Folderol

Last weekend, Marika and I went to Fête du Lac, the largest fireworks display in Europe, held over the beautiful lake. We went for the cheap seats, which didn't include seats. Most people brought blankets, but Marika found a great waterproof fold-up mat for us:

The problem came when we went to fold it back up at the end of the evening: How do you make sure the handles line up again? I managed to bumble my way to a solution, I'm sure due to years of experience doing origami with my mother, but it got me thinking about how many ways you can fold something.

This is called the Map-Folding Problem, and there is so far no known way to predict the number of ways to fold a map, aside from trying every solution. The main thing I was curious about was what a particular folding looked like once unfolded – the pattern of mountains and valleys. Naturally, I wrote a Python script to figure it out:

Alternating

Repeated

Here we're considering a 3 x 3 grid, which needs 2 folds in each direction. The color of the lines indicates the direction of the fold: red for valley and blue for mountain. We make each fold at position 1, but in the first image we alternate directions, and in the second we go twice in one direction, and twice in the other.

Looking at the unfolded images, it's reasonably easy to spot where each fold happens, but as the number of folds increases, the complexity grows rapidly. So far the result is only known up to 7 x 7: 129950723279272! At that point, your best option might be to crumple the map (or yourself) into a ball, and enjoy some fireworks: