The Difficulties of Folding Paper

So we had our tutorial this week and there was section dedicated to folding paper. At first, I was wondering whether I'm enrolled in a computer science course. But it seems that paper folding or the crease patterns in paper folding has an awful lot to do with algorithms, which seems to be at the core of computer science (which I do hope to learn about in the following weeks).

Below is my attempt to discover the algorithm in knowing the crease patterns of folding paper.

Vertex pointing up = +
Vertex pointing down = -

First Fold                                  -                                  1

Second Fold                         + - -                               3

Third Fold                       + + - - + - -                         7

Fourth Fold         + + - + + - - - + + - - + - -              15

 

The above was how I sorted out what crease patterns and how many creases there were per fold. The easiest to figure out was the number of creases just by staring at the numbers, which is 2n + 1 where 'n' is the number of folds. And it was also notable that '-' was always is in the centre


I had a few theories when I was comparing my second and third fold, but those theories fell apart when I reached by fourth fold.
Here was one of my theories that fell apart:

1. For the third fold, if we were start from the middle and compare what was on the right side to the left side, the signs were opposite of each other. At first there was one positive on the right side, and one negative on the left side. When we moved down more, the right side had 2 negative signs which is complementary to the 2 positive signs on the left. So as we move down from the middle, the signs alternate, and the number of each signs' occurrence increases by one, starting from the central '-'. Now this fold compared to the second fold was also opposite in signs to what was immediately beside the central '-'. This led to to hypothesize that the fourth folds would have a pattern like this: + - - + + - + - - + - - + + -. However this pattern is completely off from reality. Firstly, the immediate right and left signs of the central '-' is the opposite of what the third fold had. So, I needed a new hypothesis. 

After staring at the pattern for a while, I noticed that on the right of the '-' of the fourth fold was the exact same pattern as the entire third fold. Then I asked myself what relation did the right side have with the left side. Then I came to realize that I was sort of on the right track with my first theory above; the right side was opposite in signs with the left side. So, now I predict the fifth fold to be:

+ + - + + - - + + + - - + - - - + + - + + - - - + + - - + - -

The actual result (which really tested my fine motor skills):

was what I predicted! Tadah! 

I'm not sure whether this is the simplest way to go about it, but for now this is my solution.

1. The first fold will always be '-' which would always be in the center of subsequent folds
2. The next fold will contain the pattern of the immediate previous fold's pattern on the right side of the central '-'
3. On the left side of the '-' on the same fold (as 2), the pattern would be the opposite as the right side as the folds travel down from the middle towards the end