Abstract
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.
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Suggestions for Further Reading
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.)
Banchoff, T. F. (1990). “Dimension,” In On the Shoulders of Giants, Edited by Lynn Arthur Steen. National Academy Press, Washington DC.
Barcellos, A. (1984). The fractal geometry of Mandelbrot. College Mathematics Journal 15: 98–114.
Barnsley, M. (1993). Fractals Everywhere, 2nd ed. San Diego, CA: Academic Press.
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.)
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.)
Cibes, M. (1990). The Sierpinski triangle: Deterministic versus random models. Mathematics Teacher 83: 617–21.
Crownover, R. M. (1995). Introduction to Fractals and Chaos, Boston: Jones & Bartlett.
Darst, P., Palagallo, J. and Price, T. (1998). Fractal tilings in the plane. Mathematics Magazine, Vol. 71, No. 1: 12–23.
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.)
Devaney, R. L. (1989). An Introduction to Chaotic Dynamical Systems, 2nd ed. Redwood City, CA: Addison-Wesley.
Devaney, R. L. (1990). Chaos, Fractals, and Dynamics: Computer Experiments in Mathematics. Menlo Park, CA: Addison-Wesley.
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.
Dewdney, A. K. Computer recreations. Scientific American, August 1985, December 1986, July 1987, November 1987, February 1989, May 1990.
Edgar, G. A. (1990). Measure, Topology, and Fractal Geometry, New York: Springer-Verlag.
Eglash, R. (1999). African Fractals: Modern Computing and Indigenous Design. New Brunswick NJ: Rutgers University Press.
Feder, J. (1988). Fractals, New York: Plenum Press.
Froyland, J. (1994). Introduction to Chaos and Coherence, Philadelphia: Institute of Physics Publishing.
Gleick, J. (1988). Chaos: Making a New Science, New York :Penguin.
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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.)
Holmgren, R. A. (1996). A First Course in Dynamical Systems, 2nd ed., New York, Springer-Verlag.
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Jürgens, H., Peitgen, H-O., and Saupe, D. (1990). The language of fractals. Scientific American, August 1990: 60–67.
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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.’)
Peitgen, H., and Richter, P.H. (Eds.) (1986). The Beauty of Fractals: Images of Complex Dynamical Systems. New York: Springer-Verlag.
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Other Fractal Resources Software, Programs, and Websites Macintosh Software
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.
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.
PC Software
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.
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.
Websites
For a list of several websites containing fractal information, information about more fractal software, and other resources see http ://www.stolaf.edu/people/cederj/geotext/info.htm
BASIC Computer Programs
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.
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.
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.
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.
Programs for Graphing Calculators
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.
Videos
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.
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.
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 14851–0880
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.
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.
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.
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.
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Cederberg, J.N. (2001). Chaos to Symmetry: An Introduction to Fractal Geometry. In: A Course in Modern Geometries. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-3490-4_5
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