# Fractal

Informally speaking fractal is a shape that's geometrically "infinitely complex" while being described in an extremely simple way, e.g. with a very simple formula or [algorithm](algorithm.md). Shapes found in the nature, such as trees, mountains or clouds, are often fractals. Fractals show self-similarity, i.e. when "zooming" into an ideal fractal we keep seeing it is composed, down to an infinitely small scale, of shapes that are similar to the shape of the whole fractal; e.g. the branches of a tree look like smaller versions of the whole tree etc.

TODO: brief history

Fractals are the [beauty](beauty.md) of mathematics that can easily be seen even by non-mathematicians, so are probably good as a motivational example in [math](math.md) education.

Fractal is formed by [iteratively](iteration.md) or [recursively](recursion.md) (repeatedly) applying its defining rule -- once we repeat the rule infinitely many times, we've got a perfect fractal. [In the real world](irl.md), of course, both in nature and in computing, the rule is just repeat many times as we can't repeat literally infinitely. The following is an example of how iteration of a rule creates a simple tree fractal; the rule being: *from each branch grow two smaller branches*.

```
                                                    V   V V   V
                                \ /   \ /         V  \ /   \ /  V
               |     |      _|   |     |   |_   >_|   |     |   |_<
            '-.|     |.-'     '-.|     |.-'        '-.|     |.-'
   \   /        \   /             \   /                \   /
    \ /          \ /               \ /                  \ /
     |            |                 |                    |
     |            |                 |                    |
     |            |                 |                    |

iteration 0  iteration 1       iteration 2          iteration 3
```

Mathematically fractal is a shape whose [Hausdorff dimension](hausdorff_dimension.md) (the "scaling factor of the shape's mass") may be non-integer and is bigger than its [topological dimension](topological_dimension.md) (the "normal" dimension suh as 0 for a point, 1 for a line, 2 for a plane etc.). For example the [Sierpinski triangle](sierpinski_triangle.md) has a topological dimension 1 but Hausdorff dimension approx. 1.585 because if we scale it down twice, it decreases its "weight" three times (it becomes one of the three parts it is composed of); Hausdorff dimension is then calculated as log(3)/log(2) ~= 1.585.

[L-systems](l_system.md) are one possible way of creating fractals. They describe rules in form of a [formal grammar](grammar.md) which is used to generate a string of symbols that are subsequently interpreted as drawing commands (e.g. with [turtle graphics](turtle_graphics.md)) that render the fractal. The above shown tree can be described by an L-system. Among similar famous fractals are the [Koch snowflake](koch_snowflake.md) and [Sierpinski Triangle](sierpinski_triangle.md).


```
              /\
             /\/\
            /\  /\
           /\/\/\/\
          /\      /\
         /\/\    /\/\
        /\  /\  /\  /\
       /\/\/\/\/\/\/\/\
       
     Sierpinski Triangle
```

Fractals don't have to be [deterministic](determinism.md), sometimes there can be [randomness](random.md) in the rules which will make the shape be not perfectly self-similar (e.g. in the above shown tree fractal we might modify the rule to *from each branch grow 2 or 3 new branches*).

Another way of describing fractals is by iterative mathematical formulas that work with points in [space](space.md). One of the most famous fractals formed this way is the **[Mandelbrot set](mandelbrot_set.md)**. It is the set of [complex numbers](complex_number.md) *c* such that the series *z\_next = (z\_previous)^2 + c*, *z0 = 0* does not [diverge](divergence.md) to [infinity](infinity.md). Mendelbrot set can nicely be rendered by assigning each iteration's result a different color; this produces a nice colorful fractal. [Julia sets](julia_set.md) are very similar and there is infinitely many of them (each Julia set is formed like the Mandelbrot set but *c* is fixed for the specific set and *z0* is the tested point in the complex plain).

Fractals can of course also exist in 3 and more dimensions so we can have also have animated 3D fractals etc.

## Fractals In Tech

[Computers](computer.md) are good for exploring and rendering fractals as they can repeat given rule millions of times in a very short time. Programming fractals is quite easy thanks to their simple rules, yet this can highly impress noobs.

However, as shown by Code Parade (https://yewtu.be/watch?v=Pv26QAOcb6Q), complex fractals could be rendered even before the computer era using just a projector and camera that feeds back the picture to the camera. This is pretty neat, though it seems no one actually did it back then.

A nice [FOSS](foss.md) program to interactively zoom into 2D fractals is e.g. [xaos](xaos.md).

3D fractals can be rendered with [ray marching](ray_marching.md) and so called *distance estimation*. This works similarly to classic [ray tracing](ray_tracing.md) but the rays are traced iteratively: we step along the ray and at each step use an estimate of the current point to the surface of the fractal; once we are "close enough" (below some specified threshold), we declare a hit and proceed as in normal ray tracing (we can render shadows, apply materials etc.). The distance estimate is done by some clever math.

[Mandelbulber](mandelbulber.md) is a [free](free_software.md), advanced software for exploring and rendering 3D fractals using the mentioned method.

[Marble Racer](marble_racer.md) is a [FOSS](foss.md) [game](game.md) in which the player races a glass ball through levels that are animated 3D fractals. It also uses the distance estimation method implemented as a [GPU](gpu.md) [shader](shader.md) and runs in real-time.

Fractals are also immensely useful in [procedural generation](procgen.md), they can help generate complex art much faster than human artists, and such art can only take a very small amount of storage.

There exist also [compression](compression.md) techniques based on fractals, see [fractal compression](fractal_compression.md).

There also exist such things as fractal antennas and fractal transistors.