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fractal.md

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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 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.
+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 nature, such as trees, mountains or clouds, often exhibit fractal structure. Fractals show self-similarity, i.e. upon "zooming" into an ideal fractal we observe that it is composed, down to an infinitely small scale, of shapes resembling 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 they 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.
+As for the [history](history.md) of fractal theory, the mathematical interest in them seems to date back to 17th century and Gottfied Leibniz's study, although humans have been "intuitively" aware of fractal patterns for as long as anyone will remember, fractals are encountered in oldest architecture etc. At the beginning of 20th century two of the most iconic fractals were described: the Koch snowflake and Sierpinski triangle. This was followed by Felix Hausdorff's definition of fractal dimension in 1918. The word "fractal" was coined in 1975 by Benoit Mandelbrot. [Computer graphics](graphics.md) enabled by the newest technology then led to popularization and increased focus on fractals.
 
 Fractal geometry is a kind of [geometry](geometry.md) that examines these intricate shapes -- it turns out that unlike "normal" shapes such as circles and cubes, whose attributes (such as circumference, volume, ...) are mostly quite straightforward, perfect fractals (i.e. the mathematically ideal ones whose structure is infinitely complex) show some greatly unintuitive properties -- basically just as anything involving [infinity](infinity.md) they can get very tricky. For example a 2D fractal may have **finite area but infinite circumference** -- this is because the border is infinitely complex and swirls more and more as we zoom in, increasing the length of the border more and more the closer we look.
 This was famously notice e.g. when people tried to measure lengths of rivers or coastlines (which are sort of fractal shapes) -- the length they measured always depended on the length of the ruler they used; the shorter ruler you use, the greater length you get because the meanders of the details increase it. For this reason it is impossible to exactly and objectively give an exact length of such a shape.
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 ## See Also
 
+- [attractor](attractor.md)
+- [recursion](recursion.md)
 - [Lissajous curve](lissajous_curve.md)
 - [rose curves](rose_curve.md)
 - [cardioid](cardioid.md)
 - [spirograph](spirograph.md)
 - [procedural generation](procgen.md)
-- [turtle graphics](turtle_graphics.md)
+- [turtle graphics](turtle_graphics.md)