Property of some sets such as each part at different levels is similar to the whole.
This is particularly the case with fractals, whose forms do repeat in a similar way through decreasing dimensional scales.
Some examples of self-similar sets are:
- CANTOR's set, obtainable by sequential suppression of the median third of a segment (a process that could theoretically be repeated ad infinitum, which is also the case in the following examples)
- von KOCH's curves
- SIERPINSKY's sieves and carpets
- PEANO's curves
- MENGER's sponges (see J. GLEICK, 1987)
Self-similarity is a powerful algorithm for compressibility. It is related to chaotic attractors, through period-doubling sequences.
Self-similarity is also the most important property of holograms.
J. GLEICK explains that self-similarity: "… implies recursion, pattern inside pattern. MANDELBROT's price charts and river charts display self-similarity, because not only do they produce detail at finer and finer scales, they also produce detail with certain constant measurements" (1987, p.103).
As GLEICK states: "The self-similarity is built into the technique of constructing the curves the same transformation is repeated at smaller and smaller scales" (Ibid).
The first and best known example of selfsimilarity is found in the Golden Proportion of the ancient Greeks. Self-similarity is also related to chaos and renormalization.
It was independently discovered by K. WEIERSTRASS as his continuous, non-differentiable function (See hereafter).
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To cite this page, please use the following information:
Bertalanffy Center for the Study of Systems Science (2020). Title of the entry. In Charles François (Ed.), International Encyclopedia of Systems and Cybernetics (2). Retrieved from www.systemspedia.org/[full/url]
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