# Koch curve To generate the Koch curve, the rule is applied indefinitely, with a starting line segment. Note that, if the length of the initial line segment is $l$, the length $L_{K}$ of the Koch curve at the $n$th step will be

 $L_{K}=\left(\frac{4}{3}\right)^{n}l$

This quantity increases without bound; hence the Koch curve has infinite   length. However, the curve still bounds a finite area. We can prove this by noting that in each step, we add an amount of area equal to the area of all the equilateral triangles  we have just created. We can bound the area of each triangle of side length $s$ by $s^{2}$ (the square containing the triangle.) Hence, at step $n$, the area $A_{K}$ “under” the Koch curve (assuming $l=1$) is

 $\displaystyle A_{K}$ $\displaystyle<$ $\displaystyle\left(\frac{1}{3}\right)^{2}+3\left(\frac{1}{9}\right)^{2}+9\left% (\frac{1}{27}\right)^{2}+\cdots$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{n}\frac{1}{3^{i-1}}$

but this is a geometric series of ratio less than one, so it converges. Hence a Koch curve has infinite length and bounds a finite area.

A Koch snowflake is the figure generated by applying the Koch replacement rule to an equilateral triangle indefinitely.

Title Koch curve KochCurve 2013-03-22 12:05:34 2013-03-22 12:05:34 akrowne (2) akrowne (2) 8 akrowne (2) Definition msc 28A33 msc 28A80 Koch snowflake