This set is drawn using very simple geometric rules. We begin with a line in which the middle third is replaced by two line segments which form two sides of an equilateral triangle, as shown below:

By pressing the "rescale" button you may continue the process by replacing the middle third of each line segment by two sides of an equilateral triangle. This can be continued three times in this program but the Koch curve itself is the result of continuing this an infinite number of steps. The fractal is self-similar, since if we enlarged any single line segment at any step it would have the same property as the larger part.

It also has another interesting property common to fractals in that it has a 'fractal dimension'. Lines have dimension 1, squares have dimension 2, and cubes have dimension 3, but the limiting Koch function has dimension approximately 1.2618, a dimension higher than that of a line(1), but less than that of a square(2), since the figure does not fill the plane. The calculation is done in the next paragraph.

If a self-similar figure contains n copies of itself and each copy has length s, then the fractal dimension D is defined to be ln(n)/ln(1/s). In this case each segment contains 4 copies of itself and each copy is 1/3 as long as the previous length.

So D = ln(4)/ln(1(1/3)) = ln(4)/ln(3)= 1.2618595071429148