nature’s geometry

NATURE’S GEOMETRY

SOUMITRO BANERJEE∗

1. Introduction

Geometry essentially deals with two typesof entities: objects and spaces. Points, cir-cles, lines, curves, cylinders, tetrahedrons— these are examples of the objects onefinds in geometry books. These objectslive in spaces that hold them. A point(a zero-dimensional geometrical object) canrest on a one-dimensional line or a two-dimensional plane. A curve (a one- dimen-sional geometrical object) can rest on a 2-Dsurface or in a 3-D volume. We thus haveobjects embedded in spaces of dimensiongreater than or equal to that of the objects.

For much of human history, geometersmajorly studied idealized objects with reg-ular shapes: triangles, circles, spheres andrectangular parallelopipeds. This was thelegacy of Pythagoras and Euclid. The em-bedding space was always thought to beEuclidean: a flat piece of paper in case of2-D space, a volume with ‘flat’ characteris-tics in case of 3-D space etc.

Towards the end of the 19th cen-tury, there was a revolution in geometry.Lobachevsky, Riemann and others showedthat space can also be curved, and thenEinstein found a profound application ofthe new geometry in his theory of gravi-tation. At least in the domain of spacesthe narrow confines of Euclidean space wasbroken.

But objects still remained Euclidean. Westill talked about angles contained in trian-gles in Euclidean space and non-Euclideanspace. The objects of study in geometry

∗Dr. Banerjee is in the faculty of the Department ofElectrical Engineering, I.I.T. Kharagpur - 721302.

continued to be idealized ones that one canthink up in one’s head but never finds innature.

Nature abounds in irregular objects.“Mountains are not cones”, as Benoit Man-delbrot, the founder of the new geometryput it, “clouds are not spheres, lightningsare not straight lines”. Towards the end ofthe twentieth century we seem to be break-ing out of the compartment of Euclideanobjects. Geometers are now consideringthese irregular objects as valid subjects ofstudy. And that is what fractal geometry isall about.

2. Euclidean objects versusnatural objects

What distinguishes natural objects fromEuclidean objects? The first thing thatcomes to one’s mind is the irregularity ofthe shape of natural objects. But that needsto be specified in mathematical terms.

If we take a very small length of a curve,say y = f(x), it approximates a straight line.The closer we look at it, the more it losesits structure. That is how we can define thefirst derivative as a limiting value of the rateof change. Same is the character of Eu-clidean surfaces—they smoothen out intoflat planes as you take a close look.1

Here lies the main distinction betweennatural objects and the idealized objects.Natural objects never flatten out—at what-

1The derivative is

dy

dx= lim

∆x→0

∆y

∆x

This limit would exist only if the curve smoothens outinto the tangent as ∆x→ 0

Breakthrough, Vol.13, No.4, January 2009 1

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ever level of magnification you may look atthem. Think of any natural surface — thesurface of your skin or that of a tree trunk.They contain complexities within complexi-ties which come in view at higher levels ofmagnification. These are continuous butnondifferentiable surfaces.

This applies to some data sets also. Noeconomist ever tries to differentiate thecurve for share prices or exchange rates. Noelectrical engineer would think of differen-tiating the curve representing the load on apower station. One cannot take the deriva-tive of the curve for oscillation during earth-quakes. The reason is, these curves as ge-ometrical objects, are continuous but notdifferentiable anywhere. They reveal moreand more rich forms as you zoom closer.

The next major difference is illustrated byMandelbrot’s famous question, “how long isthe coastline of England?”. A little thoughtmakes it evident that the measured lengthof the coastline depends on the yardstickof measurement. If the yardstick is long,much of the detail of coastline geometrywould be missed. As smaller and smalleryardsticks are used, the creeks and bendscome in view and the measured lengthincreases. And in the limit—when theyardstick length shrinks close to zero—thelength of the coastline becomes infinitelylarge.

Yet the area of England is finite. Thus thecoastline is a curve of infinite length enclos-ing a finite area.

