in constantly oscillating patterns of matter and energy) allowed to coalesce and persist from this tendency towards the random --- towards increasing entropy?

Dynamical systems theory also deals with probability and can therefore allow us to synthesise thermodynamics and so-called "Chaos", (which is really a highly complex form of hierarchical, enfolded order that appears to be disorder). The really interesting area here though, is the entities at the transition zone between ordered, stable systems at equilibrium (maximum entropy) and "disordered" (but complex) and unstable Chaotic (minimum entropy) ones. According to the Nobel laureate Ilya Prigogine, these far from equilibrium dissipative systems locally minimise their entropy production by being open to their environments --- they export it in fact, back into their environments, whilst importing low entropy. Globally, overall entropy increase is nevertheless preserved, with the important caveat that the dissipative system concerned often experiences a transient increase (or optimisation) of its own complexity, or internal sophistication, before it eventually subsides back into the flux.

This is known as the region of alternatively, Emergence, Maximum Complexity, Self-organised Criticality, Autopoiesis, or the Edge of Chaos. (Nascent science debates nomenclature routinely - and appropriately, in this case, the crucial point being that they are all different terms for essentially the same phenomena.)

Lifeforms, ecosystems, global climate, plate tectonics, celestial mechanics, human economies, history and societies, even consciousness itself - all manifest this feedback-led, reflexive behaviour; they maximise their adaptive capacities by entering this region of (maximum) complexity on the edge of Chaos, whenever they are pushed far from their equilibrium states, thereby incrementally increasing their internal complexity, between occasional catastrophes.

Remarkably, this transition zone is mathematically occupied by The Golden Mean. This ratio acts as an optimised probability operator, (a differential equation like an oscillating binary switch), whenever we observe the quasi-periodic evolution of a dynamical system. It appears in fact, to be the optimal, energy-minimising route to the region of maximum algorithmic complexity, and to be a basin of attraction for the edge of Chaos. In this review, we shall cover some demonstrations of this behaviour, and seek to understand its role.

As far as architectural application is concerned, we must look at the temporal as well as the spatial, at how quite literally, the dynamics (of systems applications - functions) can inform the statics (forms) of building. The aesthetics of the banal imitation of some motif of fractal geometry is simply painting half the picture! From the Egyptians to the Greeks, Gothic to the Renaissance and the Modernists in the Western tradition, and especially in Hindu, Islamic, Buddhist and Meso-American aesthetics, fractal structures or rubrics (such as the regulating lines of Le Corbusier or his Modular based on the "Divine" Golden Mean), have shown that intuitively, the best architecture has understood and reproduced the true geometry of nature as more than just decoration - but as pure, optimal structure, that allows the thematic, historic and actual forces and loads being carried to be read as a tension, as a dynamic equilibrium - from the scaled arches of Roman aqueducts to the tiered flying buttresses at Chartres, to the contrapuntal three-pin arch of Grimshaw's Waterloo International in London and perhaps most appositely in recent times, the structures of Calatrava, Hopkins, Piano, Rogers and Foster; the purest architecture has been taken from nature's own template.

Aristotle implied over two millennia ago that the proper investigation required was one of telos, the "final cause" of morphology, of form being the result of the processes that engendered it.

His "final cause" of morphogenesis suggests an imperative behind any generative process that has often been interpreted as having theological (as well as teleological) connotations. Here, we shall take a more determinist route, in line with his mentor Plato's definition of the logos, as the "proportion" which was commensurate in square, which best squared the circle, or presented a unity that was more than the sum of its parts.

It will be suggested that this imperative behind form (as static, precipitate matter resulting from dynamic flows of energy) is certainly nothing to do with the metaphysical, but simply the result of the way nature minimises energy waste (entropy production), also known as the principle of least action - and that one way of mathematically representing this behaviour appears to be analogous to the dynamical behaviour of the Golden Mean.

So how does nature manifest this limiting principle in a way that still allows for the immense emergent complexity we see, or to put it another way, how can we demonstrate that least action acts as an attractor for Complexity and self-organising emergence, by symmetry-breaking to lower energy states, towards the edge of Chaos?

