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The Koch curve was first described by the Swedish mathematician Helge Koch in 1904 -- long before fractals were thought of.
To construct a Koch curve, begin with a triangle with sides of length 1, so its perimeter is of length 3.
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Onto the middle of each side, add a new triangle one third of the size of the original one.
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The length of the new perimeter is
3 x 4/3 = 4.
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The same process of adding a new triangle one third of the size of the previous ones onto the middle of each side is repeated. After n steps the perimeter has length
3 x (4/3)n.
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The perimeter of the resulting shape becomes infinitely long, but surrounds a finite area. While a simple one-dimensional line fills no space at all, the Koch curve, with infinite length crowding into finite area, does fill space: so it is more than 1- but less than 2-dimensional; in fact, it can be calculated to be of dimension 1.2618.
Benoit Mandelbrot described this 'snowflake'-shape as ''a rough but vigorous model of a coastline'' -- like a coastline, it is self-similar in such a way that it keeps 'becoming longer' the more closely one looks.
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