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Miloslav Ciz 2023-11-19 18:06:37 +01:00
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fractal.md
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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.
Fractals are the [beauty](beauty.md) of mathematics, they can impress even complete non-mathematicians and so are probably good as a motivational example in [math](math.md) education.
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*.
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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 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.
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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.