## Abstract

Self-similarity seems to be one of the fundamental geometrical construction principles in nature. For millions of years evolution has shaped organisms based on the survival of the fittest. In many plants and also organs of animals, this has led to fractal branching structures. For example, in a tree the branching structure allows the capture of a maximum amount of sun light by the leaves; the blood vessel system in a lung is similarly branched so that a maximum amount of oxygen can be assimilated. Although the self-similarity in these objects is not strict, we can identify the building blocks of the structure — the branches at different levels.

## Keywords

Brownian Motion Fractal Dimension Random Forest Percolation Threshold Fractional Brownian Motion
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.

## Reference

- 1.In: Benoit B. Mandelbrot,
*The Fractal Geometry of Nature*, Freeman, 1982, p. 1.Google Scholar - See F. M. Dekking, Recurrent Sets, Advances in Mathematics 44, 1 (1982) 78–104.MathSciNetCrossRefGoogle Scholar
- 12.
*See R*. Voss, Fractals in Nature, in: The Science of Fractal Images, H.-O. Peitgen and D. Saupe (eds.), Springer-Verlag,New York, 1988, pages 36–37.Google Scholar - 13.For a survey of the state-of-the-art of percolation theory see A. Bunde and S. Havlin (eds.), Fractals and Disordered Systems, Springer-Verlag, Heidelberg, 1991.Google Scholar
- 14.E. Guyon and H. E. Stanley (eds.),
*Fractal Forms*, Elsevier/North-Holland and Palais de la Découverte, 1991.Google Scholar - 15.See
*The Fractal Approach to Heterogeneous Chemistry: Surfaces*,*Colloids*,*Polymers*,edited by D. Avnir, Wiley, ChichesterGoogle Scholar - 19.
*Aggregation and Gelation*,edited by F. Family and D. P. Landau, North-Holland, Amsterdam, 1984.Google Scholar - 18.Of course, after the experiment, the used liquids must be disposed of properly (not in the sink). Moreover, a good ventilation of the room is recommended.Google Scholar
- 17.However, the heat of the lamp in the projector disturbs the experiment, which runs best at constant temperature (large solid zinc leaves form). It is therefore advisable to leave the overhead projector off most of the time. It is best to video film the experiment for immediate viewing on monitors of a projection unit.Google Scholar
- 18.M. Matsushita,
*Experimental Observation of Aggregations*, in:*The Fractal Approach to Heterogeneous Chemistry: Surfaces*,*Colloids*,*Polymers*, D. Avnir (ed.), Wiley, Chichester 1989.Google Scholar - 19.The model presented here originates from T. A. Witten and L. M. Sander, Phys. Rev. Lett. 47 (1981) 1400–1403 and Phys. Rev. B27 (1983) 5686–5697.MathSciNetGoogle Scholar
- 20.See the review article by H. Eugene Stanley and Paul Meakin,
*Multifractal phenomena in physics and chemistry*, Nature 335 (1988) 405–409 and the survey by A. Aharony,*Fractal growth*, in:*Fractals and Disordered Systems*, A. Bunde and S. Havlin (eds.), Springer-Verlag, Heidelberg, 1991.Google Scholar - 21.At large scales, however, the dendritic structure obtained using a small sticking probability does not look ‘thick’. The fractal dimension, measured at large scales, is independent of the sticking probability.Google Scholar
- 22.See R. F. Voss and M. Tomkiewicz,
*Computer Simulation of Dendritic Electrodeposition*, Journal Electrochemical Society 132, 2 (1985) 371–375.CrossRefGoogle Scholar - 23.For details see L. Pietronero, C. Evertz, A. P. Siebesma,
*Fractal and multifractal structures in kinetic critical phenomena*. in:*Stochastic Processes in Physics and Engineering*, S. Albeverio, P. Blanchard, M. Hazewinkel, L. Streit (eds.), D. Reidel Publishing Company (1988) 253–278.Google Scholar - 24.For a discussion of these and other phenomena from a physical point of view see the book
*Fractals*by J. Feder, Plenum Press, New York, 1988.Google Scholar - 27.See any textbook on probability theory or statistics.Google Scholar
- 28.For example, the Box-Muller method, see W. H. Press, B. P. Flannery, S. A. Teukolski, W. T. Vetterling,
*Numerical Recipes*. Cambridge University Press, 1986, p. 202.Google Scholar - 29.In many programming environments this division is internally carried out, and those random numbers are already uniformly distributed in the unit interval.Google Scholar
- 31.The method was introduced in the paper by A. Fournier, D. Fussell and L. Carpenter,
*Computer rendering of stochastic models*, Comm. of the ACM 25 (1982) 371–384.CrossRefGoogle Scholar - 32.Another advantage is that we can prescribe the values of
*X (t)*for various times t and have the random midpoint displacement compute intermediate values. In this sense, the method could be interpreted as fractal interpolation.Google Scholar - 33.It is not necessary to consider larger angles because the displacement may be positive or negative.Google Scholar
- 34.Note that this is not the generalization of Brownian motion which yields height field models of landscapes mentioned earlier on page 487 (see also section 9.6).Google Scholar
- 35.For details we refer to R. Voss,
*Fractals in Nature*, in: ‘The Science of Fractal Images’, H.-O. Peitgen and D. Saupe (eds.), Springer-Verlag, New York, 1988, pages 63–64 and B. B. Mandelbrot,*Self-affine fractals and fractal dimension*, Physica Scripta 32 (1985) 257–260.Google Scholar - 36.Several more algorithms, including pseudo code, are discussed in the first two chapters of
*The Science of Fractal Images*, H.-O. Peitgen and D. Saupe (eds.), Springer-Verlag, New York, 1988.Google Scholar - 37.See for example
*Illumination and Color in Computer Generated Imagery*,R. Hall, Springer-Verlag, New York, 1988.Google Scholar - 38.This method has been used in the opening scene of the video
*Fractals: An Animated Discussion*H.-O. Peitgen, H. Jürgens, D. Saupe, C. Zahlten, Freeman, 1990.Google Scholar

## Copyright information

© Springer Science+Business Media New York 1992