Chaos to Symmetry: An Introduction to Fractal Geometry

  • Judith N. Cederberg
Part of the Undergraduate Texts in Mathematics book series (UTM)


Although the term “modern geometries” traditionally refers to post-Euclid geometries, namely the non-Euclidean and projective geometries presented in Chapters 2 and 4, it seems ironic to describe topics formalized hundreds of years ago as “modern.” The topics presented in this chapter, on the other hand, are among those in a newly emerging area of mathematics and are honestly modern. In fact, the area known as fractal geometry is so new that its exact content has yet to be determined, let alone given a formal axiomatic structure. Thus, this chapter contains an informal presentation of concepts and themes basic to the topics currently regarded as part of fractal geometry. This presentation also attempts to convey the excitement experienced by professional mathematicians and scientists and large numbers of interested non-professionals as they discover and comprehend these new ideas and contemplate their far-reaching applications. In this vein, the presentation interweaves a number of exercises that will involve you in discovering and exploring concepts. Hopefully, in so doing, you will experience the excitement of mathematical discovery in a truly modern geometry as well as gain a deeper understanding of the affine transformations that are essential tools of fractal geometry.


Fractal Dimension Fractal Geometry Iterate Function System Sierpinski Carpet Koch Curve 
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.


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Suggestions for Further Reading