All fractal forms, inert (clouds, landscapes, galaxy clusters) or animate (plants, animals), are self-similar scaled copies of an original; chaotic systems (climate, the solar system, the stock market) also always possess this fractal quality, but taken to the paradoxical extreme of having infinite trajectories within a finite boundary. To produce these forms, a recursive feedback regime must be operating. Feedback (encoding similarities) underlies the entire subject, and is the basis of the thesis research (undertaken at the Engineering, Computer Science and Architecture Faculties, University of Westminster) that underlies this review.

This research began several years ago (during the degree at Kingston University), as an intuition that The Golden Mean, or Phi for short, (as a ratio) must have been fractal in nature. By extension, it seemed plausible that Phi may also have been embedded in higher dimensional, dynamical systems as an attractor of some kind, since complex dynamical systems always have a fractal temporal structure as they evolve over time.

A major clue leading to the above interpretations can be seen in the fact that Phi is simultaneously both an arithmetic and geometric expansion of itself and One of the simplest possible kind. This immediately places it in both the linear (arithmetic progression) and non-linear (geometric progression) realms, and as an effective bridge, operating between the two.

Virtually every aspect of fractal geometry and type of dynamical system can be expressed by variations upon the simple quadratic iterator:

X = X**2 + c

which expresses the particular type of feedback being examined, Phi can be expressed by a related but more archetypal variation to derive the Fibonacci series:

X[n+1] = X[n] + X[n-1]

which incrementally gravitates towards a particular ratio which possesses unique qualities. Numerically, it can be derived from the relation:

(1 + sqrt(5))/2.

For example, if one diminishes Phi by Unity you derive its reciprocal. Additionally, Phi is the unique ratio that fulfils:

1/Ø + 1/Ø**2 = 1

in other words, Phi is also the only possible geometric and arithmetic, expansion and partitioning of One.

This leads us to the other cardinal feature of Phi. There is only one proportional division of One possible using two terms, with the third being One itself. From Euclid's ELEMENTS Book Five, Theorem Three (Alexandria, 3rd century B.C.):

"A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less."

The Golden Mean then, is an archetypal fractal in that it preserves its relationship with itself (its inherent similarities under scaling are conformal symmetries - with topological consequences, that are invariant about themselves), in the most mathematically robust, economical but also elegant, way. It is analogia exemplified.

[Essay]

___Because with this new understanding of dimension, that is we have not just 2-dimensional thingys, & 3-dimensional thingys, but 1.2618 dimensional thingys & so on in between, and because they pervade nature, and because they share recursion as an operative element, I think any real grand unified theory must include recursion as an operative element.

 

 

I've been debating whether to mention this thought, and here (above) is something along the lines of my thoughts for the past couple o' years.

 

There are many instances in particle physics where an integer is used to express some quantized parameter (i.e. spin?).

 

Could fractals be substituted for these quantum values?

 

X = X^2-c

 

...or words to that effect?

 

...even if the above is meaningless, I'd think that somewhere within the Standard Model and/or SUSY would be some opportunities to substitute fractal expressions for 3-D quantum numbers/parameters.

:(

p.s. Thanks Tbird for the excellent post below. (Please see)

[HydrogenBond]

I used to do digital artwork, although I don't do it as much anymore. One of the tools I liked was a set of filters called KTP or Kali's power tools. One of the filters in that version of KPT was frac-splorer. This filter allowed one to create an endless supply of fractal patterns. I would use these patterns as raw materials for further processing.

 

A technique that I developed was to take a similar series of a fractal pattern into Adobe Photoshop and blend them, flatten image. This kept the basic pattern but added more layers of variety such as spheres, fibers, etc.. I would then duplicate the resulting layer, flip one layer horizontally, blend the two layers, flatten image. Then duplicate the next resulting layer, flip one vertically, blend, flatten. The result was very interesting. What would almost always appear were little figures in the center of the symmetry, still surrounded by the basic fractal patterns in the perimeter. This technique suggested that certain mathematical manipulation of fractals patterns can lead to a different level of semi-fractal pattern in the center, that seem to set the parameters for the basic layout of nature's many critters. A loose analogy is the snail's shell is fractal. Inside is the snail.

