FRACTALS IN PHYSICS - Captured Lightning are now indispensable in most natural sciences (and even in...

$: FRACTALS IN PHYSICS - Captured Lightning are now indispensable in most natural sciences (and even in school). In physics one of the main fields in which fractal geometry is widely$
FRACTALS IN PHYSICS http://appserv01.uni-duisburg.de/hands-on/files/autoren/nordm/nordm.htm

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Fractals in Physics- From Low-Cost Experiments to Fractal Geometry -

Volkhard Nordmeier

University of Essen, Germany

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. Benoit B. Mandelbrot

"Why is geometry often described as ‘cold’ and ‘dry’? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline,or t-rees. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in astraight line. More generally, I claim that many patterns of Nature are so irregular and fragmented, that, compared with Euclid - a term used(...) to denote all of standard geometry - Nature exhibits not simply a higher degree but an altogether different level of complexity." (2,Introduction).

When Benoit B. Mandelbrot (2) introduced the term fractal geometry in the middle of the seventies, probably nobody suspected the revolutionary effects of this new viewpoint on the sciences.

The term fractal (from the Latin fractus - irregular) is nowadays applied both to natural objects as clouds, mountains, plants and tomathematical things like the Koch curve, the triangular Sierpinski gasket, the Mandelbrot set or to physical phenomena (in space and time)like chaotic attractors, fractal growth patterns and to other phenomena in natural sciences, in medicine, astronomy and further research fields(see (4), (7)).

1. Theory of fractals - What are fractals

One way to define "fractal" is as a negation: a fractal is a set that does not look like an euclidean object (point, line, plane, etc.) no matter howclosely you look at it. Imagine focusing on a smooth curve (imagine a piece of string in space) - if you look at any piece of it closely enough iteventually looks like a straight line (ignoring the fact that for a real piece of string it will soon look like a cylinder and eventually you will seethe fibers, then the atoms, etc.). A fractal, like the Koch Snowflake, which is topologically one dimensional, never looks like a straight line, nomatter how closely you look. There are indentations, like bays in a coastline, look closer and the bays have inlets, closer still the inlets havesubinlets, and so on.

"Fractal" is a term which has undergone refinement of definition by a lot of people, but was first coined by B. Mandelbrot and defined as a setwith fractional (non-integer) Hausdorff dimension. While this definition has a lot of drawbacks, note that it says nothing about self- similarity -even though the most commonly known fractals are indeed self- similar.

So generally a fractal is an object which can not be completely described by means of Euclidean geometry - it has too much structure. Viszeck(7) notes on spatial fractals: "In general, we call a physical object fractal, if measuring its volume, surface or length (...) it is not possible toobtain a well converging measure for these quantities (...). "

Thus for example the coastal line of an island can be interpreted as a fractal: If one wants to measure the length of a coast, then one will seethat with finer scale the coastal length increases (the more rugged, the longer) - so the length is no suitable measure for this object, because itdepends on the scale.


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Fig. 1: The modern version of a Lichtenberg figure.

In order to describe the phenomenon fractal mathematically, numerous concepts were developed, which led to a new scientific discipline -named fractal geometry or theory of fractals.

In physics both geometrically arranged structures, which can be satisfyingly described with the help of Euclidean geometry, and unorderedstructures, which can be observed in far-from-equilibrium growth phenomena and which are common in many fields of science andtechnology exist. These patterns are examples for fractals, their structures in space and time can be described by the theory of fractals.

For the characterization of these objects we can use the terms of fractal geometry: the fractal dimension and the self-similarity of fractals.

1.1 Fractal Dimension

The fractal dimension represents (beside the self-similarity) the most substantial characterisation of a fractal. Barnsley writes: "fractaldimensions (...) are attempts to quantify a subjective feeling which we have about how densely the fractal occupies the metric space in whichit lies. Fractal dimensions provide an objective means for comparing fractals." (1, p.172)

Fig. 2: Fractal curves with different dimensions

"Fractal dimensions are important because they can be defined in connection with real-world data, and they can be measured approximately bymeans of experiments. For example, one can measure ‘the fractal dimension’ of the coastline of Great Britain; its value is about 1.2. Fractaldimensions can be attached to clouds, trees, coastlines, feathers, networks of neurons in the body, dust in the air at an instant time, the clothesyou are wearing, the distribution of frequencies of light reflected by a flower, the colours emitted by the sun and the wrinkled surface of thesea during a storm. These numbers allow us to compare sets in the real world with the laboratory fractals." (1, p.172)


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In the mathematical sense a fractal is basically to be understood as a quantity (or a set), which possesses a topological dimension DT embedded

in the Euclidean space Rn. As for example a straight line and a 'zigzag curve ' have the same topological dimension ‘one’, this kind ofdimension does not allow a differentiated description of fractal patterns.