Same is the case of all 3-D natural ob-jects enclosed by natural surfaces. Takethe structure of our lungs as an example.The task of the lungs is to absorb oxygenfrom air. In order to have a high absorptionrate, the surface area needs to be very high.Yet, the volume must be small—it has to beaccomodated within the rib cage. Thus ithas the same geometrical characteristics asthe coastline. You can multiply examples

just by looking around yourself.These objects, evidently, need a new

mathematical tool for characterization.There comes the question of dimension.

3. Dimension of geometricalobjects

It is strange that in school and college levelmathematics curricula we never learn what‘dimension’ means in geometry. Our con-ception, naturally, is mostly derived fromcommon sense: if we have a line-like objectwe say it is one dimensional, if we have asurface-like object we say it is two dimen-sional and so on.

We are accustomed to thinking of dimen-sions as integers, and have considerabledifficulty in visualizing anything otherwise.The ancients had the same sort of difficultywhen the only numbers they knew werenatural numbers. Three horses, ten men,twentifive bananas — such numbers camenaturally to them. But finally they had toshake off the narrow confines of integersand conceive fractions — in order to rep-resent length, area, weight etc.

It now appears that we again have toshake off the notion of integral dimensionsand conceive fractional dimensions whenwe face the task of representing natural ob-jects.

It must first be understood that the di-mension of any object and that of the em-bedding space are two different quantities.The dimension of the embedding space isgiven by the degree of freedom. In 1-Dspace one can move only left or right, in 2-D space one can move left-right as well asfront-back, in 3-D space one can also moveup-down. The embedding dimension, natu-rally, has to be an integer.

The dimension of an object can not be de-fined in a similar manner. It has to be de-fined according to the way it fills space. Toprobe the question, let us take a simple Eu-

2 Breakthrough, Vol.13, No.4, January 2009


Figure 1: To find how a triangle fills space, it is covered by a grid and the grid size issuccessively reduced. The boxes required to cover the object is shown in shade. In thelimit we find that the triangle fills space in the same way as the square. But the boundaryof the carbon particle seen in the electron microscope photograph does it in a differentway.

clidean object: the square. We know that itis two dimensional. But how do we obtainthe number 2 from the structure of this ob-ject?

In order to see how it fills space, wecover it with a grid as shown in Fig.1. Ifthe square has 1 cm sides and the dis-tance between two consecutive grid lines is1/10 cm, then 100 grid elements would benecessary to cover the square. If we nowreduce the grid length by half, 400 grid el-ements would be required. We see that thenumber of grid elements required to coverthe object increases as the square of the re-ciprocal of grid length.

N(ε) =(

1ε

)2

where ε is the grid length and N(ε) is thenumber of grid elements required to coverthe object (a function of ε).

If the object taken is a right angled trian-gle, the count of covering boxes finally con-verges to

N(ε) =12

(1ε

)2

And for a circle we have

N(ε) =π

4

(1ε

)2

To generalize, we can write

N(ε) = K

(1ε

)2

where K is a constant.We can extract the dimension (2 in this

case) from it as follows:

lnN(ε) = lnK + 2 ln1ε

2 =lnN(ε)

ln 1ε

− lnKln 1

ε

The second term would vanish as ε → 0.Thus the dimension D of the object is givenby

D = limε→0

lnN(ε)ln 1

ε

If instead of the known Euclidean objectswe take some natural object like a carbonparticle or a map of the Andamans, andsubject it to the above procedure, we wouldfind that the dimension turns out to be afraction. This is a general characteristicsof all natural objects as distinct from ideal-ized objects. The fractional dimension thusprovides a method of characterizing naturalobjects. In fact, such objects are defined bythis property.

Geometrical objects with fractional di-mensions are called Fractals.



4. What use is fractaldimension?

How do we quantify the fractal character ofan object? What experimental procedurewould enable us to judge if two dissimilarfractal objects are closely related or geomet-rically equivalent?

Looking at any object, we always have aintuitive feeling about how densely the ob-ject occupies space, how crooked, twisted,broken it is. Looking at a curve plottedfrom a set of data, we do feel how wavy or“noisy” it is. But these are subjective feel-ings. We need a concrete objective method-ology to assess this quality. Measurementof the fractal dimension provides the meansto achieve this end.