[Turtle]

I used to do digital artwork, although I don't do it as much anymore. One of the tools I liked was a set of filters called KTP or Kali's power tools. ...

 

You prompted a memory HB, and without regard to which side of my head it's recorded in, or you(all) record it in, here it is digitally.

 

Back when James Gleick's book Chaos came out, there was a ring-bound version that came with software. If I show a little less enthusiasm for doing fractal exploration these days, it's likely related to the untold hundreds of hours I spent with that software. Doing a quick web-search for any particulars, I find the software is available free online. :eek:

 

Chaos Downloads from Rudy Rucker's Wesbsite

[InfiniteNow]

I chanced on a PBS show last night entitled Colours of Infinity. Hosted by Arthur C. Clarke and featuring Mandelbrot himself, Steven Hawking, and Prof. Ian Stewart, it presented some breakthrough applications of fractals. Specifically, a new technique in image data compression that is phenominal! Basically, (as if!), any image is scanned or otherwise digitized and then a

a fractal equation is formulated on that image. Then taking that new fractal equation, it is applied to the zoomed in/pixelated view and the detail is 'filled' in!

 

:)

 

Sadly, my friend, it appears this was/is a local phenomenon. Upon reading your post, I was filled with a giddy excitement and launched your program schedule links. When I recognized they were specific to the pacific, of the northwest variety, I searched my local PBS schedule. But alas, it 'twas not to be. We have a wonderful Nature program on schedule, but my infinity will be colorless that day. ;)

 

Did you remember to tape it Turtle-San??

 

There was no way I could stay awake until 4 in the morning for the re-airing of the first one I missed...

 

Alas:doh: , I saw Sunday, but the 4:00 am part didn't register. In other words, I was ready to go at 4:00 am MONDAY. :shrug: Sorry I missed the mark.

I do have the homepage of Prof. Ian Stewart however and he has some cool fractal images as well as e-lectures on the subject. I think he's the one who developed the fractal compression scheme. Sorry again for my lameage.

 

Look what I found: :D

 

Fractals - The Colors Of Infinity (By Arthur Clarke) http://video.google.com/videoplay?docid=8570098277666323857

 

 

;)

[Pyrotex]

Damn! I wish I had time right now (at work!) to watch that video. But alas! Duty calls. Maybe later. At home. After I do my taxes. [sigh]

[InfiniteNow]

Damn! I wish I had time right now (at work!) to watch that video.

 

See, you're truly lucky then. It's always right now. :hyper:

[InfiniteNow]

 

 

 

I saw it, too. Great program. From the history of the idea, to the events helping it form, to the applications of it... It talked of biology, and how even the rhythms of the heart can be modelled with fractal geometry.

 

Two thumbs up. Watch below:

 

NOVA | Hunting the Hidden Dimension | Watch the Program | PBS

 

 

[Thunderbird]

I caught the show last night terrific stuff! . The trees are fractal yea , but the pattern extends to the whole forest. What next ? The universe , quantum particles ?

[maddog]

I felt bad I missed that show last week. We had just moved and had left our

TV. At least we have one now. I will have to try and watch it off of pbs.org

website.

 

I have been fascinated by fractals for quite awhile now. First thing me and

some friends looked into was to see what other algebraic systems did with

this process. First looked at Quaternions (equivalent to Vector algebra in 3D).

 

Because Quaternions are an anticommunative algebra

 

i * i = -1

j * j = -1

k * k = -1

 

i * j = k

j * k = i

 

j * i = -k ... etc

 

you get strange pictures in projective 3D that are always twisted.

 

Somebody earlier in this post mentioned Julia Sets and they simply where

the constant C in equation z <- z^2 + C is the coordinate in the iteration.

 

The equation can be anything algebraic like

 

Z <- cos^2 (Z^2) + C

 

and you will get an a fractal that will look like and earthworm about the

x axis.

 

And so on.

 

maddog