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Standard (trivial) self-similarity [1] In mathematics, a self-similar object is exactly or approximately similar to a part of itself (i.e., the whole has the same shape as one or more of the parts). Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many ...
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 Tweedie convergence theorem can be used to explain the origin of the variance to mean power law, 1/f noise and multifractality, features associated with self-similar processes. [9] Ethernet traffic data is often self-similar. [5] Empirical studies of measured traffic traces have led to the wide recognition of self-similarity in network ...
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 ...
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: ′ = [(+) +] ′ = [(+) +] =.
More formally, one describes it in terms of functions on closed sets of points. If we let d A denote the dilation by a factor of 1 / 2 about a point A, then the Sierpiński triangle with corners A, B, and C is the fixed set of the transformation d A ∪ d B ∪ d C {\displaystyle d_{\mathrm {A} }\cup d_{\mathrm {B} }\cup d_{\mathrm ...
Sierpinski triangle created using IFS (colored to illustrate self-similar structure) Colored IFS designed using Apophysis software and rendered by the Electric Sheep.. In mathematics, iterated function systems (IFSs) are a method of constructing fractals; the resulting fractals are often self-similar.
To adjust for this variation of kerf, the self-similar property of the logarithmic spiral has been used to design a kerf cancelling mechanism for laser cutters. [18] Logarithmic spiral bevel gears are a type of spiral bevel gear whose gear tooth centerline is a logarithmic spiral. A logarithmic spiral has the advantage of providing equal angles ...