Fractals: Basics

Short: Fractals are geometric structures whose patterns repeat in similar form when zoomed in, a property called self-similarity. They appear both in mathematics and, surprisingly often, in nature.

In more detail: Benoit Mandelbrot coined the term in 1975 and developed the Mandelbrot set, named after him, one of the best-known fractals.

In detail

• Self-similarity: A section of a fractal often looks remarkably similar to the whole pattern, no matter how much you zoom in.
• Mandelbrot set: This complex fractal structure arises from a simple mathematical formula applied repeatedly to itself.
• Fractal dimension: Fractals often have a non-integer dimension lying between classic geometric dimensions.
• Natural fractals: Lightning, coastlines, fern leaves, and snowflakes show fractal patterns, even though they’re not mathematically perfect.
• Koch curve: A simple example where a triangle is repeatedly supplemented with smaller triangles, creating an infinitely long edge with a finite area.
• Applications: Fractal algorithms are used in computer graphics to generate realistic landscapes, clouds, or plants.
• Chaos theory connection: Fractals are closely linked to chaos theory, as both deal with seemingly unpredictable but underlyingly ordered systems (primzahlen-und-ihre-bedeutung, fibonacci-folge-und-goldener-schnitt).