A common mathematical definition of the fractal dimension D comes from Hausdorff and Besicovitch (see (2)): the so called fractional orHausdorff-Besicovitch dimension, a positive non-integer number with D > DT. Mandelbrot writes: "A fractal is by definition a set for whichthe Hausdorff-Besicovitch-dimension strictly exceeds the topological dimension. (...) Every set with a non-integer D is a fractal." (2, p.65)

We might say the fractal dimension interpolates between the topological dimensions. In the geometrical sense for example it indicates for 1 <D < 2 how smooth or rough a curve turns out. For D » DT = 1 the curve is not structured, it resembles a distance or a straight line.

The larger D becomes with D > DT, the more structured the form of the curve will be, a possible approximation with straight lines will bemore and more difficult. If D finally converges to almost the value DT + 1 = 2, then the curve possesses such a structured shape that it couldbe understood as an object with a topological dimension of two, the curve almost becomes a surface and fills out the space in which it liescompletely (see Fig. 2).

Each fractal possesses a fractal dimension but basically only mathematical fractals can be described well by the fractalHausdorf-Besicovtich-dimension. A lot of methods exist for determinating or computing the dimension of natural or physical fractals, whichcan be understood as more or less numerical ‘estimations’ of the real Hausdorff-Besicovitch-dimension. These methods or the computedfractal dimensions are called effective dimensions (named after Mandelbrot).

To compute natural fractals the following effective dimensions are at present mostly used in physics: the box counting dimension or boxdimension, the correlation dimension and the mass dimension.

For the estimation of effective dimension of mathematical fractals are helpful: the original Hausdorff-Besicovitch dimension, the(self-)similarity dimension (for self-similar fractals) and the capacity dimension (and a lot of other ones).

1.2 Self-Similarity

In contrast to the fractal dimension which characterizes each fractal, not every fractal has to be self-similar. Descriptive formulatedself-similarity means that if you enlarge a part or a sections of a self-similar fractal this enlargement 'resembles' the original fractal structure.Or: In different 'zones' the basic structure of the total object is preserved, these fractals are scale independent. One could call this characteristicof self-similar fractals a ‘structure preservation'.

Most of the mathematical fractals are self-similar. Usually they can be described by iterative algorithms, the resulting fractal (after infinityiterations) has structures on all scales. Natural or physical fractals are usually self-similar only over a few scales and they are self-similar onlyin a statistical way.

For example if one regards a fern (in biology), then the observable self-similarity is available only over some few orders of magnitude: Usuallythe leaf of a leaf of a fern already does not already possess the fan-like structure of the total fern. In addition, many other natural fractals, forexample the form of diverse plants or leafs, fractal mountains or coastlines can be described in their apparently complex structures by meansof self-similarity.

The emerging of self-similarity even enables us to simulate nearly authentic natural patterns or objects. Famous computer methods are the IFS(developed by M. Barnsley, see (1)) which for example can generate a (almost) perfect simulation of a fern and the L-systems (developed byLindenmayer, see (3)) which for example allow us to create whole bushes and trees with leaves.

2. Fractals of spatial structure and form - fractal growth phenomena

Nature surrounds us with a countless number of shapes and forms. Yet, even today, relatively little is known about how these shapes andforms are created. Nevertheless, we can make some elementary observations about these shapes and forms. For example, the shape of anyobject tends to depend strongly on the nature of the process from which it originated. As described above fractals were invented by Benoit B.Mandelbrot (only about twenty years ago) to characterise complex shapes and patterns in nature and natural sciences. The history of fractals isshort, but fractals have already been applied throughout diverse fields of science. Fractals are now indispensable in most natural sciences (andeven in school). In physics one of the main fields in which fractal geometry is widely applied are far-from-equilibrium growth phenomena (see(9)). A broad class of growing patterns can be characterized by an open branching and (statistical) self-similar structure. Such objects can bedescribed in terms of fractal geometry.

