This model is a perfect example of the weird and wonderful things you can do with origami. The model itself is pretty simple: you fold two diagonals and n²-1 evenly-spaced concentric squares that you then shape as pleats: mountain, valley, mountain, valley, etc. The description is more complicated than the folding itself. But what makes it so fascinating is the saddle-like superstructure that emerges as you fold. You can try and hold the paper flat (with the pleating), but it will automatically pop back into this saddle-shape (which is reminiscent of a hyperbolic paraboloid, hence the name). So basically, what you’ve created is a self-folding model. How cool is that?
If, somehow, this has yet to blow your mind, here’s something guaranteed to do so: strictly speaking, this model doesn’t exist. About six years ago, Erik Demaine and some colleagues discovered* that, from a purely mathematical standpoint, this crease pattern yields an impossible structure. You have to add or subtract creases to make it foldable. The many years of people folding this model thus poses a bit of a conundrum which still hasn’t been solved. Demaine’s best guess is that there are many unseen microfolds in the paper which allow it to be expressed in physical reality, but it’s still an open question.
This is one of the many reasons I love origami: there seems to be an infinite capacity for discovery and creativity within the scope of something so seemingly simple as folding paper.
*If you would like some sources, here is Erik Demaine’s talk where I heard about the impossibility of this model. I’ve linked to the part where he discusses the hyperbolic paraboloid, but the whole talk is fascinating. If you would like to read the published paper on this subject, you can find it here.