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# 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. 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.
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 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, they can impress even complete non-mathematicians and 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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[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.
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.