The Mandelbrot/fractal

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I look for simple explanations online and I don’t understand.

In: Mathematics

The mandelbrot fractal appears when you try to plot the answer to a specific infinite series.

Take any complex number Z0 (wich has a real part X and an imaginary part Y so Z=X+iY with i×i=-1)

Then repeat this calculation:

Z_(n+1)=(Z_n)²+Z0

So basically every step you square the number and then add the number you started with.

Repeat this infinite times, and depending on what starting number you picked you either get a finite (it converges) or infinite (it diverges) value.

Usually you then colour the starting point in black for converge, or in a colour thats brighter the faster it diverges.

The mandelbrot fractal shows that the question if a certain starting value diverges or not is extremely hard to answer. There are infinitely complex areas, where no matter how far you zoom in you will find more details. (And therefore changing the input value 100 digits after the decimal point *can* radically change the output in unpredictable ways)

Its an apparently simple mathematical process. You take a number and square it, then square it again and so on. One of two things will happen. It will get very very large 2, 4, 16, 256, … or it will get small 1/2, 1/4, 1/16 …. depending if your start point is above or below 1. Easy enough for real numbers.

However if you do the same for “complex” numbers, things get complicated. Complex numbers involve “i”, the square root of -1 so look like (x + y*i) and can be represented on a flat plane with x and y axes instead of a line like “real” numbers. The Mandlebrot set is the border between the start points on the plane that stay small and those that cross the border and grow uncontrollably.

Its really the black area in the middle, numbers that stay small after an infinite number of squarings, but obviously you can’t do that. So you calculate it a specific number of times and colour the points that cross the border before that time with one colour, the group that are still OK after the next set of calculations with another colour and so on. You build up a contour map of how long each start point survives before growing.

A fractal is basically a structure that builds off of itself in smaller, repeating patterns. One real world example is trees…. start with a trunk, then it expands into branches. Which expand into smaller branches. Which expand into smaller branches. Circulatory systems are the same way; starting with large main arteries and branching out smaller and smaller segments. Lightning, snowflakes, river systems, etc can all be described in terms of fractals.

Mandelbrot fractals are clever math formulas that have similar repeating structures.

The Mandelbrot process is a set of instructions for walking around a 2D plane (they use complex numbers to make the description of the math very succinct, but you don’t have to). If you start somewhere on the plane and start following these instructions, one of two things can happen – you either wander around your starting point forever, or you run off into the distance and never return.

The Mandelbrot set itself (the black part in the pictures) is all the starting points where you would end up wandering close by. The multi colored parts are starting points where you would run off and never return. The colors are assigned by how quickly you run off.