Update

fractal.md

@@ -6,6 +6,9 @@ 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 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.
+
 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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