The curve of Helge von Koch (1904)

A line is divided into three equal sections. The central one is removed, and two lines of equal length are put in its place. In this way the figure below is generated.

In the center is now an equilateral triangle, with its lowest side removed. What would happen if you were to divide every line into three pieces, remove the central piece, and replace it with two other ones of equal length just like was done with the first line? You would see the situation below.

If one did the same again:

After 100 iterations I had the following picture:

Now suppose that the length op the first line was 1 dm (=10cm). Then the length of the segments in the second picture would be 1/3 dm. The total length of the segments would be 4*1/3 dm=4/3dm. After the second iteration there are already 16 segments in the curve. With each iteration a segment gets 3 times as small, but 4 times as many segments are created. The total length of the 16 segments is therefore (4/3)

After 100 iterations I have 4

The length of the 'curve' is (4/3)

If you were to stick the end of the curve to the earth, and were to pull the other end out into space, to try and pull out the 'winkles' in the curve, it would only be smooth after you were a good way past the moon. Is this the longest curve on the Internet?

If you were to continue iterating into infinity, the curve would become infinitely long. This curve is called "Koch's curve"; you can't draw it, which is why it only exists in our imagination.

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