UNIT 47 - FRACTALS

UNIT 47 - FRACTALS

Compiled with assistance from Brian Klinkenberg, University of British Columbia

  • A. INTRODUCTION
  • B. SOME INTRODUCTORY CONCEPTS
  • C. SCALE DEPENDENCE
  • D. SELF-SIMILARITY AND SCALING
  • E. ERROR IN LENGTH AND AREA MEASUREMENTS
  • REFERENCES
  • EXAM AND DISCUSSION QUESTIONS
  • NOTES

    UNIT 47 - FRACTALS

    Compiled with assistance from Brian Klinkenberg, University of British Columbia

    A. INTRODUCTION

    Why learn about fractals?

    Length of a cartographic line

    Where did the ideas originate?

    B. SOME INTRODUCTORY CONCEPTS

    Euclidean geometry

    C. SCALE DEPENDENCE

    Determining fractal dimension

    Some questions

    D. SELF-SIMILARITY AND SCALING

    Self-similarity

    Scaling

    E. ERROR IN LENGTH AND AREA MEASUREMENTS

    REFERENCES

    Only a very small portion of the literature is presented here. For further references you should refer to the Goodchild and Mark (1987) paper; recent issues of Water Resources Research and Science also contain relevant papers

    Burrough, P.A., 1981. "Fractal dimensions of landscapes and other environmental data," Nature 294:240-242.

    Buttenfield, B., 1985. "Treatment of the cartographic line," Cartographica 22:1-26.

    Goodchild, M.F., 1980. "Fractals and the accuracy of geographical measures," Mathematical Geology 12:85-98.

    Goodchild, M.F., and Grandfield, A.W., 1983. "Optimizing raster storage: An evaluation of four alternatives," Auto-Carto 6(2):400-407.

    Goodchild, M.F., and Mark, D.M., 1987. "The fractal nature of geographic phenomena," Annals AAG 77(2):265-278.

    Hakanson, L., 1978. "The length of closed geomorphic lines," Mathematical Geology 10:141-167.

    Lovejoy, S., 1982. "Area-perimeter relation for rain and cloud areas," Science 216:185-187.

    Mandelbrot, B.M., 1977. Fractals: Form, Chance and Dimension, Freeman, San Francisco.

    Mandelbrot, B.M., 1982. The Fractal Geometry of Nature, W.H. Freeman and Co., New York.

    Milne, B.T., 1988. "Measuring the fractal geometry of landscapes," Applied Mathematics and Computation 27:67- 79.

    Peitgen, H.-O. and D. Saupe (Eds.) 1988. The Science of Fractal Images, Springer-Verlag, New York.

    Richardson, L.F., 1961. "The problem of contiguity," General Systems Yearbook 6:139-187.

    Unwin, D., editor, 1989. Special issue on fractals. Computers and Geosciences 15(2).

    Woronow, A., 1981. "Morphometric consistency with the Hausdorff-Besicovich dimension," Mathematical Geology 13:201-216.

    EXAM AND DISCUSSION QUESTIONS

    1. Although fractal concepts are important in understanding the error associated with pixel-based area estimates, little has been said about the relationship between fractals and area estimates obtained from vector-based systems. Why? (i.e., would the area of an enclosed figure change significantly? It is expected that the area shouldn't change significantly, as the self-similar detail should increase the area as much as it decreases the area.)

    2. Define "fractal". Include in your description terms such as scale dependency, self-similarity and scaling.

    3. Discuss some of the ways in which fractals have changed our way of looking at phenomena. Based on your readings, provide examples from a variety of fields.

    4. Theoretically, fractal behavior applies to a phenomenon across all scales. Practically, of course, there are limits to the application of self-similarity to natural phenomena. Where do you think some of these limits occur? (i.e., between what scales do you think portions of coastlines, for example, exhibit self-similar behavior.) What are the implications with respect to the generalization of cartogrpahic lines, if we observe definite limits to the self-similar behavior of cartographic features?


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