The method can easily be guessed fromthe definition of dimension given in the lastsection. We divide the embedding spaceinto a number of equal “boxes”. In case of2-D space, we divide the sheet of paper intosmall squares as in a graph paper. In caseof 3-D space we divide it into cubes.

Then we count how many of these ele-mental boxes are required to cover the ob-ject. This is our N . The side of the box isε. Subsequently, reduce the size of the boxin steps and repeat the procedure of count-ing. Then plot D versus ε. The plot approxi-mates a horizontal straight line as ε tends tozero. When a reasonable approximation isobtained, the point where it cuts the y-axisgives the dimension of the object (Fig.2).

But what use in this new piece of infor-mation? It actually quantifies the surfacecharacteristics. And if any of the propertiesof the body is determined by the character-istics of the surface, fractal dimension cansupply vital information.

For example, the fractal dimension of thesurface of the carbon particles in auto-mobile exhaust is related to the afficacywith which the particle will attach with thebreathing ducts and cause harm.

1.0

2.0

Dim

en

sio

nEpsilon

straight line

square

parabola

trianglefractal object

Figure 2: Measurement of the fractal di-mension of a natural object.

When two surfaces make electrical con-tact (as in a switch), all the points nevertouch each other. The amount of actualelectrical contact is determined by the frac-tal character of the surfaces, and that isquantified by the dimension of the two sur-faces.

Some silt particles float in river water andsome settle quickly. The precipitation obvi-ously depends on the size and specific grav-ity of the particles. But these two parame-ters being equal, the more crooked the sur-face, the more it will get carried by flowingwater. This property is again quantified bythe fractal dimension.

When metals, semiconductors or alloyscrystallize, the crystal grains have bound-aries that are fractals. Scientists havefound a number of properties of such mate-rials that are related to the fractal characterof the grains.

Any surface is fractal. However muchyou may polish and smoothen a surface,at some level of magnification irregularitiesmust show up. And the light absorptionproperty of the surface is determined notonly by the property of the material but also



by the character of the surface. This, again,can be quantified by fractal dimension.

When rock structures form in various ge-ological processes, the characteristic sig-nature of the formative process is left inthe geometry of the rocks. And theseare nothing but fractals. You may lookat large mountain ranges, large boulders,small pebbles or miniscule grains that be-come visible when you venture inside therocks by cutting them — you see structuresinside structures — that wait to tell youmuch about their character.

The afficacy of solid catalysts in help-ing or retarding a reaction depends on howmuch surface area is exposed to the reac-tants. Same is the case for the electrodesin electrochemical reactions. None of thesesurfaces are smooth, and it is easy to seehow their behaviour would be related to thefractal dimension of the surfaces in ques-tion.

The veins in the plant leaves, similarly,are fractals. The arteries and veins in yourbody are fractals. Tumours and cancer cellsare fractals. The economists’ data set forinflation rates are fractals. And in all thesecases, the fractal dimension gives a directmeasurement of the character of the object.

5. Mandelbrot and Julia sets

Can we generate geometrical objects withsuch characters mathematically? BenoitMandelbrot considered this problem firstand came up with a solution. He found thatsuch structures can not be generated by theknown mathematical procedures of writingequations and functions. For it, one has tofollow the procedure of repeating the sameoperation again and again.

Suppose there is a system whose status ischanging continuously. In scientists’ per-lance it would be called a dynamical sys-tem, and it is easy to appreciate that all nat-ural systems are dynamical systems. Sci-

entists try to express the changes or “dy-namics” by equations so that the status atany point can be computed from its history.

Generally the dynamical equations maybe quite complicated. But simplified ver-sions often reveal properties observed incomplicated systems as well. Let us take asimple dynamical system expressed by theequation:

Zt+1 = (Zt)2 + C

where Zt is the state of the system at thet-th instant and Zt+1 is the state at the nextinstant. The Z ’s and C are complex num-bers which have a “real part” and an “imag-inary part” expressed as a + ib; and i isthe “imaginary” number

√−1. Such com-

plex numbers can be plotted as points ona “complex plane” with the real part alongthe x-axis and the imaginary part along they-axis. Dynamics of the complex numberZ can then be viewed as the changing po-sition of a point on a sheet of paper — thecomplex plane.

