2. A. Ineffective Ways to Measure

Length of the Koch curve

On the left is a summary of the data so far, and an extrapolation. On the right is a graph of Ln vs n.

n length of a segment number of segments Ln 0 1 1 L0 = 1 1 1/3 4 L1 = 4/3 2 1/9 = 1/32 16 = 42 L2 = 16/9 = (4/3)2 3 1/27 = 1/33 64 = 43 L3 = 64/27 = (4/3)3 ... ... ... ... n 1/3n 4n Ln = (4/3)n

The length of the Koch curve is greater than each Ln, so greater than every number. That is, the Koch curve has infinite length.

Here's an interesting corollary to the infinite length of the Koch curve.

Not only does the Koch curve itself have infinite length, but measured along the curve, the length of any pair of points in the Koch curve is infinite.

Do you see why?

To make these computations concrete, note that if the original L0 is 1 meter, then

length of a segment number of segments length of the curve L24 3.5(10-12) m 281, 474, 976, 710, 656 1 km L128 8.5(10-62) m 115, 792, 089, 237, 316, 195, 423, 570, 985, 008, 687, 907, 853, 269, 984, 665, 640, 564, 039, 457, 584, 007, 913, 129, 639, 936 1 light year

Yet both have a horizontal extent of only one meter.

Return to Ineffective Ways to Measure.