Fractal-oscillation attraction

[KALSTER]

First, a quote from the Wiki page on Fractals:

 

A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured."

 

A fractal often has the following features:

It has a fine structure at arbitrarily small scales.

It is too irregular to be easily described in traditional Euclidean geometric language.

It is self-similar (at least approximately or stochastically).

It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).

It has a simple and recursive definition.

 

And an example:

 

The Mandelbrot set is a famous example of a fractal.

 

A closer view of the Mandelbrot set.

 

Now, imagine a 3D version of a fractal created by an oscillating disturbance at its centre. The oscillation means that the orientation of the fractal (into its mirror image) changes with a certain frequency and that the speed at which the newly created fractal moves away from its source is governed by the medium of propegation it finds itself in. The shape, size and type of the fractal is determined by the particular shape, size and type of the disturbance. It is possible for the shape of two fractals to be equal in every way except for the orientation being in the opposite direction (i.e. it points outward instead of inward).

 

My question is: Is it possible for the interaction between these two fractals, equal except for orientation, to exert an attractive force on each other? Would two of the exact same fractals then exert a repulsive force on each other?

[Thunderbird]

First, a quote from the Wiki page on Fractals:

 

A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured."

 

A fractal often has the following features:

It has a fine structure at arbitrarily small scales.

It is too irregular to be easily described in traditional Euclidean geometric language.

It is self-similar (at least approximately or stochastically).

It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).

It has a simple and recursive definition.

 

And an example:

 

The Mandelbrot set is a famous example of a fractal.

 

A closer view of the Mandelbrot set.

 

Now, imagine a 3D version of a fractal created by an oscillating disturbance at its centre. The oscillation means that the orientation of the fractal (into its mirror image) changes with a certain frequency and that the speed at which the newly created fractal moves away from its source is governed by the medium of propegation it finds itself in. The shape, size and type of the fractal is determined by the particular shape, size and type of the disturbance. It is possible for the shape of two fractals to be equal in every way except for the orientation being in the opposite direction (i.e. it points outward instead of inward).

 

My question is: Is it possible for the interaction between these two fractals, equal except for orientation, to exert an attractive force on each other? Would two of the exact same fractals then exert a repulsive force on each other?

The opposite orientation, but identical form are the two fundamental equal but opposite elements that underlie all emerging system.

 

 

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. Nigel Reading

 

[Buffy]

Now, imagine a 3D version of a fractal created by an oscillating disturbance at its centre. The oscillation means that the orientation of the fractal (into its mirror image) changes with a certain frequency and that the speed at which the newly created fractal moves away from its source is governed by the medium of propegation it finds itself in. (Emphasis Buffy)

You need to be careful here: an oscillation may in fact produce a "frequency" that is so long that for all intents and purposes the output is random. It is also sensitive to the quantization of the scale of the fractal instantiation.

 

This will work for simple fractals like those T-bird shows, but this is exactly the kind of "additional input" into a seemingly well-behaved system that turns it chaotic....So:

My question is: Is it possible for the interaction between these two fractals, equal except for orientation, to exert an attractive force on each other? Would two of the exact same fractals then exert a repulsive force on each other?

Maybe, it depends on the fractals, their orientation and oodles of initial conditions!

 

Augurs and understood relations have by magot-pies and choughs and rooks brought forth the secret'st man of blood, :confused:

Buffy

[KALSTER]

Good, legit points.

 

You need to be careful here: an oscillation may in fact produce a "frequency" that is so long that for all intents and purposes the output is random. It is also sensitive to the quantization of the scale of the fractal instantiation.

 

This will work for simple fractals like those T-bird shows, but this is exactly the kind of "additional input" into a seemingly well-behaved system that turns it chaotic.

The medium I am thinking of is perfectly uniform, with only variations in "density" for lack of a better word. Given this, along with zero friction, do you think it would at least be less prone to chaos? Plus, the source would be going at a very high frequency.

 

Another thing to consider, is that when two similar source/fractal systems interact and they are attracted to each other, some momentum would have to be lost which would translate into the source losing frequency. Conversely, when two repulsive systems interact they should both gain momentum, since this is a zero friction scenario. Does that follow you think?

[Thunderbird]

Good, legit points.

 

The medium I am thinking of is perfectly uniform, with only variations in "density" for lack of a better word. Given this, along with zero friction, do you think it would at least be less prone to chaos? Plus, the source would be going at a very high frequency.

 

Another thing to consider, is that when two similar source/fractal systems interact and they are attracted to each other, some momentum would have to be lost which would translate into the source losing frequency. Conversely, when two repulsive systems interact they should both gain momentum, since this is a zero friction scenario. Does that follow you think?

Sounds like the structure of an atom....

[Little Bang]

I suspect that the charge of an electron and proton is a fractal of four waveforms and of course the mass is determined by frequency.

[KALSTER]

Sounds like the structure of an atom....

Don't jump the gun yet;).

 

So, any thoughts?

[KALSTER]

From the Wiki page on Fractals:

 

"Approximate fractals are easily found in nature. These objects display self-similar structure over an extended, but finite, scale range. Examples include clouds, snow flakes, crystals, mountain ranges, lightning, river networks, cauliflower or broccoli, and systems of blood vessels and pulmonary vessels. Coastlines may be loosely considered fractal in nature."

 

The images in the OP was only to get a casual reader an idea of what a fractal is.

 

I was thinking of fractals in the form of self-similar reducing eddies in a medium, that is, reducing as the energy is dissipated with distance by forming multitudes of smaller and similar eddies (if that makes sense). This hypothetical medium would have no internal friction as one would expect from water, say.

By oscillating I mean that they form mirror images of each other along an axis running through the source, roughly in the same manner a sine-wave would.

 

So assuming that nature can not be infinitely reduced (the space-time fabic for instance), I guess this fractal would have to terminate after a certain number of instances. Maybe even straight down to the planck length.

 

For simplification, think of a wave-form being emitted around an axis of symmetry, similar to a sine wave, but instead of the regular positive and negative curves, you have (alternating between each side of the axis) spirals/eddies curling in the direction of the disturbance. This single arm comes into contact with the arm of another disturbance. The direction of rotation of the spirals/eddies will clash head-on with the spirals/eddies of the other arm. Now extrapolate this to a 3D situation with large numbers of arms from the two disturbances interacting with each other. The disturbances should be attracted to each other, no?

 

Just to be clear: I am not making any claims, just doing a mind experiment.

[KALSTER]

I made this same thread on another forum and it got Really Interesting over there!