If we choose a value of C and a startingpoint Z0, we can calculate subsequent val-ues of Z and observe the dynamics. We findthat for some values of Z0 the system re-mains bounded, for some other values Z in-creases without bounds, i.e., runs towardsinfinity. If we now plot those values of Z0 forwhich the system remains within bounds,we get a set of points making up a picture.This turns out to be a fractal—the Julia Set(Fig.3).

We can also vary C while taking the start-ing point Z0 same in all computations. Wefind that for some values of the parameterC the system remains within bounds andfor some other values it doesn’t. We canagain plot on a sheet of paper the locationsof those values of C which make the sys-tem bounded. We again get a fractal—theMandelbrot set (Fig.4).

The Julia set and the Mandelbrot set havebecome sort of a universal representation of



Figure 3: Examples of Julia Sets

Figure 4: The Mandelbrot set. In this ge-ometrical object, however small part of thefigure you may want to observe, it revealsrich internal structures. The same is truefor Julia sets and all other fractal objects.

fractal geometry. Most books, articles andpopular expositions on fractal geometry be-gin with them and end with them, empha-

sizing their symmetry, beauty, complexityand all that, obtained from the simple equa-tion Zt+1 = Z2

t + C. And in most cases thebasic idea is lost: they are derived from asimplified representation of a natural pro-cess, a dynamical system.

Nature is full of dynamical systems. Andfor each one there would be certain param-eters and certain initial conditions. The les-son that we learn from Mandelbrot’s work isthat for every dynamical system there existfractals in the space formed by the systemparameters and in the space formed by theinitial conditions. Most dynamical systemsin nature, in fact, work on fractal geometry.

Let us take an example from engineer-ing. A ship is stable in the upright positionand tends to come back to this position ifslightly disturbed from it. But with exces-sive amount of tilt, it may capsize. Thus,depending on the initial condition of thetilt, its state may be bounded (upright po-sition) or can escape without bounds (cap-size). A real ship in an ocean would con-tinuously be bombarded by waves, and itsdynamics would depend on the intensity of



the waves (let it be denoted by F ). Now wewrite down the equation of motion of thesystem2, and find out which initial condi-tions keep the system bounded. We cannow plot these initial conditions in a planewith the tilt angle in the x-axis and the rateof change of tilt angle in the y-axis. We geta picture. If we now change the value ofF in steps and draw the picture again, wefind that it changes dramatically with in-creasing wave intensity. The boundary ofthe zone for stable initial conditions can notbe properly discerned and the initial condi-tions from which the system collapses getmixed up with it. The picture becomes afractal (Fig.5). While rocking in the wave,if the state of the ship ever touches any ofthe white dots, it will capsize. Thus frac-tal structures are of vital importance to en-gineers. And it is generated by the samemethodology as Julia sets.

6. Iterative Function Systems

There is an entirely different approach tothe problem of modelling real-life objectswith mathematical methods. Developed byProfessor Michael Bernsley of Georgia Insti-tute of Technology, USA, this method useselegant mathematical reasoning to generateimages of natural objects on the computerscreen.

Any black and white two-dimensional pic-ture is nothing but a set of black dots in awhite background. It can be viewed as a setof points in a 2-D space. Now, any point inthe 2-D space can be located with the helpof two real numbers: the first number giv-ing the distance that we go along the x-axis

2Neglecting unnecessary details, the equation maybe written as

x+ βx+ x− x2 = F sinωt

where β represents the frictional damping and wavesof intensity F strike the ship with a frequency ω. Forthe computations that generated the pictures, we tookβ = 0.1 and ω = 0.85

Figure 5: The set of initial conditions forwhich a ship remains stable becomes frac-tal under certain conditions. This is thereason for many cases of ship capsize.

and the second number for the distance wego parallel to the y-axis in order to reachthe point. Hence a sheet of paper, in theview of mathematics, is a space formed bytwo real numbers. Any picture drawn on apaper is nothing but a set of points in thatspace, that is, a collection of pairs of tworeal numbers.

How can you reach a point starting fromanother point? Here comes the concept offunctions. We know the functions of a realnumber, like f(x) = a.x+ b. Here, x is a realnumber and and f(x) is also a real number.So both the starting point and the destina-tion are elements of the real line (Fig.6). Thefunction (or mapping) carries a point on thereal line to another.

Figure 6: A function takes one point on thereal line to another.

There can be such functions in the 2-Dspace also — carrying a point on a sheet of



paper to another. Such a simple function ofthe form

x2 = ax1 + by1 + e

y2 = cx1 + dy1 + f

is called affine transformation.We can apply such transformations on all

the points of a figure. Thus, while we canget one point from another by application ofthe transformation, we can also get a wholegeometrical figure from another using thesame transformation. And by application ofthe transformation again and again we geta “sequence” of figures.

The question is, will such a sequence leadus anywhere? We know that a convergentsequence of numbers (like 1, 1

2 ,14 , ...) always

has a specific number as the limit point,and for any given number we can alwaysdefine a suitable converging sequence togive that number. If the sequence of fig-ures is made convergent, it will also havea particular figure as its limit. This is infact guaranteed by a theorem called “Ba-nach fixed point theorem”.

In order to get such a meaningful se-quence we only need to ensure that the se-quence of figures is convergent. That is en-sured if two points obtained from the trans-formation are closer to each other than theoriginal pair of points. Such transforma-tions are called contraction mappings.

Thus we can get any figure by suitablydefining a contraction mapping. For com-plicated constructions, we only have to de-fine a number of such transformations,each giving a figure. The resultant figurewould simply be the combination (or union)of these sub-figures.

In this method it is immaterial fromwhich figure we start. We can as well startfrom a square. On repeated applicationof the transformations (iterations) the fig-ure will gradually change shape before youreyes and “converge” on to the figure you

Figure 7: Successive application of thefunction system transforms a square into afern. All information about this immenselycomplicated picture is contained in just 24numbers. For any figure, it is possible todefine similar iterated function systems.

want. The collection of contraction map-pings that generate a particular picture iscalled the “iterative function system”. Thepicture of the fern shown in Fig.7 was gen-erated from a square, by iterating a func-tion system comprising four affine transfor-mations defined by a,b,c,d,e and f as fol-lows.

fn a b c d e f1 0 0 0 0.16 0 02 0.85 0.04 -0.04 0.85 0 1.63 0.2 -0.26 0.23 0.22 0 1.64 -0.15 0.28 0.26 0.24 0 0.44



It has been shown that all geometrical fig-ures that can be drawn on a piece of pa-per can be generated from such transfor-mations. We only have to find the trans-formations for any particular figure. Themathematics to do this has also been de-veloped. We can thus squeeze the informa-tion contained in pictures and photographsin the from of just a few numbers.

This has immense technological conse-quence. For example, when spacecrafts farin space send photographs, the picture isdivided into grids and the grey level in eachgrid has to be coded and transmitted toearth. For good pictures this needs longtransmission time as a huge amount of in-formation has to be sent. Now the fractalimage compression offers the possibility ofcompressing the image into a few numbersenabling very fast transmission. Programsfor coding the image into such numbers hasbeen developed and may soon become aninternational protocol for satellite transmis-sion.

7. Last Words

Much of the popular literature on fractalspresent it as a mathematical game of gen-erating fancy pictures in a computer. Incontrast, this article presents it as a neces-sary tool to model the real geometry of na-ture. In fact in recent years the discovery offractal geometry has caused a sea change inthe geometers’ approach. We are no longerthinking up “perfect” mathematical shapesand specifying apriori what nature must belike. We are now taking real lessons fromnature.

This has led to new ways of measur-ing the complexity of natural objects. Andwe find that the fractal dimension quan-tifies diverse phenomena in nature, whichhave real scientific and technological signif-icance.

In an attempt to model the natural ob-

jects scientists have discovered that verysimple rules, applied in an iterative way,can generate extremely complicated struc-tures. Does it hold a clue to the immensecomplexity of natural objects? It has al-ways intrigued scientists how the informa-tion stored in a single molecule—the DNA—can generate an unimaginably complex en-tity like a human body. Now we know it ispossible, mathematically.�


nature’s geometry

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