1: Everything you need to know about the Mandelbrot Set (Pure Mathematics and Art)

The Mandelbrot Set is a set of complex numbers commonly used in art, due to its mathematical beauty. It was first defined by Robert W. Brooks  and Peter Matelksi in 1978, and the first visualisation was made by Benoit Mandelbrot in 1980. 

Credit: American Scientist

A lot of websites, videos, applets, and so on do a repeated zoom into the boundary of the Mandelbrot Set. But in mathematical terms, we define the Mandelbrot Set as the set of complex numbers which (under iteration) stays bounded within two. Here’s how it’s mathematically represented (don’t skip this, it’s easy math):

f_c (z) = z² + c, f_c (z)<= 2

Let’s take an example here to understand this. 

If c=-2, 

f_-2 (0) = 0² + (-2) = -2 (You start with z=0 as default)

f_-2 (-2) = (-2)² + (-2) = 4+ (-2) = 2 (You take the result of the previous function and input it as z in the next iteration)

f-₂ (2) = (2)² + (-2) = 4+ (-2) = 2

We know that the result of the previous function (2), upon another iteration will yield (2), which then yields 2 - and so on. Since the result constantly yields 2, c = -2 is a part of the Mandelbrot Set. But it doesn’t necessarily need to repeat a single value on every iteration (like 0.0625 - which does not become large enough to leave the set, or 0 - which does not change at all on iteration, or -1, which alternates between 0 and 1).

Alternatively, let’s take an example of something that is not inside the Mandelbrot Set. Say, 1.

If c=1, 

f₁ (0) = 0² + (1) = 1

f₁ (1) = (1)² + (1) = 1+ 1 = 2

f₁ (2) = (2)² + (1) = 4+ 1 = 5

f₁ (5) = (5)² + (1) = 25+ 1 = 26 …

Each iteration gets larger and the iterative function tends to infinity, eventually. It’s also worthy to note that once you leave a circle relation drawn on a complex plane with radius 2, then no value can ever be part of the Mandelbrot set and will tend to infinity for sure.

So you know what happens on the inside of the Mandelbrot Set, and you know what happens on the outside. But what is the Mandelbrot Set known for among artists and the general populace uninterested in mathematics? Its boundary! 

How does the boundary of the Mandelbrot Set work? It follows the same iterations of the same function. No change. The biggest difference, however, is that when we iterate the function a part of the circle drawn by the function may start to leave the Mandelbrot Set (or else they iterate over themselves). This leads to the development of shapes that show smaller structures and you can zoom in infinitely. Unlike the Mandelbrot Set, this region has its Hausdorff Dimension (or topological dimension) as 2 instead of 1 (Shishikura, 1994). Hence, unlike the Mandelbrot Set, it is harder to understand the parameterization of the boundary of the set. Consequently, I shall not be explaining the math of this any further.

Something I found extremely surprising when I first read about the Mandelbrot Set is that the art we see about the set is simply the result of a basic complex function (explained earlier).  On these images, the stability of the fractal is shown by different colours. Stability essentially means that for a minor change in the number of for a higher number of iterations, the number will definitely stay within the Mandelbrot Set. You can read a bit more about chaos theory and the butterfly effect if you want to understand these topics in alternative contexts. Different artists and coders use different colours to represent degrees of stability, but black is almost always used to show the main, stable components of the set. This art is the perfect representation of the beauty of math. Two artists’ works I particularly like are William Latham and Scott Draves. 

Credit: Scott Draves, Softology

I really hope everyone enjoyed this week's blog! Drop a comment if you have anything you want me to cover, or if I made a mistake, or if you simply enjoyed the post.

 

Here’s a cool website to check out if you want to know the iterations of a value you’re wondering is in the Mandelbrot Set or not: https://math.libretexts.org/Bookshelves/Analysis/Complex_Analysis_-_A_Visual_and_Interactive_Introduction_(Ponce_Campuzano)/05%3A_Chapter_5/5.05%3A_The_Mandelbrot_Set 

Here’s some nice Mandelbrot Set art: https://fineartamerica.com/art/mandelbrot+set, https://www.anart4life.com/the-mandelbrot-set-and-fractal-art/#:~:text=Fractal%20Art%2C%20also%20called%20Mandelbrot,images%2C%20animations%2C%20and%20media and https://web.mit.edu/jorloff/www/chaosTalk/mandelbrot/davideck-js/MB.html (to zoom in)

Pi and the Mandelbrot Set: https://youtu.be/d0vY0CKYhPY (Shoutout Numberphile, one of the best resources for mathematics)

01001100 01101001 01110110 01100101 00100000 01001100 01101111 01101110 01100111 00100000 01100001 01101110 01100100 00100000 01010000 01110010 01101111 01110011 01110000 01100101 01110010 00100001 (No, your computer isn't hanging).

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