Tutorial: Designing Pleated Forms 2

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:

Vase design

Bowl design.

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.

Lengths of paper segments

Lengths of paper 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:

Gore crease pattern

Gore crease pattern

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:

Math tutorial vase

Finished bowl

13 responses to “Tutorial: Designing Pleated Forms 2

  1. What an excellent tutorial. I defenitly am going to try this wonderful bowl. Thank you very much!!!

  2. Danielle, thanks so much! I’m glad you liked it!

  3. Hello sir/Mam
    Please let me know in detail about this paragraph or give me a tutorial.
    “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.”

  4. Deepak, I would be happy to help you. Could you be a little more specific about what questions you have? Also, have you read my previous tutorial? That one may help with answering some of your questions.

    • Hello mam i am making a urn design but i want to make it exact diamention so as told us in your last tutorial that figure out the shape of the curved to stright lines. Please kindly let me show by tuttorial on graph paper about this lines “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”
      Please let me know….
      Thanks
      Deepak

    • Hello men i am going to make a urn sketche with diamentionaly which total height is 242mm and meximum diameter 166.11 mm and other destail of this urn you can go on this link : https://www.facebook.com/photo.php?fbid=638436822835561&set=a.389095231103056.105076.100000077472172&type=1&theater
      This skatch i want make on graph paper and make plating or gores.

      • Deepak, I don’t have time to do all the math for you, but I’ll try to point you in the right direction. If this is your first time folding in this style, I would suggest using something like 8 or 10 gores and simplifying the shape a bit.

        As far as figuring out where the curves go: I’ll use the diameter at the top rim from the image you posted as an example. Let’s say (for example) that your valley folds are spaced 5 cm apart (each gore is 5cm wide), so the middle of the gore is 2.5 cm from the valley fold on the left edge. At the top rim of the vase, the mountain fold should be (101.5/166.11)*2.5 cm = 1.5 cm from the center of the gore, or (2.5 cm + 1.5 cm) = 4.0 cm from the valley fold on the left edge of the gore.

  5. Hello Rebecca !
    What you can do with paper is astonishing! I really love the way you bend it and paint it ♥ To create my first range of lamps, I would like to draw my inspiration from your organic art, if I may. If i manage to make one with my own design in polyethylene (which is much thicker and stiffer than paper), it will be called Rebecca in tribute to your method. Will you allow me?

    Meanwhile I have a question: right after your last pattern you said “At the very bottom edge, the radius is zero, so the curved fold is 1 cm from the edge”. But what I can see is a 5-square radius, right? So the crease should be 1.5cm away from the left edge of the gore, shouldn’t it? I am lost.

    • Hi Stephane,

      Thanks so much! Absolutely, you can create your own designs inspired by my work. I would love to see what you create!

      Re: your question, you’re correct that the radius is 5 squares where the sides of the bowl hit the ground. That’s the 1.5 cm mark about 3/4 of the way down the crease pattern. But I’ve also given the bowl a flat base so you could actually use it. That’s why the radius is zero at the very bottom of the paper. I hope that helps!

  6. Oh my God, you are right!! I didn’t pay attention, sorry.
    I will show you some pieces or at least tell you if it didn’t work 🙂
    Thank you Rebecca, I’ll get back to you in a few days!

  7. Hi Rebecca,
    As promised, I get back to you with some news. Polypropylene is much much thicker and stiffer than paper, as I said. I am not very happy with the way it bends (actually, it DOES NOT bend well) especially as I covered it with scrim. Sometimes it is curved, sometimes it’s flat… I can not create the shapes I want with this material.
    My first “success”, which can not be my Rebecca yet: http://prntscr.com/3gkb7a
    http://prntscr.com/3gk164
    http://prntscr.com/3gkbib
    Hope you enjoy it anyway.

    • Stephane, yes, the material makes a huge difference! I haven’t tried folding any plastics, so I don’t have any advice there. Even in paper, it took me months to get comfortable with these curved-crease models. But it looks like you’re at least off to a nice start!

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