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.
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: