## Theory

Essential properties of fractals in mathematics are self-similarity and often existing scale invariance. In the Romanesco Broccoli, self-similarity and scale invariance are evident in the fact that the conical bulges appear again and again in smaller forms. With mathematical fractals these properties can be exactly fulfilled.

In many cases fractals are attributed to "irregular geometry". But this concept of irregularity is not meant here. Rather, the term is derived from irregular complex functions, which are used here to generate the images.

This is done by iterative processes, similar to the formation of the Mandelbrot set, also known as the "Apfelmännchen" in german. The latter is formed by iterations with a regular function, the quadratic function.

The functions I use are not holomorphic, not analytic or even regular in the sense of the theory of functions of one complex variable: they are irregular. Further information can be found in the following two articles.

#### Fraktal brochure (PDF-Download)588 KB

Images of chaos - calculated fractals

#### Irregular fractals skript (PDF-Download)2 MB

In nature there are shapes that are very similar to the fractals defined in mathematics. For example, in Romanesco Broccoli, structures of self-similarity can be found in multiple repetitions.