Calculations at a Koch curve. The Koch curve is built by dividing a line into three equal segments, putting an equilateral triangle above the middle segment and removing the bottom line of this. This is repeated with the four new lines, and so on, infinitely. This makes the Koch curve a fractal with the Hausdorff dimension ln(4)/ln(3)≈1,26186 and infinite length.
The single steps are called iterations. Here, the length of a Koch curve after n iterations can be calculated. Its height is the height of the first equilateral triangle. Enter the number of iterations and one of the two lengths. Choose the number of decimal places, then click Calculate.
Formulas:
m = l * (4/3)n
h = √3 / 6 * l
Lengths and height have the same one-dimensional unit (e.g. meter). The number of iterations is dimensionless and an integer.
The name fractal was Benoît by Benoît Mandelbrot, it comes from the Latin word fractus, which means broken. This refers to the Hausdorff dimension of such structures, which, unlike the geometric shapes we are used to, is not an integer. To calculate the Hausdorff dimension for the Koch curve, see above. If it is tripled in each direction, this curve becomes four times as large. A rectangle, for example, becomes nine, i.e. 3² times as large, if it is tripled in each direction, and is therefore two-dimensional. A cuboid, on the other hand, would be 27, i.e. 3³ times as large, and is therefore three-dimensional.
However, the famous Mandelbrot set named after Mandelbrot is also a fractal, although not only the set itself, but also its boundary line has the Hausdorff dimension 2, although it is not known whether the boundary has an area or not. But this is only a glimpse into the world of fractals, into which the Koch curve represents a good introduction.