I realized the other day that a zine pattern looks a lot like a space filling curve, just less consistent. The Peano curve bears a striking resemblance to a zine pattern.
I've been trying to figure out what branch of mathematics this stuff is, and I think it's somewhere in the graph theory area.
I was able to make some progress in sequence validation. I wrote an excel sheet that takes a sequence of folds, e.g. AVAIAVAI, a starting position, and a starting heading, and tells you whether or not there are any conflicts, whether the sequence makes a loop, and whether there are any bubbles (pages that aren't used), as well as the width, height, and number of pages. This is all accomplished by imagining a grid and calculating how each fold moves the path of the zine along the grid.
I'm certain there's a better way to do this, but I don't know it yet.
The calculation section of the sheet has columns for the step, the fold, the x and y positions, and the direction of the line. These coordinates and direction are all concatenated together to create a string, and those strings are then all compared to every string before them to check for any duplicates, which would mean an invalid pattern.
Each line looks at a fold in the sequence and changes the coordinates and direction accordingly.
The resulting table of coordinates can be plotted on a graph and connected with lines to show the flow of the folds, with each line segment representing a fold.