Recently I have folded several test pieces and a finished model incorporating a diagonal shift element. Here are several crease patterns showing how that element works, along with some notes and folding hints below:
For each design, the “curves” marking the top and bottom of the diagonal shift are based on sine curves. The sine curves are offset by one gore (the top curve is shifted one gore right relative to the bottom gore). This allows all the mountain folds to cross the centers of their gores at exactly the same height. Without this offset, the crease pattern will not collapse correctly.
One of the biggest challenges in designing these forms is figuring out how far apart the two sine curves need to be. I wrote an Excel spreadsheet to automatically calculate the correct distance based on the angles and distances in the crease pattern.
Of course, this element can be incorporated into more complex models, like I did recently. I’m working on folding more models of this sort.