  1. Abbott, E. A. (1991). Flatland, Princeton NJ: Princeton University Press. (A reprint of the classic introduction to the fourth dimension together with a must-read introduction by Banchoff that explains the social satire Abbott uses.)Google Scholar
  2. Banchoff, T. F. (1990). “Dimension,” In On the Shoulders of Giants, Edited by Lynn Arthur Steen. National Academy Press, Washington DC.Google Scholar
  3. Barcellos, A. (1984). The fractal geometry of Mandelbrot. College Mathematics Journal 15: 98–114.MathSciNetMATHCrossRefGoogle Scholar
  4. Barnsley, M. (1993). Fractals Everywhere, 2nd ed. San Diego, CA: Academic Press.MATHGoogle Scholar
  5. Barnsley, M., and Hurd, L. (1993). Fractal Image Compression. Welles-ley, MA: AK Peters, Ltd. (Presents the mathematics behind the CD Microsoft Encarta, a multimedia encyclopedia for which all the pictures are fractals.)MATHGoogle Scholar
  6. Bunde, A. and Havlin, S. (Eds.) (1994). Fractals in Science New York: Springer-Verlag. (Includes PC or Mac diskette of interactive programs for fractal models.)MATHGoogle Scholar
  7. Cibes, M. (1990). The Sierpinski triangle: Deterministic versus random models. Mathematics Teacher 83: 617–21.Google Scholar
  8. Crownover, R. M. (1995). Introduction to Fractals and Chaos, Boston: Jones & Bartlett.Google Scholar
  9. Darst, P., Palagallo, J. and Price, T. (1998). Fractal tilings in the plane. Mathematics Magazine, Vol. 71, No. 1: 12–23.MathSciNetMATHCrossRefGoogle Scholar
  10. Davis, D. (1993). The Nature and Power of Mathematics. Princeton NJ: Princeton University Press. (Writing for the liberal arts student, Davis provides substantial introductions to non-Euclidean geometry, number theory, and fractals. The fractal chapter (Chapter 5) focuses on their appealing beauty and includes several relevant BASIC programs.)MATHGoogle Scholar
  11. Devaney, R. L. (1989). An Introduction to Chaotic Dynamical Systems, 2nd ed. Redwood City, CA: Addison-Wesley.MATHGoogle Scholar
  12. Devaney, R. L. (1990). Chaos, Fractals, and Dynamics: Computer Experiments in Mathematics. Menlo Park, CA: Addison-Wesley.MATHGoogle Scholar
  13. Devaney, R. and Keen, L., (Eds.) (1989). Chaos and Fractals: The Mathematics Behind the Computer Graphics, proceedings of Symposia in Applied Mathematics, Vol. 39, AMS.MATHGoogle Scholar
  14. Dewdney, A. K. Computer recreations. Scientific American, August 1985, December 1986, July 1987, November 1987, February 1989, May 1990.Google Scholar
  15. Edgar, G. A. (1990). Measure, Topology, and Fractal Geometry, New York: Springer-Verlag.MATHGoogle Scholar
  16. Eglash, R. (1999). African Fractals: Modern Computing and Indigenous Design. New Brunswick NJ: Rutgers University Press.MATHGoogle Scholar
  17. Feder, J. (1988). Fractals, New York: Plenum Press.MATHGoogle Scholar
  18. Froyland, J. (1994). Introduction to Chaos and Coherence, Philadelphia: Institute of Physics Publishing.Google Scholar
  19. Gleick, J. (1988). Chaos: Making a New Science, New York :Penguin.Google Scholar
  20. Goldenberg, E. P. (1991). “Seeing beauty in mathematics: Using fractal geometry to build a spirit of mathematical inquiry.” MAA Note: Visualization in Teaching and Learning Mathematics , Washington, DC: MAA. Google Scholar
  21. Hastings, H., and Sugihara, G. (1993). Fractals: A User’s Guide for the Natural Sciences, Oxford: Oxford University Press. (The introduction contains a brief chronology of fractals, and Chapter 3, “Dimension of Patterns,” provides a a well-written extension of ideas presented in this text.)Google Scholar
  22. Holmgren, R. A. (1996). A First Course in Dynamical Systems, 2nd ed., New York, Springer-Verlag.MATHCrossRefGoogle Scholar
  23. Hutchinson, J. E. (1981). Fractals and self similarity. Indiana University Mathematics Journal, Vol. 30, No. 5:713–747. (Provides the mathematical foundations for IFSs.)MathSciNetMATHCrossRefGoogle Scholar
  24. Jürgens, H., Peitgen, H-O., and Saupe, D. (1990). The language of fractals. Scientific American, August 1990: 60–67.CrossRefGoogle Scholar
  25. Lauwerier, H. (1991). Fractals: Endlessly Repeated Geometrical Figures, Princeton NJ: Princeton University Press.MATHGoogle Scholar
  26. Lorenz E. (1993). The Essence of Chaos, Seattle, WA: University of Washington Press.MATHCrossRefGoogle Scholar
  27. Mandelbrot, B. B. (1983). The Fractal Geometry of Nature. Rev. ed. New York: W. H. Freeman & Co.Google Scholar
  28. Mandelbrot, B. B. (1984). On fractal geometry and a few of the mathematical questions it has raised. Proceedings of the International Congress of Mathematicians, August 16–24 (1983):1661–1675. Warsaw: Polish Scientific Publishers.Google Scholar
  29. Peitgen, H., and Saupe, D. (Eds.) (1988). The Science of Fractal Images. New York: Springer-Verlag.MATHGoogle Scholar
  30. Peitgen, H., Jürgens, H., and Saupe, D. (1992). Chaos and Fractals: New Frontiers of Science New York: Springer-Verlag. (Contains most of the two-volume text Fractals for the Classroom.’)Google Scholar
  31. Peitgen, H., and Richter, P.H. (Eds.) (1986). The Beauty of Fractals: Images of Complex Dynamical Systems. New York: Springer-Verlag.MATHGoogle Scholar
  32. Peterson, I. (1984). Ants in labyrinths and other fractal excursions. Science News 21: 42–43.CrossRefGoogle Scholar
  33. Richardson, L. F. (1922). Weather Prediction by Numerical Process. Republished by Dover Publications (1965).MATHGoogle Scholar
  34. Robinson, C. (1995). Dynamical Systems: Stability, Symbolic Dynamics, and Chaos, Ann Arbor, MI: CRC Press.MATHGoogle Scholar
  35. Rodriguez-Iturbe, I. (1997). Fractal River Basins: Chance and Self Organization. Cambridge: Cambridge University Press.Google Scholar
  36. Ruelle, D. (1991). Chance and Chaos. Princeton, NJ: Princeton University Press.Google Scholar
  37. Steen, L. A. (1977). Fractals: A world of nonintegral dimensions. Science News 112: 122–123.CrossRefGoogle Scholar
  38. Stewart, I. (1987). The two-and-a-halfth dimension. The Problems of Mathematics. Oxford: Oxford University Press.Google Scholar
  39. Stewart, I. (1989). Does God Play Dice? The Mathematics of Chaos. Oxford: Blackwell.MATHGoogle Scholar
  40. Wegner, T., and Peterson, M. (1991). Fractal Creations. Mill Valley, CA: The Waite Group Press.Google Scholar

Other Fractal Resources Software, Programs, and Websites Macintosh Software

  1. Fractal Attraction. (1992). Kevin D. Lee, Yosef Cohen. Uses IFS (iterated function systems) to generate fractals with either random or deterministic algorithms. The IFS codes can be specified either by making numerical entries in a spreadsheet-like display or by using click and drag techniques. Allows interactive application of the collage theorem to create fractal generation of predetermined images. Includes an informative 80-page manual. Currently out of print, but well worth the effort to find.Google Scholar
  2. FractaSketch 2.0 (1998). Peter Van Roy. Creates fractals at specified levels by iterating a template, i.e., by replacing each segment in the prior level with a copy of the template, and displays the fractal dimension of the resulting fractal. Templates canbe createdby click and drag techniques. Available from Dynamic Software, PO Box 13991, Berkeley, CA 94701.Google Scholar

