November 28, 2023

# The Mandelbrot set marks the boundary between order and chaos

The Mandelbrot set marks the boundary between order and chaos: the spectrum of science

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## The world of Freistetter formulas: The thin line between order and chaos

Simple rules can create complex and unmanageable systems. In these cases we are dealing with the fascinating phenomenon of deterministic chaos.

My real introduction to science was chaos theory. When I met her, I had already completed five semesters of studying astronomy. But even though I was definitely interested in space and celestial bodies back then, somehow that last spark of enthusiasm was missing. At least until I was confronted with this mathematical formula in a course:

It’s not particularly complicated. It is a simple iteration rule for complex numbers for example AND C. Include the initial value of the iteration for example0 = 0, it’s very easy to see what happens. You just have to keep multiplying a complex number by itself and by the previously chosen number C add.

The interesting aspect of the formula becomes apparent when you see how it applies to values ​​larger than No behaves. In some cases it will for example becoming bigger and bigger. However, if you choose a different value for Cso the values ​​for remain for example limited, no matter how far you push the iteration. Now mark these values ​​by Cwhere the sequence does not grow beyond any limit, like points in the plane of complex numbers, a surprising picture emerges.

It is known as the Mandelbrot set, named after the mathematician Benoît Mandelbrot, who studied it in depth. The mob is sometimes called the apple man, and with a little imagination, it really does look like an apple. But only at first glance! In fact, there is a large contiguous area within which iteration remains limited. But many large and small “gems” are sprouting from this region – and this is far from the end.

The most legendary mathematical tricks, the worst obstacles in the history of physics and all sorts of formulas of which almost no one can see the meaning hidden inside them: these are the inhabitants of the world of Freistetter’s formulas.
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The variety of shapes presented here can hardly be described in words. From the buds grow “antennae” and structures that look like seahorses and, upon closer inspection, dissolve into rays and spirals. On a smaller scale, figures suddenly appear that look like copies of Mandelbrot’s original set. The variety of structures and their complexity are endless, no matter how much you zoom in on the image.

### Between art and mathematics

The different behavior of the iteration sequence can be classified even more precisely and represented in different colors. The resulting images therefore appear to come more from art than mathematics. Indeed, the aesthetics of the Mandelbrot set have continually inspired art and are also why it is more present in the public eye than other mathematical objects.

Personally, I was particularly fascinated by their complexity. Simply put, the Mandelbrot set represents the set of starting values ​​for which ordered behavior develops over time. The set of remaining numbers shows a chaotic progression. However, the boundary between order and chaos in the Mandelbrot set is nested so complexly and confusingly that an arbitrarily small change in the initial value can be sufficient to transition from one state to another.

This is precisely the most striking feature of complex dynamical systems – and it was precisely what inspired me enormously then. I subsequently used the knowledge of chaos theory in my research work, for example to understand the movement of planets and asteroids, which can often be as complex as the Mandelbrot set. I wonder if I would have abandoned astronomy if I hadn’t encountered chaos.

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#### Marle

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