Fractals. An overview of their discovery, principles, and applications

Pre-University Paper from the year 2019 in the subject Mathematics, grade: 13 Punkte, , language: English, abstract: They surround us every day in a wide variety of situations, are the key to modern communication society, and a fundamental building block of all life: fractals. Their patterns and principles govern not only many other mathematical fields, but also biological processes such as evolution, which has produced forms of fractal geometry in nature, right up to the technological innovations that made possible the spread of our communication and information tools like mobile phones, which we take for granted today.
But despite their importance to our lives, science was for a long time unable to describe these sometimes so obvious and everyday forms, or even to recognize their structure. Not only did this discovery open up entirely new fields of study in mathematics, but biology, IT, and many other fields also made unimagined progress through the expansion or revision of current theories. However, a crucial question now arose: What practical problems can these new findings solve for research and thus for humanity? And also: What opportunities do they also offer for modern societies?
The following paper addresses these questions using a few examples, illustrating the impact and influence of fractal geometry in our everyday lives. To do so, we will first discuss the background to its discovery, then examine some basic principles-including what constitutes a fractal-and introduce the most well-known fractal, the Mandelbrot set. Finally, we will provide an outlook on the application of fractals.