Complete Explanation of the Mandelbrot Set

Original Post Date: Jun 13, 2014
Original Post:

I tried explaining the Mandelbrot Set completely, in a way an 8th grader could understand.

I wrote this in reply to someone explaining how the Mandelbrot Set, the subject of my icon, is generated. I have decided to repost it here.

Basically, mathematicians found that there seemed to be no way to represent the square root of a negative number, as any number, positive or negative, squared was a positive number. They then came up with the idea of the poorly named imaginary numbers and the equally poorly named complex plane. Imaginary numbers are just as real as real numbers, and the complex plane isn’t all that complicated.

√-1 became known as i. Any number could be tagged with i, and it became an imaginary number. Basically, i² was -1, i³ was -i (i² × i), and i⁴ was 1 (i² × i²). Then it looped back, with the fifth power being i, the sixth being -1, and so on. But where could this be represented on the number line? There was no space left or right.

Because it took four “turns” to get back to itself in powers, mathematicians deduced that i would be a quarter turn above the number line… that is, add a line perpendicular to the number line to get something that looks a lot like the Euclidean plane we’re familiar with, with the imaginary numbers along the “y” axis and real numbers along the “x” axis. This became known as the complex plane. It worked a lot like the normal coordinate system, except a point represented a number; instead of (3, -2i) it would be 3+2i. These numbers would be treated as individual entities. When multiplied with another, you would use the distributive property and the properties of i described above. It’s essentially a different type of number line—really, a number plane. It is important to remember each point (called z, on the complex plane, as opposed to x and y) represents one number.

Now you need to understand the concept of iteration. It is very simple. It is a feedback loop. Plug a number into a function. You get an output. Plug this new number back into the function. Repeat. Each time you do this, it’s called an iteration. It compounds onto itself. Let’s say, you have a constant c within the bounds specified below. Now pick a point called z on the complex plane within a window of [-2,2,-2i,2i] (essentially, its bounds). Now plug it into the function z²+c. The constant c will remain the same throughout the entire operation. This will give you a new z. Is this new z out of bounds? No? So it is still in what is called the Julia Set. There are Julia Sets for all constants within the specified bounds. Place a dot there. If it is out of bounds, don’t place a dot there. Plough the new z back into the function for the second iteration? Is the point still in bounds? Follow the same rules as before. Keep doing this over and over again for more iterations.

What is interesting is that, the more iterations done, the more intricate the shapes become. This is due to the notion of chaos theory—how the tiniest of changes, over time, can lead to the biggest of changes. It can mean that a given point is or isn’t in the set after an iteration, even if it is less than a billionth away from the other number that is in the set. Why 2 as the bounds, though? Well, after that, numbers tend off to infinity. 2² is 4. 4² is 8. 8² is 16. And so on. Fractals use a method called color speed to tell how fast a point is tending off to infinity. A number like 2 will speed off to infinity faster than a number like 0.5. Depending on the rate of “acceleration,” a particular color is assigned, from 0 to a certain number less than the iteration depth, to the iteration depth (the color of the actual set). If the number never escapes to infinity, that point is in the set.

Now you may wonder, what if c wasn’t a constant? What if c was the initial z for that selected point z? Everything else is done the same, except for how c is now the initial z for that given point (i.e. it is that point). If you think about it for a second, you’ll realize that it encompasses all constants for all Julia Sets, even for values there seem to be no Julia Sets for. Take the point -1, for example. -1²+-1 is 0. 0²+-1 is -1. It latches onto itself and never escapes to infinity. Take 0. 0²+0 is 0. Note that I use real numbers for simplicity. All complex numbers in the bounds mentioned can be used, too. But these points, no matter how many iterations are done, will never reach infinity. These points are deeply ingrained into what is my profile picture… the Mandelbrot Set. Like the Julia Set, the more iterations done, the more finely detailed the edge becomes. Chaos Theory is why it’s so infinitely intricate, as described above.

Benoit Mandelbrot sadly died in 2010, but was thankfully aware of the song Jonathan Coulton (who also wrote the Portal credits songs) wrote about his most famous discovery bearing the same name. If you’re still confused, I know of a nice short video that explains all of this very nicely. In fact, what I talked about was essentially a summary of what that clip talked about, even though I knew how it worked beforehand. The clip is part of a full documentary on topology, and all of it is worth a watch. It is part four of the “Dimensions: A Walk Through Mathematics” documentary, viewable on YouTube.

It may not appear simple at first, but it is to a mathematician. The fact that something infinitely intricate could come about by simply iterating a complex number in the function z=z²+z until it no longer was contained of a four by four bounding box is simply astounding. The Mandelbrot Set is considered among mathematicians to be the most beautiful in all of mathematics, and for fair reason.

And people wonder why I’m so turned on by math.

Yes, math turns me on. Deal with it. Sooner or later I’m going to repost my justification for such feelings. But that’s besides the point.

Ahh, the M-set is so beautiful.

Something strange. I actually knew what the M-Set was before I actually knew about it. It doesn’t make sense. I must have seen it *somewhere* briefly. But that day when my friend showed me a picture of the M-Set and asked me if I knew what it was… I instantly recognized it, and said “Oh, that’s the Mandelbrot Set! Wait, HOW DID I KNOW THAT?” I also knew what a fractal was, but I think that’s due to a faint memory of ViHart and Numberphile. But to my understanding Numberphile didn’t mention the Mset until very recently, and ViHart never mentioned it (oddly enough… she goes on and on and on about fractals, though). I’ll never solve this enigma, it seems. Though in 7th grade, a friend told me one of my drawings looked like a fractal, so I named it “Flower Fractal.” I still have it somewhere. Maybe. I think. YAY 5:40AM! Time to go to school, soon!

Complete Explanation of the Mandelbrot Set

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