In origami there are two ways of folding a piece of paper. The usual way is to fold the paper upwards and if unfolded the crease is called a valley fold and from the side it looks like this: \/ If the paper is folded backward instead the crease is called a mountain fold and looks like this: /\.

If you take a strip of paper and fold it in half twice (left over right) and then unfold it without flipping you will get a figure that looks something like this from the side:

/\_/

Which consists of the following sequence of folds:

/\ \/ \/

Describe the sequence of folds if the original strip of paper had been folded over 10 times and then unfolded.

My description is not going to be complete, and I am not using the  /\  \/ graphics here either. But first, an overview of my structure.

The layers of paper are 2^n, where n is the number of times that the paper is folded over.

When opened out there will be 2^n segments separated by 2^n-1 creases some of which will be mountains and others will be valleys.

My Process
I am going to consider mountain creases as '1' and valleys as '0'.

Firstly I have one flat sheet of paper.  I am considering a valley at the left edge of it,  "0_". If I add another sheet to the left side of the '0' I have the \/ structure. This also represents one fold.

Now I place another '0' at the left of this arrangement and copy the right side to the left as if in a mirror, except that 0's become 1's and 1's become 0's.  So my creases after two folds are represented by 100.

Add another '0' to the left side and flip the right side as indicated above.

Continue until you have 1024 segments or 1023 creases.  The 511 1's and 0's sequence will be a 'mirrored inverse' of the 511 sequence on the right side of the 512th fold which is a '0'.

Wonder what this binary number becomes in base 10?

I had thought to program this operation but I have a feeling that I may not be allowed to write a string length up to 512 characters.  

Posted by brianjn on 2006-07-19 20:31:34