Abstract
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. Just imagine a box of nails which is poured out onto a table.
Chaos is the score upon which reality is written.
Henry Miller
Nothing in Nature is random... A thing appears random only through the incompleteness of our knowledge.
Spinoza
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References
The discovery was made by the botanist Robert Brown around 1827.
M. F. Barnsley, Fractal modelling of real world images, in: The Science of Fractal Images, H.-O. Peitgen and D. Saupe (eds.), Springer-Verlag, New York, 1988.
Barnsley explains the success of the chaos game by referring to results in ergodic theory (M. F. Barnsley, Fractals Everywhere, Academic Press, 1988). This is mathematically correct but practically useless. There are two questions: One is, why does the properly tuned chaos game produce an image on a computer screen so efficiently? The other is, why does the chaos game generate sequences which fill out the IFS attractor densely? These are not the same questions! The ergodic theory explains only the lauer, while it cannot rule out that it may take some 1011 years for the image to appear. In fact, this could actually happen, if computers lasted that long.
This was first observed by Gerald S. Goodman, see G. S. Goodman, A probabilist looks at the chaos game, FRACTAL 90 — Proceedings of the 1st IFIP Conference on Fractals, Lisbon, June 6–8, 1990 (H.-O. Peitgen, J. M. Henriques, L. F. Penedo, eds.), Elsevier, Amstedam, 1991.
M. F. Bamsley, Fractals Everywhere, Academic Press, 1988.
We need the above construction using the e-collar of the attractor because the area of the attractor itself may not be meaningful. For example, the area of the Sierpinski gasket is zero.
J. Hutchinson, Fractals and self-similarity, Indiana University Journal of Mathematics 30 (1981) 713–747.
Details have appeared in the paper Rendering methods for iterated function systems by D. Hepting, P. Prusinkiewicz and D. Saupe, in: FRACTAL 90 - Proceedings of the 1st IFIP Conference on Fractals, Lisbon, June 6–8, 1990 (H.-O. Peitgen, J. M. Henriques, L. F. Penedo, eds.), Elsevier, Amsterdam, 1991
See S. Dubuc and A. Eiqortobi, Approximations of fractal sets, Journal of Computational and Applied Mathematics 29 (1990) 79–89.
See G. H. Golub and C. F. van Loan, Matrix Computations, Second Edition, Johns Hopkins, Baltimore, 1989, page 57.
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© 1992 Springer Science+Business Media New York
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Peitgen, HO., Jürgens, H., Saupe, D. (1992). The Chaos Game: How Randomness Creates Deterministic Shapes. In: Fractals for the Classroom. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-2172-0_6
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DOI: https://doi.org/10.1007/978-1-4757-2172-0_6
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