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Self-similar solutions appear whenever the problem lacks a characteristic length or time scale (for example, the Blasius boundary layer of an infinite plate, but not of a finite-length plate). These include, for example, the Blasius boundary layer or the Sedov–Taylor shell .
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 ...
Self-similar processes are stochastic processes satisfying a mathematically precise version of the self-similarity property. Several related properties have this name, and some are defined here. A self-similar phenomenon behaves the same when viewed at different degrees of magnification, or different scales on a dimension.
An object whose irregularity is constant over different scales ("self-similarity") is a fractal (examples include the Menger sponge, the Sierpiński gasket, and the Koch curve or snowflake, which is infinitely long yet encloses a finite space and has a fractal dimension of circa 1.2619).
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: ′ = [(+) +] ′ = [(+) +] =.
The self-similar solution exists because the equations and the boundary conditions are invariant under the transformation ,,, where is any positive constant. He introduced the self-similar variables Developing Blasius boundary layer (not to scale). The velocity profile ′ is shown in red at selected positions along the plate.
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.