Strange Loops // Mandelbrot

When talking about feedback loops, the annoying squeals from audio feedback are normally the first thing that comes to mind, but there are many other feedback loops. I would like to mention a few here to expound on my interest in these strange recursive systems.

In mathematics, self referencing systems are bountiful, so I will only focus on three theorems and/or sets. Mandelbrot’s set, the Fibonacci sequence and Gödel’s incompleteness theorems. Benoit Mandelbrot, Leonardo Bonacci (known as Fibonacci) and Kurt Gödel were some of the mathematicians behind these concepts. Even their names hint at the beauty that lies at the heart of their ideas. I will briefly and very simplistically explain what each of these theorems/sets are and why they interest me. To be clear, I am not a mathematician, but I do understand the basic reasoning behind them.

The Mandelbrot set.

Named in recognition of mathematician Benoit Mandelbrot, the Mandelbrot set forms a part of complex dynamics. Adrien Douady and John H. Hubbard established many of its fundamental properties and are credited with starting the proper mathematical investigation of the Mandelbrot set. So what is it?

In mathematical terms, the Mandelbrot set is the set of complex numbers c for which the function fc(z) = z² + c does not diverge when iterated from 0, i.e. when an iterative sequence remains bounded. In simple terms, it is the set of values you get when using the above formula recursively with complex numbers that do not diverge, i.e. when you plug the output value of successive iterations back into the input starting from zero and the set doesn’t expand in size.

For example, letting c = 1 gives the sequence 0, 1, 2, 5, 26,…, which tends to infinity. As this sequence is unbounded and expanding, 1 is not an element of the Mandelbrot set. On the other hand, c = −1 gives the sequence 0, −1, 0, −1, 0,…, which is bounded, and so −1 belongs to the Mandelbrot set.

So why is this interesting? Well, when you plot these two sets of numbers (Mandelbrot and non-Mandelbrot set) graphically and you assign colours to the speed of divergence (how fast the size increases) you end up with fractals! Infinitely zoomable recursive images that are not only fun to look at, but also think about. Below is a diagram of the Mandelbrot set. You can zoom in infinitely, delving deeper and deeper into squiggly mathematical shapes. How can you start with a finite plane (picture) and have an infinite border? Think about it!

 

Mandel_zoom_00_mandelbrot_set

Infinitely zoomable Mandelbrot set.

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