The Origami Forum

http://www.langorigami.com/science/math ... ctions.pdf

In this article, there's an equation for finding the maximum number of folds n required using binary folding algorithm to fold a fraction by a given accuracy e:

n = ceiling ((log2 1/e)-1)

Does anybody know if there is a existing proof of this published by Mr. Lang?

what this binary algorithm does is it essentially provides you with the approximation of a prime division of a power of 2
for example- the first few binaries for 1/3 will provide you with these divisions
1/2
1/4
3/8
5/16
11/32
21/64
the algorithm for this was reverse engineered from this method, so just folding using the binary method is pretty much the only proof of it you will get

its hard to explain in words, but once you fold it, it will make perfect sense to you

Maybe it's easier to think about it in the reverse direction, i.e. given a certain number n of parallel folds, what accuracy e can you obtain?

With n=1 folds you can obtain accuracy to within e=1/4 the side of the paper.
With n=2 folds you can obtain accuracy to within e=1/8, and so on. So, by inspection...

e = 1/2^(n+1)

Picking the largest integer n bigger than the inverse of this yields the formula given.