Examples for fractal structures can be observed in various experiments on growth of unstable interfaces including viscous fingering inHele-Shaw-cells, dielectric breakdown or electrochemical deposition.

This hands-on workshop will introduce you to the fractal geometry as a topic of natural and physical sciences. You will have the opportunity


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to handle easy to do 'hands-on' experiments (in order to show fractal growth) und you will take home a Hele-Shaw-cell, littleelectrodeposition-cells and probably some other examples of real fractals.

2.1 Solnhofen chalks

Fig. 3: MnO2 dendrites grown on Solnhofen chalks. One of the most familiar examples of fractal patterns are the manganese oxide / iron oxide ‘pseudo-fossils’ (the so called MnO2 dendrites,Eisen-Mangan-Abscheidungen) that are found at the joint surfaces in sedimentary rocks such as the Solnhofen chalks (SolnhofenerPlattenkalke) (see Fig. 3).

2.2 Dielectric breakdown - steel balls in oil

Fig. 4: Petri-dish with ring electrode and position of the tip elektrode.

A round glass plate (petri-dish) with a diameter of approximately 14 cm is filled up to about two millimeters with a non leading, quite viscousliquid. For this attempt castor-oil (Rizinusöl) is suitable, whereas Glycerin or silicone oil are not. On the interior rim of the petri-dish anaccordingly long metal strip or ring (e.g. of copper) is attached and contacted with a well leading junction point for the link of a cable (see Fig.4). During the experiment this ring electrode is grounded. Small steel balls (diameter max. 1-2 millimeter) are now distributed (e.g.statistically) into the castor-oil. With the experimental setup described here about 500 balls are needed. (Later the in oil 'inserted' balls can be'rescued' with the help of a small magnet.)

You position the tip elctrode over the center of the surface of the petri-dish (see Fig. 4, e.g. a sharpened metal bar of some millimetersdiameter). This electrode is given a potential of approx. 15 to 20 kV with the help of a suitable voltage supply (such a high voltage devicemight be available in well equipped school collections.) For working with such high voltages one has to pay attention that the ring electrodeand the experimenter are sufficiently grounded and that the tip electrode doesn’t touch the castor-oil. In order to avoid sparks(Funkenüberschläge) the electrode should not be placed nearer than about a centimeter to the surface of the liquid. Because of voltagedifference between tip and ring electrode charges will now be 'sprayed' down on the oil. You can hear a hissing noise and see how the oilsurface is indented underneath the tip electrode by the ‘charge rain’. The well leading steel balls come into motion and thus they try to gather


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and transport (over connected balls) the charges to the grounded ring electrode. Meanwhile a spontaneous pattern formation takes place, thesteel balls form chains which are fractaly branched in a relatively short time (like the Lichtenberg figures mentioned above or the dendriticiron-manganese deposits on the Solnhofen chalks, see Fig. 1 or Fig. 3). Here one can observe fractal growth. The fractal growth process canbe particularly impressively observed, if the steel balls are distributed (evenly) at the rim of the petri-dish. For good experimental results about700 to 800 balls are needed. The dendrites grow now from the border of the plate towards the tip electrode (see Fig. 5).

Fig. 5: Evenly distributed steel balls at the rim grow to fractal patterns.

2.3 Electrochemical deposition

The formation of fractal patterns can also be observed by the growth of metallic forms through electrochemical deposition. Depending on thematerial being deposited and the electrochemical conditions, a wide variety of patterns can be generated. The principle experimental setupcorresponds to the experiment presented in the previous paragraph, to the steel balls in oil. The petri-dish (prepared with the ring electrode) isnow filled with an ionic solution. It is to be made certain that ring and solution are of the same metal, since the ring electrode serves as ananode (so ions are separated from the metal into the solution). Copper/copper sulfate (Cu/CuSO4) or zinc/zinc sulfate (Zn/ZnSO4) aresuitable.

The tip electrode is attached in the center of the petri-dish as in the above experiment (see Fig. 4, Fig. 6). However it is now immersed into theliquid and serves as a cathode.

