Maze1125 said:

So the title of this was "We're going to tell you what fractals are because people often get them confused." ("What the heck is a fractal?") and then immediately we get told that the article is going to do the exact opposite of that...

For anyone who actually cares what a fractal *really* is:

A fractal is a shape that has such infinite complexity that it starts to take on properties of a dimension higher than that used to create it. This means that these shapes have non-integer dimensions. (The term "fractal" refers to their fractional dimensions.)

For example the Koch snowflake [https://en.wikipedia.org/wiki/Koch_snowflake] is created from a 1-dimensional line but ultimately has a dimension of 1.2619

Yep.

What the article also doesn't mention is that terrain modelling is often done using fractal geometry because trees, mountains, coastlines, river systems all exhibit fractal like patterns, although they're rougher than the examples like the Mandelbrot set or the Koch Snowflake. For instance, a mountain can be drawn by starting with a triangle and breaking it down into smaller triangles as per Samus' example. But because this is taking place in 3d as opposed to just on paper, if those smaller patterns are rotated slightly by random, it creates an effect of terrain fluctuating that mimics natural patterns.

What makes this relevant to

*procedurally* generated content is that the equations used to define the fractal patterns respond to a related brach of mathematics commonly known as the 'butterfly effect': that is, changing the initial condition even slightly leads to dramatically different output. This allows programmers to generate random content across a broad area without having obvious repetition at a macro level. For instance, two mountains in

*Minecraft* that are right next to each other in-game might look quite different to the eye, but the random values that are put into the equation that generates them might only be separated by the tiniest of values.

This is arguably the coolest thing about fractals and chaos theory: a complex system can be randomised using a single value. An entire Minecraft map can be recreated for someone else provided they have the same seed. The underlying equations will respond exactly the same.