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Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales. [2] Self-similarity is a typical property of fractals. Scale invariance is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is similar to ...
Self-similarity, which may include: Exact self-similarity: identical at all scales, such as the Koch snowflake; Quasi self-similarity: approximates the same pattern at different scales; may contain small copies of the entire fractal in distorted and degenerate forms; e.g., the Mandelbrot set's satellites are approximations of the entire set ...
The self-similar solution of the second kind also appears in different contexts such as in boundary-layer problems subjected to small perturbations, [8] as was identified by Keith Stewartson, [9] Paul A. Libby and Herbert Fox. [10] Moffatt eddies are also a self-similar solution of the second kind.
The terms fractal dimension and fractal were coined by Mandelbrot in 1975, [16] about a decade after he published his paper on self-similarity in the coastline of Britain. . Various historical authorities credit him with also synthesizing centuries of complicated theoretical mathematics and engineering work and applying them in a new way to study complex geometries that defied description in ...
Statistical Self-Similarity and Fractional Dimension", published on 5 May 1967, [12] Mandelbrot discusses self-similar curves that have Hausdorff dimension between 1 and 2. These curves are examples of fractals, although Mandelbrot does not use this term in the paper, as he did not coin it until 1975. The paper is one of Mandelbrot's first ...
Unlike self-similar sets, the Hausdorff dimension of self-affine sets depends on the position of the iterated elements and there is no formula, so far, for the general case. (+) 1.8617: Pentaflake: Built by exchanging iteratively each pentagon by a flake of 6 pentagons.
A Koch curve is self-similar. It is sometimes said that fractals are scale-invariant, although more precisely, one should say that they are self-similar. A fractal is equal to itself typically for only a discrete set of values λ, and even then a translation and rotation may have to be applied to match the fractal up to itself.
A self-similar subset of a metric space (X, d) is a set K for which there exists a finite set of similitudes { f s} s∈S with contraction factors 0 ≤ r s < 1 such that K is the unique compact subset of X for which A self-similar set constructed with two similitudes: ′ = [(+) +] ′ = [(+) +] =.