Fig.6: Electrodeposition.cell Fig. 7: Pattern formation during the electro-chemical deposition of zinc (Zn/ZnSO4).


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Fig. 8: Fractal growth at a ring and a parallel attached cathode.

By applying a voltage of some volts (5V to 15V, depending upon the size of the dish) small deposits can already be observed at the cathode

after a short time (see Fig. 7). The cations (e.g. Zn2+) move to the cathode and deposit themselves there. Because the solution and theelectrodes are from the same material, the concentration reduction is levelled by the slowly decomposing anode. The depositions increase fastto a fractal branched fractal structure, which has a statistical self-similarity. The applied voltage should not be selected too high (<15V), because otherwise thermal effects can destroy the fine fractal branches.

It is also possible to control the growth process by means of constant currents (for instance 0.2A to max. 0.5A for the above mentioned setup).So it is possible to control the growth rate directly: the more current, the faster the patterns are formed. The depostion rate and the ionicconcentration are also the key control parameters for these experiments.

The described experimental setup can be amended in various ways (see Fig. 7): the electrodeposition with parallel attached electrodes is alsoparticularly interesting - here fractal 'hedges' grow (see Fig. 7). Almost two-dimensional electrodeposition experiments can be carried out underbetter controlled conditions using thin cells (see Fig. 9, Fig. 10).

Fig. 9: Thin radial and longitudinal electrodeposition cell.


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Fig. 10: Different growth patterns of the electrodeposition in thin cells.

2.3 Viscous fingering

A further example for fractal growth behavior can be observed in hydrodynamics: if a low viscous fluid displaces a more viscous one, theinterface between these fluids can have a very complex geometry. For the observation and investigation of such processes the British shipengineer Henry S. Hele-Shaw already developed equipment (the Hele-Shaw-Cell) briefly before the turn of the century (1898). Thisequipment permits us to observe viscous fingering in a thin layer in an almost two-dimensional geometry.

Fig. 11: The Hele-Shaw-cell.

Fig. 12: Different growth patterns at the viscous fingering

The Hele-Shaw-cell (see Fig. 11), consists of two parallel plates (e.g. plexiglass), which are separated by a narrow gap from each other. A veryviscous fluid is included between these plates, another less viscous liquid is now injected into it. After a very short time the interface betweenthe fluids gets instable and (statistically) self-similar fractal branched patterns (the Hele-Shaw-fractals) will grow into the more viscous fluid.

The control parameters of the viscous fingering are the viscosities of the fluids, the interface tension, the injection rate, the wetability and thegeometry of the plates and the geometric anisotropies.

3. Fractal dimension and self-similarity of natural and physical fractals

The Hele-Shaw-fractals are particularly suitable for the introduction to the basic concepts of fractal geometry: both the mass distribution andthe irregular border of these objects can be characterized by means of fractal dimension, besides the growing patterns are self-similarconcerning their growth and the development of the individual branches.

3.1 Mass dimension

As pointed out the fractal dimension gives a response to the question: How densely does the regarded object occupy the surrounding space inwhich it is embedded?


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As the length of a coastline depends on the used length-scale, the occupied area of a Hele-Shaw-fractal isn’t a suitable quantity to characterizeits form or structure because the measured area is also scale dependent. Instead of examining the area or surface of these patterns, it isphysically more meaningful to regard the mass distribution - both sizes are equivalent in this experiment. If one now regards the massdistribution M(r) of the approximately radial-symmetrical Hele-Shaw-figures as a function of the radius vector r then the following relation is

obvious:

The exponent DM can then be interpreted as a fractal dimension - the mass dimension. You get the following equation for these mass fractals

with DE as the Euclidean space dimension (DE = 2 in R2): .

Fig. 13: Viscous fingering patterns with different fractal dimensions

You may then interpret mass fractals as a set of cohered or connected points, so their fractals dimension exceeds their topological dimension.While a distance (or a straight line, with ‘substance’) has a fractal dimension of DM = 1 because the mass increases linear with the distance tothe center, a circle (with DM = 2) behaves accordingly squarely. Viscous fingering patterns grow fractal with a non-integer mass exponent(see Fig. 13). (Computing the fractal box dimension gets the same results, see e.g. (7), (9).)

