The Chaos Game: How Randomness Creates Deterministic Shapes

  • Heinz-Otto Peitgen
  • Hartmut Jürgens
  • Dietmar Saupe


Our idea of randomness, especially with regard to images, is that structures or patterns which are created randomly look more or less arbitrary. Maybe there is some characteristic structure, but if so, it is probably not very interesting — like a box of nails poured out onto a table.


Random Number Affine Transformation Iterate Function System Markov Operator Contraction Ratio 
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  1. 1.
    The discovery was made by the botanist Robert Brown around 1827.Google Scholar
  2. M. F. Barnsley, Fractal modelling of real world images in: The Science of Fractal Images, H.-O. Peitgen and D. SaupeGoogle Scholar
  3. This was first observed by Gerald S. Goodman, see G. S. Goodman, A probabilist looks at the chaos game, in: Fractals in the Fundamental and Applied Sciences, H.-O. Peitgen, J. M. Henriques, L. F. Peneda (eds.), North-Holland, Amsterdam, 1991.Google Scholar
  4. 17.
    J. Hutchinson, Fractals and self-similarity, Indiana University Journal of Mathematics 30 (1981) 713–747.CrossRefGoogle Scholar
  5. 15.
    This procedure was suggested among other pseudo-random number generators by Ian Stewart, Order within the chaos game? Dynamics Newsletter 3, Nos. 2 and 3, May 1989, 4–9. Stewart ends his article: `I have no idea why these results are occurring […] Can these phenomena be explained? […]’ Our arguments will give some first insight. They were worked out by our students E. Lange and B. Sucker in a semester project of an introductory course on fractal geometry.Google Scholar
  6. 19.
    t is important to do the histogram computation using double precision calculations. Otherwise it is very likely, that the iteration for the logistic equation will run into a periodic cycle of a low period (perhaps even less than 1000), and, as a consequence a histogram based on such an orbit would be a numerical artifact. This effect and the topic of histograms will be continued in chapter 10.Google Scholar
  7. 20.
    This experiment approximates the so-called natural measure of the quadratic iterator. See chapter 10 for more details.Google Scholar
  8. 22.
    For an introduction into the topic of random number generation see D. E. Knuth The Art of Computer Programming, Volume 2, Seminumerical Algorithms, Second Edition, Addison-Wesley, Reading, Massachusetts, 1981.Google Scholar
  9. 23.
    Details have appeared in the paper Rendering methods for iterated function systems by D. Hepting, P. Prusinkiewicz and D. Saupe, in: Fractals in the Fundamental and Applied Sciences,H.-O. Peitgen, J. M. Henriques, L. F. Peneda (eds.), North-Holland, Amsterdam, 1991.Google Scholar
  10. 26.
    See S. Dubuc and A. Elqortobi, Approximations of fractal sets, Journal of Computational and Applied Mathematics 29 (1990) 79–89.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1992

Authors and Affiliations

  • Heinz-Otto Peitgen
    • 1
    • 2
  • Hartmut Jürgens
    • 3
  • Dietmar Saupe
    • 4
  1. 1.CeVis and MeVisUniversität BremenBremenGermany
  2. 2.Department of MathematicsFlorida Atlantic UniversityBoca RatonUSA
  3. 3.CeVis and MeVisUniversität BremenBremenGermany
  4. 4.Department of Computer ScienceUniversität FreiburgFreiburgGermany

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