von Koch Curve
This applet shows the von Koch curve as the user iterates through the stages. It also calculates the length of and the area under the curve. By building the curve for infinite iterations, a complete von Koch curve would be formed. By observing the graph of length, it can be seen that the length of a von Koch curve would be infinite. By looking at the graph of area, it can be seen that the area enclosed by the curve if the bottom line was to be redrawn would be finite. That area is equal to the square root of 3, over 20 or about 0.0866. The von Koch curve is a fractal. Successively finer structures are self-similar. By zooming in on a portion of the curve, the same structure is observed. Furthermore, the curve has a fine structure at any level of magnification; the triangular forms continue to an infinitesimally small scale. The von Koch curve is displayed in two dimensions as a curve lying in a plane. These dimensions provide extra information, though; the position and shape of the curve is known, so only the position along the curve is needed to fully describe a point. Unlike a simple line segment, a single value cannot describe this position. As the length of the von Koch curve is infinite and the distance between any two points along the curve is infinite, no single value can describe a point's position. The dimension of this curve is log(4)/log(3) which is about 1.26.
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