Fold a Hyperbolic Paraboloid

Some of you may be familiar with the paper folding exercises that were once taught at the Bauhaus, including the design for a simple hyperbolic paraboloid. This project is a riff on those experiments using a folding pattern of my own design. It's a bit more difficult to execute, but the end result is quite neat.

This project consists of two main steps. First, you need to pre-crease the paper with diagonal and orthogonal grids. Steps 1 - 8 cover this process, but it's more or less self-explanatory so feel free to fold the grid however you like and jump ahead to what comes next. The takeaway is that you want to end up with the paper divided by creases into sixteen sections in both directions diagonally and in one direction orthogonally and with a single crease dividing the paper in two in the other direction orthogonally. You'll eventually need more than a single crease in this direction, but it gets a little more complicated so wait on that, or jump ahead to Step 9 to get the details.

Once the grid of pre-creases is finished, you'll use it to fold the paper into a hyperbolic paraboloid.

Step 1: Material

First things first, you need a sheet of origami paper. I'm using a small sheet of gold foil paper, but any origami paper will do. Whatever you use, it needs to be durable enough to withstand being folded and unfolded a number of times.

Step 2: Fold and Unfold the Paper in Half

Begin by folding the paper in half, colored side out, and giving it a good crease. Next, unfold the paper and smooth it flat.

Step 3: Fold in Half the Other Way

Next fold the paper in half in the other direction, white side out. Unfold and smooth flat.

Step 4: Fold in Quarters

In the same direction, fold both edges of the paper in to the central crease created in the previous step. Unfold and smooth flat. The paper should now be subdivided into four sections of equal width.

Step 5: Fold Diagonally

Fold and unfold the paper in half diagonally. Repeat along the other diagonal.

Step 6: Fold the Four Corners in to the Center

Fold each of the four corners into the center of the paper. Unfold and smooth flat.

Step 7: Continue Dividing Diagonally

Continue subdividing the paper until you have sixteen equal divisions in each direction diagonally. It might look a little daunting, but won't take more than a few minutes.

A word of advice- adding all of these folds to the paper distorts it slightly, and your grid will turn out better if you work simultaneously in both directions rather than folding all sixteen divisions along one diagonal before starting on the other.

When you're finished, smooth the paper flat.

Step 8: Continue Dividing Orthogonally

Returning to the orthogonal direction in which you had previously divided the paper into four equal subdivisions, continue dividing first into eight and then sixteen subdivisions.

Unfold and smooth the paper flat.

Step 9: Complete the Grid

This last step is a bit tricky, because the paper is not being evenly subdivided.

First, notice that the paper is now divided into a diagonal grid of tiny squares. All the way back in Step 2, you created a single center fold in the paper. Returning to this direction, we are going to add four additional folds. Each one should be separated from the others by one and a half of the tiny squares. These should all be valley folds, folded with color side in.

Unfold and smooth flat. You're finished with pre-creasing the grid!

Step 10: Collapse the Central Crease

From here on out, we're going to be using the creases that we've created to fold the model into its final form.

To begin, use the orthogonal creases which run at right angles to the ones that you created in the previous step to fold alternating mountain and valley folds all the way across the paper. Press tightly and then spread the paper back out a bit, but not completely.

Now, use the centermost crease from the previous step to create a valley fold while simultaneously creating locking folds on the underside of the paper. This is a bit tricky, so take a look at the embedded video to see exactly how it is done. When you're finished with this step, the paper should look like a series of ridges with a ninety degree bend across its middle.

Step 11: Fold Down Along the Adjacent Diagonal Creases

Reverse all of the ridges by folding the paper along the zig-zag of diagonal creases immediately adjacent to where you were working in the previous step. This is another tricky fold, but a careful look at the photos should tell you everything you need to know.

Once you've finished all of these folds, press tightly to set them in place and spread the paper back out a bit.

Step 12: Continue Activating the Grid of Pre-creases

Repeat the last two steps at each of the four creases created in Step 9, each time first folding up along the crease while simultaneously creating locking folds underneath and then reversing the ridges by folding down along the adjacent zig-zag of diagonals. Work all the way out to either end of the paper.

As you fold, periodically flip the piece over to check what's happening underneath. It's easier to see the folds falling into place on the underside of the paper.

Step 13: Sculpt the Paraboloid

Once you've activated all of the creases, give the model one final firm press to lock the folds in place. Spread it back out slightly and the paper should naturally spring into a hyperbolic paraboloid. Sculpt it a bit to get the shape you want.

Step 14: Done!

You're finished!

This is a fun pattern to play around with, and you can experiment with folding it at different scales, with different types of paper, etc. Anything works. It might be helpful to start out by folding a simple version like the coral colored piece in the photo, just to get a better feel for how the folds work. Have fun!