3.2 Border dimension

Not only a coastline but also the border of a viscous fingering pattern can be regarded as a fractal curve. In that case the fractal dimension tellsus something about the irregularity or the roughness of the object. The following figure (Fig. 14) shows the mass and rim dimensions ofdifferent Hele-Shaw-fractals (computed as box dimensions). Both dimensions have nearly the same value (DM » 1.7) when the patterns arevery crumpled. This corespondents to the fractal dimension of a simulated growth pattern (DLA-Simulation in Fig. 14 on the right, see(7)-(9)).


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Fig. 14: Mass and border dimensions of different Hele-Shaw-fractals.

3.3 Self-similarity

Fig. 15: Statistical self-similarity of a Hele-Shaw-fractal..

While the terminology of self-similarity is accurately applicable to mathematical fractals (as for example to the Koch curve) it is more difficultto apply to natural or physical fractals. With the Hele-Shaw-fractals however the characteristic of properties of self-similarity can be shown ina very descriptive way: both the individual branches and parts of these branches (etc..) are similar to other branches or to parts of otherbranches (in the statistical sense). If one cut a branch out and increases or reduces it accordingly then it could hardly be differed from theorigin branch or other parts of the total pattern (see Fig. 15).

The viscous fingering patterns have a further self-similarity: as a circle always resembles another circle (circles can be transferred simply intoanother by reduction or enlargement) so the different growth stages of the Hele-Shaw-fractals are self-similar in time. By scaling shots ofindividual growth stages to the same size (see Fig. 16) the absence of a 'natural' scale becomes quite particularly clear: after exchangingindividual shots no one could say (at first sight) which picture showed the older or younger Hele-Shaw-fractal.

The global structure of these patterns is temporally preserved during the growth process (by means of the fractal geometry). This can beshown with the help of the fractal dimension which remains constant with growth.

4. Pattern formation far from equilibrium - how fractals grow


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Fig. 16: Self-similarity in time: significant similarity between different growth stages of a Hele-Shaw-fractal at appropriate scaling. "The development of full understanding of the universe around us, in terms of the basic properties of fundamental ‘particles’ and theirinteractions, has long been a dream of the physicist. (...) during the past one or two decades, substantial progress has been made (...). To thelarge extent, this progress has been made by giving up the idea that an understanding of complex systems can be based on an ever moredetailed knowledge of their microscopic components and focusing instead on the ‘universal’ properties that all materials possess in common,irrespective of their atomic and molecular structure (...)." (9, preface)

The pattern formation of fractal growth phenomena could be discussed on the basis of partial differential equations (boundary value problemsof potential equations, solution of the Laplace equation etc.), but this mathematically very difficult methods represents an insurmountable wayfor the understanding of the nevertheless quite descriptive micro-molecular processes of the growth phenomena for the use in physicseducation.

The elementary processes of pattern formation during fractal growth are characteristic for the cooperation of law and coincidence, a typicaleffect of nonlinear systems which can lead to the reinforcement of coincidental fluctuations (see (5)).

4.1 Electrochemical depostion

If one e.g. regards the electrodeposition process and observes the individual diffusing ions (e.g. the Zn2+- cations) at the beginning of the

experiment - as through a strongly magnifying glass -, then some ions are immediately deposited as zinc (Zn2+ + 2e ? Zn) at different placesat the cathode due to the electrical force. So (statistically distributed) smallest 'hills' and 'valleys' grow at the surface of the cathode. The furthersequentially arriving ions prefer to depose themselves at the hills. The probability they will go (or diffuse) deep into the valleys becomessmaller with increasing growth. You can say: 'hills grow faster than valleys '. The growth of the individual hills is now affected by statistics. Ifa particle deposits at the edge of the tip of a hill then there will be a little deformation which will carry on growing more then the rest of thehill. If another particle deposits at another ‘edge’ of this tip, there will be a growth instability. As a consequence the further growth of the hillwill take place at the individual finest tips - so the tip of the hill gets some little branches (or tips) - and so on (this process is called tip splitting). For the simulation of this simple aggregation process Witten and Sander (8) suggested 1981 the DLA-Modell (diffusion limited aggregation), which can be implemented very efficiently and impressively as a computer experiment (see Fig. 17).

Fig. 17: Computer simulations to the DLA-Modell.


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4.2 Viscous fingering

At first sight the observed flow figures of the Hele-Shaw-cell probably do not correspond to the expectations of the observer - why isn't themore viscous medium displaced evenly and continuously (circularly) by the injected fluid? (If one exchanges the viscosities and injects themore or at least equally viscous fluid into the second less viscous liquid, then a circular surface will grow as expected.)

Actually the border front starts moving evenly in the initial stage of growth (because of the pressure of the injected fluid). The surface tensionkeeps the boundary surface stable. It would be energetically inconvenient to indent the border front under the effect of surface tension -boundary surfaces are therefore generally also minimal surfaces. The described behavior however changes rapidly: with increasing growth theboundary cannot withstand the high pressure of the flowing liquid and (due to coincidental fluctuations) smallest indents spontaneously form.In these places the front is broken through - you might say ‘the enclosed fluid runs out’. Thus individual fingers grow into the more viscousmedium. Each child knows the phenomenon of the glass of juice or the plate of soup that is tool full: the boundary surface first curves beyondthe rim of the glass or plate and then it suddenly breaks through and overflows. Schlichting (6) compares this with the bursting of a dike:"Auch in diesem Fall gibt der Deich den anstürmenden Wassermassen nicht als Ganzes nach, sondern an Stellen, die z.B. aufgrund einesMaulwurfbaus äußerlich nicht wahrnehmbar gestört sind." (Translation: Also in this case the dike does not give way to the attacking water asa whole, but maybe at places which are outwardly not perceptibly disturbed, e.g. due mole burrowing."

However the fluid cannot completely run out, because the growth front is locally kept round by the surface tension. On the one hand theliquid experiences an increased counter-pressure in the places in which the boundary surface is most strongly curved. On the other hand theforce necessary for the displacement rises with the increasing boundary surface (a high viscosity can be interpreted as a strong internalfriction). You can see by more exact observation that growth only takes place at the tip of the fingers and the adjustment of the fingers doesn’tchange anymore. The fingers serve as channels for the friction-weak flow of the injected fluid towards the actual growth zones.

The growth at the tips of the fingers works similarly: The boundary surface does not evade to the pressure of the flowing fluid as a whole andthe increasing growth instability leads to the emergence of smallest indents. In this critical state individual fingers grow into the more viscousmedium. This type of self organizing leads to further growth instabilities and thus to constant new tip splitting of the individual fingers - theviscous fingering. The surface tension prevents the fingers from suddenly increasing boundlessly and from boundless tipp-splitting of thesefingers.

It appears that the elementary processes and mechanisms of viscous fingering are affected by the cooperating of law and coincidence -similarly to the electrolytic deposition.

Literature

Michael Barnsley: Fractals everywhere. Boston Academical Press, Inc., 1988.1.Benoit B. Mandelbrot: The fractal geometry of nature. W.H.Freeman and Company, New York, 1983. (german Translation: Die fraktale Geometrieder Natur. Birkhäuser Verlag, Basel u.a., 1987.)

2.

Przemyslaw Prusinkiewicz, James Hanan: Lindenmayer Systems, Fractals and Plants. Lecture Notes in Biomathematics, Nr. 79, Springer Verlag, NewYork, 1989.

3.

H. J. Schlichting: Schöne fraktale Welt - Annäherung an ein neues Konzept der Naturwissenschaften. In: MNU 45/4 (1992) S.202.4.H. J. Schlichting: Energie, Entropie, Synergie - Ein Zugang zur nichtlinearen Physik. In: MNU 46/3 (1993) S.138.5.H. J. Schlichting: Fraktales Wachstum, .. .In: PhidS, 3/31 (1993) S. 113.6.Tamás Vicsek: Fractal Growth Phenomena. World Scientific Publishing, Singapore, 2. Aufl., 1992.7.T. A. Witten, L. M. Sander: Diffusion-Limited Aggregation - A Kinetic Critical Phenomenon. In: Phys. Rev.Lett. 47 (1981) S.1400.8.P. Meakin: Fractals, scaling and growth far from equilibrium. Cambridge University Press, Cambridge, 1998.9.

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