The Koch Curve and Snowflake

This fractal is generated by iteration as follows. The initiator, the initial or $k=0$ step, is a unit line element. The first step, $k=1$, called the generator, involves removing the middle one-third of the unit line and replacing it with two line segments each one-third in length as shown in the figure. The figure now contains four equal line segments. In the next step, $k=2$, each of the four line segments is replaced by the (scaled) generator, leading to a figure with $16$ segments. The procedure is repeated endlessly, $k \to \infty$, to generate the Koch curve.



Note that the Koch curve, with its exact self-similarity at all scales, is obtained only after an infinite number of iterations, at the $k = \infty$ step. The figures leading up to the Koch curve are called prefractals and at large $k$ these already look like the final fractal because of the finite resolution of our eyes. How long is the Koch curve? Clearly at each step in its generation, the length increases by a factor of $4/3$ as a line segment of one-third unit is replaced by two of equal length. Therefore the length of the prefractals diverges as the number of steps increases, leading to an infinite length for the Koch curve! The reason for this is of course obvious since the Koch curve is not smooth but infinitely 'kinky' (In fact the curve consists entirely of corners !) Three Koch curves can be fitted together to form a Koch snowflake. (Try it. Does the result look like a real snowflake ?) Alternatively, one can start with an equilateral triangle and apply the generator of the Koch curve repeatedly to each line segment. Again, the Koch snowflake has an infinite length, but its area is bounded by that of the circle that circumscribes the original triangle. The Koch snowflake, with its infinite length but finite area is an unusual object from the viewpoint of Euclidean geometry where objects occupy finite lengths, areas and volume in finite space. In order to characterize such objects we have to generalize our notion of dimension.