SELF-SIMILARITY
| Collection | International Encyclopedia of Systems and Cybernetics |
|---|---|
| Year | 2004 |
| Vol. (num.) | 2(2) |
| ID | ◀ 2986 ▶ |
| Object type | Methodology or model |
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 self-similarity 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).
See also
also: ELLIOTT waves, Fields within fields