PC Software

  1. James Gleick’s CHAOS: The Software. (1990). Originally written by Josh Gordon, Rudy Rucker, and John Walker for Autodesk, Inc. to accompany Gleick’s book, it includes the following programs: The Mandelbrot Sets, Magnets and Pendulum, Toy Universes, The Chaos Game, Strange At-tractors and Fractal Forgeries. For the current web address from which the software can be obtained, see the website below.Google Scholar
  2. FRACTINT. A freeware fractal generator created and constantly upgraded and improvedby the Stone Soup team. For the current web address from which the software can be obtained see the website below.Google Scholar


  1. For a list of several websites containing fractal information, information about more fractal software, and other resources see http :// Scholar

BASIC Computer Programs

  1. Bannon, Thomas J. (1991). Fractals and Transformations. Mathematics Teacher 84: 178–85. Programs incorporating IFS algorithms for graphical display of the Dragon curve, the Koch curve, and the Sierpinski triangle. True BASIC programs generating the Sierpinski triangle using both deterministic and random algorithms.Google Scholar
  2. Davis, Donald (1993). The Nature and Power of Mathematics. Princeton University Press. Programs for iterating logistic functions and for graphical display of orbits, the Lorenz attractor, Julia sets, and the Mandelbrot set.MATHGoogle Scholar
  3. Devaney, Robert L. (1990). Chaos, Fractals, and Dynamics: Computer Experiments in Mathematics. Menlo Park CA Addison-Wesley. Programs for computing orbits by iterating functions and for graphical display of Julia sets, the Mandelbrot set, and the Sierpinski triangle.MATHGoogle Scholar
  4. Peitgen, Heinz-Otto, Jürgens, Hartmut, and Saupe, Dietmar (1992). Chaos and Fractals: New Frontiers of Science, New York: Springer-Verlag. Programs for graphical iteration, the Koch curve, the Sierpinski triangle, the chaos game for the fern, and for graphical display of Cantor, Mandelbrot, and Julia sets.Google Scholar

Programs for Graphing Calculators

  1. Peitgen, Heinz-Otto, Jürgens, Hartmut, and Saupe, Dietmar (1991–2) Fractals for the Classroom—Strategic Activities, Volume 1. New York: Springer-Verlag and NCTM. Programs for simulating the chaos game for both the CASIO and Tfexas Instrument graphing calculators.CrossRefGoogle Scholar


  1. The Beauty and Complexity of the Mandelbrot Set: School Edition (1989; 47 min.). A richly illustrated video-lecture by John H. Hubbard of Cornell University. The talk describes iteration and its use in creating pictures of the Julia set and the Mandelbrot set. Available from The Science Television Co., PO Box 2498, Times Square Station, New York, NY 10108.Google Scholar
  2. Chaos, Fractals and Dynamical Systems (1989; 63 min.). ISBN 1–878310–00–3. Robert Devaney. Tells the mathematical story behind chaos, fractals, and dynamical systems. Computer-generated diagrams and graphs give a visual introduction to the concepts. Available from The Science Television Co., P O Box 2498, Times Square Station, New York, NY 10108.Google Scholar
  3. Focus on Fractals (1990; 23 min.). An “entry level introduction” with some narration featuring footage from the Dr. John Hamal Hubbard Dynamical Systems Laboratory. Contains four zooms of the Mandelbrot set, two Julia set promenades, and a 3D rendition of the Lorenz attractor. Available from Art Matrix, PO 880NA, Ithaca, NY 14851–0880Google Scholar
  4. Fractals: An Animated Discussion (1990; 63 min.). Heinz-Otto Peitgen et al. A combination of animated sequences and interviews with Benoit Mandelbrot and Edward Lorenz, accompanied by music composed according to fractal principles. Available from W.H. Freeman, New York.Google Scholar
  5. Fractals: the colors of infinity (1994; 30 min.) An explanation of the Mandelbrot set and the revolutions in thought resulting from its discovery. Includes interviews with Benoit Mandelbrot, Michael Barnsley and Ian Stewart. Available from Films for the Humanities and Sciences, P.O. Box 2053, Princeton NJ 08543–2053.Google Scholar
  6. The Hypercube: Projections and Slicing (1978; 15 min.). A Banchoff/Strauss Production that uses computer graphics to describe the three and four dimensional cubes and the forms they create when rotated around various axes; discusses perspective and shows shapes that evolve when a three and four dimensional cube is sliced at various points. Available from International Film Bureau.Google Scholar
  7. Mandelbrot Sets and Julia Sets (1990; 2 hrs.). Animated zooming (no narration) into the Mandelbrot set, from the Cornell National Supercomputer Facility and the Dr. John Hamal Hubbard Dynamical Systems Laboratory. Available from Art Matrix, PO 880, Ithaca, NY 14851–0880.Google Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Judith N. Cederberg
    • 1
  1. 1.Department of MathematicsNorthfieldUSA

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