Fractal Folds Lab Answers

1. (b) The edges include all the cuts, so the easy answer is that the edges have infinite length.

For possible later use, we find the length of the edges at step n. Refer again to the picture of the first few steps to find the lengths of the folds.

Here is a tabulation of the lengths of the folds.

step	number of folds	length/fold
1	1	1 = 20
2	3	1/2 = 2-1
3	9	1/4 = 2-2
4	27	1/8 = 2-3
	...
n	3n-1	21-n

So the total length of the folds is

1 + 3/2 + 9/4 + 27/8 + ... + 3^n-12^1-n + ...

The sum is a geometric series with ratio 3/2, hence it diverges.

In the n^th step the edges have length

(3^n-1/2^n-2) + (3^n-1sqrt(2)/2^n-1) + (3^n-1/2^n-1) = (3ⁿ + sqrt(2)3^n-1)/2^n-1.

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