This page shows how to construct an origami hyperbolic parabola. This shape is also called a saddle or saddle surface sometimes. This model is not my own creation. Instructions for this model can be found in Paul Jackson's book "The complete origami course", where it says that a recent rediscoverer of the model was John Emmet. It's not clear where this model originates, though it crops up in various places, including in the work of the Bauhaus artists.
Take a square and crease the diagonals.
Then you have to fold the edge into the center. Only crease the section between where the creases diagonal are:
Open out, and crease the opposite side:
Then fold over the top edge to the 1/4 mark, and then the 3/4 mark, (so you'll be making folds at the 1/8 level, and the 3/8 level):
Then repeat the other side:
And you keep on going; next you fold to the 1/8 level, then the 3/8 level, then the 5/8 level, then the 7/8 level
Next, do the same thing for the other direction to get:
Then turn it over, and make folds in between each of the ones already made, but in the other direction
You end up with something like this:
The next step is to fold all the creases. It's a bit like folding a fan on each edge.
This stage is a little difficult to describe in more detail. Perhaps the following avi files might help. These show versions of the hyperbolic parabola with 4 concentric squares, and 8 concentric squares (which is the case described above). For the photo at the top you actually have to fold 16 concentric squares, which is done in a similar way; if you can fold the parabola for 4 and 8 concentric squares, you should also be able to manage 16; the more folds, the smoother the final piece of origami, but also the more acurately you need to fold. Images: 1a) (4 squares) Finishing off the above folding, 1b) (4 squares) Forming the parbola shape, 2a) (8 squares) Forming the parbola shape 1, 2b) (8 squares) Forming the parbola shape 2, 2c) (8 squares) Forming the parbola shape 3.
And you end up with something like this: