The Fractals in Nature

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The Fractals in Nature

MAT 351Spring 2006

Dr. Stephen C. Preston



Introduction

There is common something in the design of the natural objects. It is a principleof self-similarity and a fractal to express this mathematically.1 Natural objects can berealistically described with fractal-geometry method, where procedures rather thanequations are used to model objects. Fractal-geometry representations for objects are

commonly applied in many fields to describe and explain the features of natural phenomena. In graphics application, fractal representations are used to model terrain,clouds, water, trees and other plan life, feathers, fur, and various surface textures,sometimes just to make pretty pattern and art.2 In other study phenomena (not for visualization), fractal patterns have been found in the distribution of stars, river islandsand moon craters; in rain-field configurations; in stock market variations; in music; intraffic flow; in urban property utilization; in physical chemistry; and in the boundaries of convergence regions for numerical-analysis techniques.3

A fractal object has two basic characteristics: infinite detail at every point, and acertain self-similarity between the object parts and the overall features of the object.4 Theself-similarity properties of an object can take different forms, depending on the

representation we choose for the fractal. We describe a fractal object with a procedurethat specifies a repeated operation for producing the detail in the object subparts. Naturalobjects are represented with procedures that theoretically repeat an infinite number of times, and then Natural objects are displayed with a finite number of steps.

In order to provide a basis for the applications of fractals to experimental resultsthis paper started with a discussion of simple fractals and of the fractal dimension. Therelated concept of the scaling or similarity dimension is also discussed.

Fractal and The Fractal Dimension

Fractals were discovered and described by a French mathematician, BenoitMandelbrot. Mandelbrot offers the following tentative definition of a fractal in 19825:

A fractal is by definition a set for which the Hausdorff-Besicovitch dimension6 strictly

exceeds the topological dimension.

After this, he has retracted this tentative definition and proposes instead the following7:

A fractal is a shape made of parts similar to the whole in some way.

According to Mandelbrot, a neat and complete characterization of is still lacking.8 Thereare some problems to define the fractal such that "Dimension" is not unique; there areseveral ways to express the self-similarity of objects9; the first definition is too restrictiveand requires a definition of the term set, Hausdorff-Besicovitch dimension (D) andtopological dimension (DT), which is always an integer.10 It excludes many fractals thatare useful in physics.

A fractal object has two basic characteristics: infinite detail at every point, and acertain self-similarity between the object parts and the overall features of the object.11 The self-similarity properties of an object can take different forms, depending on therepresentation we choose for the fractal. In mathematics, a class of complex geometricshapes that commonly exhibit the property of self-similarity, such that a small portion of



it can be viewed as a reduced scale replica of the whole.12 To obtain this, we can select asection of the fractal for display within a viewing area of the same size. We then carry outthe fractal-construction operations for that part of the object and display the increaseddetail for that level of magnification. As we repeat this process, we continue to displaymore and more of the object’s detail. Because of the infinite detail inherent in the

construction procedures, a fractal object has no definite size. When the detail in an objectdescription is included more, the dimensions increase without limit, but the coordinateextents for the object remain bound within a finite region of space. Natural objects arerepresented with procedures that theoretically repeat an infinite number of times, and then Natural objects are displayed with a finite number of steps.Thus, we can characterize theamount of variation in object detail with a number called the fractal dimension. Fractalmethods have introduced the useful for modeling a very wide variety of natural phenomena.

An Expression for The Fractal Dimension

The amount of variation in the structure of a fractal object can be described with

a number D, called the fractal dimension, which is a measure of the roughness, or fragmentation, of the object.13 There are some methods to generate a fractal objects. Asone method of them, there is an iterative procedure that uses a selected value for D.

An expression for the fractal dimension of a self-similar fractal constructed witha single scalar factor, r, is obtained by analogy with the subdivision of a Euclidean object,shown in Figure 1 in Figure page. Figure shows that, with s=1/2, the unit line segment isdivided into two equal- length subparts. For the same scaling factor, the square is dividedinto four equal-area subparts, and the cube is divided into eight equal-volume subparts.For each of these objects, E D sn ⋅ =1 is obtained from between the number of subparts andthe scaling factor. In analogy with Euclidean objects, the fractal dimension D for self-similar objects can be obtained from D sn ⋅ =1. Solving this expression for D, the

fractal similarity dimension, we haveD=

( ) s

n

1ln

ln.14

For a self-similar fractal constructed with different scaling factors for the differentsubparts of the object, the fractal similarity dimension is obtained from the implicitrelationship

∑=

n

k

D

k s1

=1

, where sk is the scaling factor for subpart k15.In figure 1, we considered subdivision of simple shapes. If we had more

complicated shapes, including curved lines and objects with nonplanar surfaces,determining the structure and properties of the subparts is more difficult. For generalobject shapes, we can use topological covering methods that approximate object subpartswith simple shapes. A subdivided curve, for example, could be approximated with smallsquares or rectangles. Covering methods are commonly used in mathematics field todetermine geometric properties, such as length, area, or volume, of a complex object by



summing the properties of a set of smaller covering objects.16 Topological covering concepts were originally used to extend the meaning of

geometric properties to nonstandard shapes. An extension of covering methods usingcircles or spheres led to the notion of a Hausdorff-Besicovitch dimension, or fractionaldimension. The Hausdorff-Besicovitch dimension can be used as the fractal dimension of

some objects, but in general, it is difficult to evaluate. More commonly, an object’s fractaldimension is estimated with box-covering methods using rectangles or parallelepiped.The area inside the large irregular boundary can be approximated by the sum of the areasof the small covering rectangles.17

