# Self-likeness

A selection from the Mandelbrot crowd

Self-similarity in the narrower sense is the property of objects, bodies, quantities or geometric objects to have the same or similar structures on a larger scale (i.e. when enlarged) as in the initial state. This property is examined, among other things, by fractal geometry , since fractal objects have a high or perfect self-similarity. The Mandelbrot set is strictly and unlike often not even like to read opinions: Generally, one each cutout of the frame in any magnification view with sufficient resolution, from which point he comes.

In a broader sense, the term is also used in philosophy as well as the social and natural sciences to denote fundamentally recurring structures that are nested in themselves.

## Fractal geometry

Self-similarity using the example of the Sierpinski triangle

Of exact (or strict ) self-similarity is the speech, when repeatedly receive the original structure at infinite magnification of the examined object without ever obtaining an elementary fine structure. Exact self-similarity can practically only be found in objects generated mathematically (e.g. by an iterated function system ). Examples are the Sierpinski triangle , the Koch curve , the Cantor set or, trivially, a point and a straight line .

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

## nature

Romanesco inflorescence with fractal structures and Fibonacci spirals

Real existing examples would be e.g. B. the ramifications of blood vessels , fern leaves or parts of a cauliflower (this is very clear in the Romanesco variety), which are very similar to the cauliflower head in a simple enlargement. In real examples, of course, the enlargement cannot be continued to infinity , as is the case with ideal objects.

Any images of the real world also have self-similarities, which z. B. be used in the fractal image compression or the fractal sound compression .

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

Self-similarity is a phenomenon that often occurs in nature. A characteristic number for the recurring self-similarity is the golden ratio .

The trajectories of a Wiener process and the broken Brownian movement are also self-similar.

## literature

• Henning Fernau: Iterated functions, languages ​​and fractals . BI Wissenschaftsverlag, Mannheim - Vienna - Zurich 1994, ISBN 3411170115 .