Rebecca Gieseking

In my last tutorial, I talked about how pleated folding works.  Here I’ll focus on how to find the right dimensions for a pleated vase or bowl.  I will be using some basic geometry and algebra for the calculations, but I’ll give some hints along the way to make the math as easy as possible.

Planning a Shape

The first step in the design process is choosing the shape for the origami form. Drawing the shape on a grid will make it easier to figure out the dimensions later. Here’s the shape I designed for this tutorial:

Some hints for choosing a shape:

1.In general, the simpler the shape, the easier it will be to fold.
2. Convex curves (like the body of the bowl) are easier to fold than concave curves (like the neck of the bowl).
3. Curves that stay close to vertical are easier and don’t have to be nearly as precise as curves that are close to horizontal.

Gore Number and Width

First we’re going to decide on a number of gores and on the width of each gore based on how big the final model will be. The width of the paper will be the circumference of the shape at its widest point, plus one extra gore for overlap.

For my design, each square in the pattern above will be 0.5 cm, so the radius of the largest circle is 5 cm. That means the circumference is (5 cm)(2)(3.14) = 31.4 cm, which I will round up to 32 cm. The circumference then divides evenly into 16 gores that are 2 cm wide. When we add one extra gore for overlap, that gives a total of 17 gores that are 2 cm wide, for a total width of 34 cm.

Paper Length

The next step is to calculate the length of the paper. Along the length of the paper, the paper will go from the center of the base to the top rim. We can estimate the length of the curve by converting it to a series of straight line segments.  Then we can use the Pythagorean Theorem to calculate the length of each line segment. If the height of the line segment is h and the width is w, then the length is √(h² + w²).  To get the total length of the paper, we just add up the lengths of all the segments.

For my design, the length of each line segment is shown in the picture above. The total length is 1.4 cm + 3.2 cm + 0.5 cm + 4.3 cm + 2.5 cm = 11.9 cm. Paired with the width we calculated above, the paper will be 34 cm wide x 11.9 cm tall.

Creating the Curved Form

The final step is to figure out the shape of the curved fold in each gore. Here we’ll use the same straight line approximation of the curve as we did above. At each end of each line segment, we can calculate how far the curve should be from the edge of the gore based on the radius of the form at that point. Where the radius is zero, the curve should be at the exact center of the gore at that point. Where the radius is at its maximum, the curve will touch one edge of the gore. Where the radius is half of its maximum, the curve will be halfway between the center of the gore and the edge, or a quarter of the way from one edge of the gore.

Here are the dimensions for my design:

Since my gores are 2 cm wide, the center of the gore is 1 cm from its left edge. At the very bottom edge, the radius is zero, so the curved fold is 1 cm from the edge. Because the radius at the widest point is 10 squares, each square corresponds to 0.1 cm. From those measurements, we can calculate the width at each point just by counting squares.

At this point, we have figured out all of the dimensions that go into the crease pattern. The folded model from the design looks like this:

In my last tutorial, I gave step-by-step instructions on how to fold a vase using a pleated folding technique.  In this tutorial, I’ll talk about how to design your own pleated vases and bowls.  This post will focus on how pleated folding works, and later I’ll write a tutorial on how to calculate the correct dimensions for any shape.  To start, here’s the pleated vase from my last tutorial:

Why does this crease pattern fold into that shape?  Let’s start from very simple crease patterns and work our way up.  Imagine that you took a long piece of paper and accordion-folded it, using equally spaced mountain and valley folds. All of the layers of paper end up on exactly top of each other to make a very narrow strip of paper.

Now imagine that instead of the folds being equally spaced, the pattern goes something like this: large gap, mountain fold, small gap, valley fold, repeat.  If this pattern is folded, the layers won’t end up exactly on top of each other.  The paper will be somewhere between the length it was when the folds were equally spaced and the length of the original strip of paper.  As the small gap gets smaller and the large gap gets larger, the folded strip of paper gets longer.

Now let’s take those three examples and attach one end of the paper to the other end to make a tube.  In all three examples, the paper starts off the same length.  In the first example where the folds were equally spaced, the paper forms a very narrow tube.  The more unequally spaced the folds are, the wider the tube is.

From these simple examples, we can understand how the pleated forms work.  Just like the accordion-folded examples, the crease pattern below is based on alternating mountain folds and valley folds.  In this example, the valley folds are straight and the mountain folds are curved.  Either the mountain folds or the valley folds need to be straight lines; otherwise the form doesn’t collapse cleanly.  We can think of the shape as a tube like our last set of examples, but in this case the width of the tube changes.

At the bottom edge of the crease pattern, the mountain and valley folds are equally spaced.  Based on our accordion-folded example, that means the tube should be very narrow.  And that’s what happens: at the center of the base, the layers of paper all match up, closing off the base.  About a quarter of the way from the top edge, the mountain and valley folds in each pair are almost on top of each other.  Here we expect the tube to be very wide.  Again, that’s what happens: the place where the mountain and valley folds are closest together becomes the widest part of the vase.  At the top edge of the crease pattern, the mountain and valley folds are somewhere in between – not evenly spaced, but not on top of each other.  The top edge of the crease pattern becomes the rim of the vase, which is intermediate in width.

Here’s one more example.  In the crease pattern below, the folds are equally spaced at both the top and bottom edges, and there are two places in between where the mountain folds curve so far out that they touch the valley folds.  So in this model, both the top and the bottom are very narrow and there are two widest points in between.

Stay tuned for my next tutorial on how to figure out the dimensions for the crease pattern.

Here I’m going to show how to fold a simple vase using a pleating technique.  I call it pleating because the shape is constructed by a series of mountain fold/valley fold pairs, just like pleats in fabric.  This is the crease pattern I’ll be working from:

The crease pattern has seven identical rectangles, which I call gores.  In the final form, the two end gores will be glued on top of each other to turn the paper into a tube, so only six gores will be visible.  I designed this model to be simple enough to be folded from printer paper, but almost any paper should work.

Pre-creasing the crease pattern

The first step is to fold along all the lines in the crease pattern.  The blue lines indicate valley folds, and the red lines indicate mountain folds.  This step will make it easier to collapse the vase into its final shape.

The mountain folds are a bit more challenging because they are curved.  I usually hold the paper up and pinch along the curve using both hands.

After all the folds are pre-creased, the model should look something like this:

Collapsing the form

At this stage, all the major folds are in place.  All that is left is to collapse the vase into its final shape using those creases.  The first step is to glue the paper into a tube. Glue the front of the gore on one end of the paper to the back of the gore on the other end of the paper.  It only takes a tiny bit of glue – I used too much here, which wrinkled the paper.  The paper should end up as a tube with six gores visible.

Once the glue is dry, it’s time to start collapsing the shape.  I find it easiest to start collapsing from the base.  Fold the paper along the creases you already made.  In the very center, all the layers of paper should start to align:

For this model, it’s a bit tricky to get the base to stay closed. (With more gores and nicer paper, the base locks into place very easily.)  Pinch the six corners of the paper that are sticking out to help lock the base in place.

Here’s what the finished base looks like:

The last step is to collapse the top edge of the vase.  Again, fold along the pairs of creases that are already there.  To get the paper to stay folded, fold the corner of the flap under.  (Alternatively, you can glue the flaps down.)

Here’s the fully collapsed top of the vase:

The finished vase

Stay tuned for my next tutorial on how to design your own pleated origami models!