Fractals in Nature and How to Measure Them

by Konrad Sandau
University of Applied Sciences Faculty of Mathematics and Science Darmstadt, Germany

Extended Abstract to the workshop 'Fractals in Biology', Santa Fe Institute, 29.11. - 2.12. 2000

GENERAL REMARKS

Fractality is a mathematical concept which seems to fit some structures in nature. Numerous scientists use derivatives of this concept, and therefore it is bound to happen that the term 'fractal structure' is used in different meanings. It either is used to denote the fractal set itself, or the generating system of the fractal set, where the generating system is based on a suitable construction rule which usually works inductively from one generation level to the next.

Historically, F. Hausdorff (1919) introduced the term fractal dimension in the sense of noninteger dimension and with the given definition the age of fractality has begun. Consequently, a set that can be assigned a fractal dimension is called a fractal set. One can determine the fractal dimension of the set by observing optimal covering systems of fractal sets with decreasing diameters. It should be mentioned that several different definitions of fractal dimension were created since Hausdorff's paper and their relations are not simple. But for self similar sets most of these definitions lead to the same dimension number (Sandau, 1996).

Looking at the properties of fractal sets more generally, they can be considered as boundary sets that connect (or divide) neighbored systems. Approximations of such sets can be observed in the real world, for example in coast lines, rough landscape surfaces, or in the "endpoints" of some bronchial or vascular trees.

GENERATION OF SELF-SIMILAR FRACTAL SETS

Because many fractal sets are described by generating systems, one often denotes the generating system of a fractal set as "fractal-like". The construction rules may depend on the generation level. Construction rules, which are independent of the generation level are called self-similar and the resulting fractal set is also called a self-similar set. A more precise definition of self-similar can be found in Hutchinson (1981). Mathematicians created a lot of self-similar fractal sets (Peitgen & Saupe, 1988), like the Cantor dust shown in Figure 1 together with its generator.

Fig. 1 Dust1.gif

Figure 2a shows a symmetric branching system which obviously is self-similar The tree structure may be denoted as fractal-like, but the corresponding fractal set is the set of endpoints (!) which is approximately shown in Figure 2b.

Fig. 2 Branch2a.gifBranch2b.gif

Moreover, fractal sets need not to be self-similarly constructed - even then it is sometimes possible to determine the fractal dimension of the set. In the case of the set, shown in Figure 3, the construction is based on a random process, but at each branching point the diameter exponent (Suwa & Takahashi, 1971) is constant and the parameters of the generating rule are chosen so that the diameter exponent is equal to the fractal dimension of the endpoint set.

Fig. 3 Branch3.gif

Such branching patterns (Figure 3) look rather naturally, and therefore many authors tried to use such generating systems as models for branching systems in nature (see, e.g., Bittner, 1991, Van Beek 1989, West et al. 1997). In most cases, the main aspect was to fit the geometry and eventually afterwards to consider the physics of internal flow ( of blood in the arterial trees, of air in the bronchial tree and so on). Generating systems like these, and in particular self-similar systems confirm observed allometric relations because of their construction laws.

GENERATION OF NON-FRACTAL BRANCHING SYSTEMS

In contrast to such models one can start with quite different premises. A branching system is a transportation tool which supports endpoints. The endpoints may be alveoli in a lung, capillaries in an organ or springs in a landscape. The implicated branching systems then are the bronchial tree, the arterial tree, or the river system. Obviously the transport direction may be different in the different cases. Starting with a given set or with a growing set of endpoints and introducing l o c a l knowledge, the branching system grows even without such restrictive construction rules as given by self-similarity. This is shown here in two simulations. The first is a simulation of the development of an arterial tree on the chorioallantoic membrane (CAM) of a chicken egg. It starts in one source point and then grows up by expanding the border. Growth is driven by local information about pressure, blood velocity etc., implemented in a procedure including some randomness. For details see Sandau & Kurz (1994). The development is shown here as a little movie in Figure 4.

Fig. 4 Camfilmb.gif

Similar premises can also be used in other systems. This shall be shown here for a river system (reverse transport direction), where a smooth surface with a known gradient is assumed and provides some l o c a l information. The superelevated shape is depicted in Figure 6. In this landscape, a set of springs with constant flow is simulated by a Matern hard-core point process (Stoyan et al., 1995). Including some randomness one gets a river system as can be seen in the movie in Figure 5. The resulting system is rather similar to natural river systems, which are periodically originated on smooth sandy landscapes. Observe e.g. tideways at the gulf of Carpentaria (Australia).

Fig. 5 Riverfilmb.gif
Fig. 6 Rivshad3.gif

In both examples there is no fractality by construction or by result, however one can find many allometric relations which are (approximately) fulfilled. For details in the case of the CAM see Kurz & Sandau (1997).

MEASURING FRACTAL DIMENSION

If a set is given in a binary image, one can always measure its fractal dimension. Several methods estimate the fractal dimension in the given range of magnification limited by the resolution of the digital image. Different methods exist and are based on the different definitions of fractal dimension. Several methods use a regression along the range of possible magnifications in the image (Falconer, Stoyan & Stoyan, 1994). An often-used method of this type is the so called box-counting method (bcm). Using bcm, it is shown here briefly that methods using regression techniques have some serious disadvantages and an extension of this method is recommended. In bcm, the number of boxes of a regular grid with boxes of side length q, intersecting the set of interest, are counted. The logarithm of this number is plotted versus log(q) in a so-called "log-log-plot". In case of self-similar sets the graph has globally a constant slope which is directly related to the fractal dimension. However, looking to the theoretical graph of a self-similar set, like the Cantor dust in Figure 1, it is a step function as shown in Figure 7 (thick plotted, piecewise constant function).

Fig. 7 Dustslop.gif

Depending on the used box sizes, one can measure different slopes and will therefore end up with different estimations of the fractal dimension which lie in this case in the interval [1.111;1.667].

The theoretical piecewise constant function also explains an often discussed artifact (see, e.g., Bittner et al., 1989, Brune et al., 1994): the data points of the regression line, taken at linearly growing box sizes, show a linear part and a constant part. For illustration of this effect, the Cantor dust is measured with suitable chosen data points(Figure 8).

Fig. 8 Dustlog2.gif

Finally it shall be emphasized here that the characterization "fractal dimension" can be used as a measure to describe a kind of complexity of a set, and that it has nice properties, which are:

• (i) Motion invariance

• (ii) Scale invariance

• (iii) Monotone function: E subset of F implies dim(E) <= dim(F)

• (iv) Maximum property: dim(E unified F) = max(dim(E), dim(F))

No technique for measuring fractal dimension in a binary image can perfectly fulfill these properties, but a good technique should be made to approximate them rather closely. The scale invariance cannot be fulfilled because of the bounded range of magnification. The motion invariance is troubled by discretization effects. A major problem of the bcm, caused by the regression, is that it does not hold (iii) and (iv). This can be seen very easily by taking the Cantor dust and superimposing a noise consisting of randomly distributed points in the image. These points are now part of the set and therefore the measured dimension should increase. However, for a Cantor dust having a fractal dimension of 1.513 the measurements (bcm) were 1.503 without noise and 1.419 with noise.

A method, closer to the desired theoretical properties, was developed by the author (Sandau, 1996) and is particularly recommended for investigations comparing experimental and control groups in the same sytem. This was applied in a first study by Sandau & Kurz ( 1997) and Kurz et al. (1998).

SUMMARY

In the light of this brief sketch, much of the published data on fractality in biological systems need to be re-evaluated - either because self-similarity was identified with fractality, or because inadequate measurements were performed, or both.

REFERENCES

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