Self-similarity

Self-similarity in the narrower sense is the property of things, bodies, quantities or geometrical objects on a larger scale, ie at magnification to have the same or similar structures as the initial stage. This property is investigated, inter alia, of the fractal geometry, as fractal objects have a high and a perfect self-similarity. The Mandelbrot set is, strictly speaking, and in contrast to often to read opinions not self-similar: In principle one can view each section of the rim in any magnification with sufficient resolution, the point from which it originates.

In a broader sense, the term is also used in the philosophy and the social and natural sciences, in order to fundamentally recurring to refer to nested structures in themselves.

Fractal geometry

From exact ( or strict ) self-similarity is mentioned, if at infinite magnification of the investigated object always returns to the original structure is obtained without ever getting an elementary fine structure. Exact self-similarity can be found practically only mathematically (eg, by an iterated function system) generated objects. Examples are the Sierpinski triangle, the Koch curve, the Cantor set or trivially a point and a straight line.

The Mandelbrot set and Julia sets are self-similar, but not strictly self-similar. Strict self-similarity implies scale invariance, and can be, among others, with the help of the characteristic exponents of the underlying power law (scale Act) quantify.

Nature

Real existing examples would include the branching of blood vessels, fern leaves or parts of a cauliflower ( which is in the variety Romanesco very clearly ) that the cauliflower head are very similar in single magnification. In real examples, the magnification can of course not continue to infinity, as would be the case with ideal objects.

Also any pictures of the real world have self- similarities, which are used eg in the case of fractal image compression, or fractal sound compression.

The recurrences denote the call or the definition of a function by themselves, are self-similar accordingly.

The self-similarity is a phenomenon that often occurs in nature. A typical number for the recurrent self-similarity is the golden ratio.

The trajectories of a Wiener process and fractional Brownian motion are self-similar.