selbstähnlichen

Selbstähnlich is a German word that translates to "self-similar" in English. It describes a property of an object or a set where parts of the object are similar to the whole object, but at different scales. This means that if you zoom in on a self-similar object, you will see smaller versions of the entire object repeated. This characteristic is a fundamental concept in fractal geometry.

Fractals are often self-similar. A classic example is the Koch snowflake. Each side of the Koch snowflake is constructed by replacing the middle third of a line segment with two line segments of the same length, forming an equilateral triangle pointing outwards. This process is repeated infinitely, resulting in a curve that is infinitely long but encloses a finite area. If you examine any small portion of the Koch snowflake's edge, it will resemble the shape of the entire edge, just on a smaller scale.

Self-similarity can be exact or statistical. Exact self-similarity means that the smaller parts are perfect replicas of the whole. Many mathematical fractals exhibit exact self-similarity. Statistical self-similarity, on the other hand, means that the smaller parts share statistical properties with the whole, but are not exact copies. This type of self-similarity is often observed in natural phenomena such as coastlines, mountain ranges, and the branching patterns of trees or blood vessels. The concept of selbstähnlich is crucial for understanding complex systems and patterns in various scientific disciplines.