I see YouTube videos all the time of Mandelbrot zooms, but what are they exactly?



I see YouTube videos all the time of Mandelbrot zooms, but what are they exactly?

In: Mathematics

The Mandelbrot set is a group of numbers that satisfy a condition. Start with a number C and Z=0, apply z²+C. Repeat with the new value as Z over and over.

A number is in the Mandelbrot set if it stays contained through infinite iterations. Instead, it may diverge and go off to infinity.

The set is typically visualized including the complex numbers, in the form of a+b*i* (*i*=√-1).

When is visualized, typically they color the outside of the set based on how many iterations it takes until the absolute value of the digit becomes greater than 2, at which point it is guaranteed to go off to infinity.

It’s an equation that plots a pattern. It happens to be a fractal pattern – which means it looks the same at whatever scale you view it.

So add some colours and it looks great when you zoom in further and further.

The Mandelbrot set is a function that can be printed on graph paper, exactly like simpler functions like y=x^2.

Unlike simpler functions, the Mandelbrot set is infinitely complicated. No matter how close you view it, there is always more detail that you can show. So when a video “zooms” in on a Mandelbrot set, it’s just showing different levels of detail in a video format.

Graphs like the Mandelbrot set that have this infinite complexity are called “fractals”. While the Mandelbrot set is the most famous one, others exist.

So, the original mandelbrot picture is two colors – black and blue.

This is plotted on the complex plane, which is a fancy way of saying two axes (like X and Y).

The black color means whatever number at that point satisfies some particular iterative equation. This equation involves squaring a number over and over again and seeing if it approaches infinity.

For numbers less than 1, greater than -1, and some “imaginary numbers” (Square root of -1), you can repeat this squaring, and never hit infinity. These values are colored black on the original mandelbrot picture. The rest are blue.

The multiple colors in the zoom you see are “different levels of stability”. An arbitrary value will be decided that will act as “the threshold for infinity”, and each color represents the number of iterations before that threshold is crossed.

Now, the extreme complexity and beauty of the mandelbrot zoom stems from “Chaos Theory”, and is a great example of it.

Chaos is the term given to systems that experience drastic changes in output based on small changes in input. [This Gif](https://gfycat.com/honoredfelineguppy) shows a double pendulum in which a couple of different ones are starting at near the same spot and same speed, but very quickly spread apart from one another as though they had no relationship to one another to begin with.

The mandelbrot zooms show how changing the starting value in the equation by even the tiniest amount can lead to a drastic change in how quickly that value blows up to infinity, as illustrated with the rapid changing of colors.