Molecular Fractal Surface

Using above methods, the surface can be rough and even fractal down to themolecular size range. 18 Adsorbing molecules of different size on the surface and“counting” their number may measure the area of surfaces in a Hausdorff-Besicovitchdimension. Surface areas are usually determined by measuring adsorption isotherms.19 One measures the number of moles n of the molecules that are adsorbed on the surface as

a function of the pressure P at a given temperature T: n = f T (P). Other methods used tointerpret adsorption isotherms have been introduced.The surfaces of most materials are fractals, that is, at this range, surface

geometric irregularities and defects are characteristically self-similar upon variations of resolution.20 The whole range of fractal dimension, 2 D<3. The specific surface of thesample depends on the size of the molecules used. With a length scale δ given in terms of the adsorption area

σ 0 = δ 2

the amount adsorbed on a sample with a fractal surface must have the form21 n ~ δ –D = σ –D/2. (1)

As an example of this type of behavior consider figure 222,which shows the results for theadsorption of spherical alcohols from toluene on porous silica gel. The mole numbersobserved follow the relation (1) with D= 3.02 +/- 0.06 over the range σ 0 = 18 to 35 Ǻ2 or δ in the range 4.2 to 5.9 Ǻ.23 This is an extreme value of the fractal dimension of asurface. In thermodynamics terms the surface terms contribute as much as the volumeterms. A monolayer of adsorbed molecules in this system amounts to a bulk phaseinterrupted by an increasing number of ever smaller voids.

As another example, Bale and Schmidt proposed that microscopic porositymight be fractal in 1984.24 A method is developed for analyzing the outer part of thesmall-angle x-ray or neutron scattering curve for porous scatterers in which the pore boundaries can be described by fractals. When the results are applied to the scatteringdata from a lignite coal, the fractal dimension of the boundary surface of the pores in

this coal is found to be 2.56 ±0.03.

25

They gave a derivation of an expression for theX-ray scattering intensity from fractal pore surface,

S(q) ~ q(D-6)

where the scattering vector q is given by q =2

sin4 θ

λ

π . (θ is the scattering angle and λ is

the wavelength of the radiation used.)26 ,shown Figure 3.The observed fractal behavior would not cover many orders of magnitude in



length scales, and the validity of the determined fractal dimensions may be questioned.However, the realization that molecular fractal surfaces exist also will have a significantimpact on many fields that relate to surfaces, such as catalysis, wetting and powder technology.27

Figures

Figure 1: Subdividing a unit line (a), aunit square (b), and a unit cube (c). TheEuclidean dimension is corresponding asDE in figure.

Figure 2: Measuredmonolayer molenumbers n on poroussilica gel as a functionof moleculecross-section δ (Ǻ2)

D = 3.02+/- 0.06



Figure 3: Measured monolayer mole numbers n of tertiary amylalcohol as a function of adsorbent particle diameter (2R)

(µm).

1 H. Aki. Fractal . Accessed April 2006 <http://kamakura.ryoma.co.jp/~aoki/paradigm/furactal.htm>2 H. Aki.3 H. Aki.4 Assignment 6: Recursion. Accessed April 2006.<http://www4.comp.polyu.edu.hk/~csgngai/courses/2003-04/COMP201B/asgns/asgn6/>5 J. Feder. Fractal. Plenum Press. New York. 1988. pp 116 the Hausdorff dimension is an extended non-negative real number (that is a number in the closed infiniteinterval [0, ∞]) associated to any metric space.7 J. Feder, pp118 J. Feder, pp119 Fractal . Accessed April 2006. <http://ja.wikipedia.org/wiki/%E3%83%95%E3%83%A9%E3%82%AF%E3%82%BF%E3%83%AB>10 J. Feder, pp1111 Assignment 6: Recursion. Accessed April 2006.<http://www4.comp.polyu.edu.hk/~csgngai/courses/2003-04/COMP201B/asgns/asgn6/>12 Fractals. Accessed May 2006. <http://kosmoi.com/Science/Mathematics/Fractals/>13 Fractals. Accessed May 2006. <http://kosmoi.com/Science/Mathematics/Fractals/> 14 Fractal & the Fractal Dimension. Accessed May 2006. <http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html> 15 J. Feder, pp22 16 Fractal . Accessed April 2006. <http://ja.wikipedia.org/wiki/%E3%83%95%E3%83%A9%E3%82%AF%E3%82%BF%E3%83%AB>17 Hausdorff dimension. Accessed May 2006. < http://en.wikipedia.org/wiki/Hausdorff_dimension>18 J. Feder, pp236 19 D. Avnir, D. Farin, and P. Pfeifer. “Molecular Fractal Surfaces”. Nature 308, 261 - 263 (15 March

1984) Accessed May, 2006. < http://www.nature.com/nature/journal/v308/n5956/abs/308261a0.html>20 D. Avnir, D. Farin, and P. Pfeifer.21 J. Feder, pp23722 D. Avnir, D. Farin, and P. Pfeifer. 23 J. Feder, pp237 24 J. Feder, pp239 25 Harold D. Bale and Paul W. Schmidt. “ Small-Angle X-Ray-Scattering Investigation of Submicroscopic

Porosity with Fractal Properties” Phys. Rev. Lett. 53, 596–599 (1984)[Issue 6 – 6 August 1984 ].26 J. Feder, pp239



27 J. Feder, pp229

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The Fractals in Nature