In my recent oval models, I have been combining segments of different sizes of circles. Here I’ve taken the same idea of combining portions of circles in a different direction. This corrugation is based on three equally sized semicircles that alternate directions like in the letter S. As the outer circles get larger, the inner circle gets smaller, and vice versa.
S-curve corrugation (back)
In this model, the outer edges were challenging to fold because there is not much tension to hold the paper in place. This design could be extended indefinitely as a series of repeating curves, or it could be combined with circles of other sizes to create a closed form like in my oval forms.
Building on my recent oval test model, I folded this bowl based on an oval. Again, the oval is made of portions of circles of two sizes. Here, the radius of the small circle is 2″ at the widest point, and the radius of the large circle is 6″ at the widest point. The collapse of the bottom half of the form was fairly straightforward, but the top part was more challenging because the form wanted to become more circular. Overall, I think ovals will work fairly well for relatively short, simple forms but will be challenging for taller forms and more complex forms. Even though I think a lot of interesting things could be done with ovals, I’m using the ovals mostly as a building block toward some of my other ideas that are in the works.
You have to be a member of OrigamiUSA to access the article. If you’re not a member and want to know what the article is about, most of the content overlaps with my three recent tutorials on the concepts, math, and folding techniques behind pleated origami forms.
The seed pod bowl is one of my most popular models, and I’ve gotten more requests for its crease pattern than for any of my other models. I finally drew the crease pattern:
Crease pattern for seed pod bowl (click to enlarge)
As with all my crease patterns, the red lines are mountain folds and the blue lines are valley folds. The top side of the paper will become the inside of the form.
A few notes:
The mountain/valley pattern on this model is the opposite as it is for most of my models. Viewed from the inside of the final form, the straight folds are mountains and the curved folds are valleys.
Since this model does not have a flat base and is not a tube, the collapse process is a bit different that my other models. Briefly, I used a combination of tape and glue to hold the creases in place at the two ends of the form, and wet-folded to create the curved form.
Given the crease pattern as drawn, the model will be curved all the way around and not sit flat. I fixed that problem by making a small dent where I wanted the base (not shown on the crease pattern).
I’ve experimented a bit with breaking away from circle-based designs previously in models like my seed pod bowl and spiral bowl, but this design is the furthest I’ve deviated from a circle. The oval is made of portions of two sizes of circles, one with a radius of 1 inch and the other with a radius of 4 inches. The two ends are each 3/8 of a circle (135 degrees) with the small radius, and the two flatter sides are each 1/8 of a circle (45 degrees). That means that the four segments add up to 360 degrees, one full rotation.
The flat base on this shape (shown in the image above) doesn’t close nearly as nicely as most of the circular designs I fold. To get the base to stay flat, I had to wet-fold it and weigh it down while it dried. My circular bases usually stay put without any wet-folding. Even though the edges of the paper line up correctly, it’s easy to open a wide gap. Without adding glue, anything I put inside this model would probably just fall through the bottom.
Despite the extra challenges, I think this approach is ready to use on more complicated models. The model fits together exactly how I expected it to based on the math, and the folding process is certainly doable.
I recently wrote two tutorials on the basic concepts and the math behind designing crease patterns for pleated forms. Here I will talk about how to fold the form from the crease pattern, adapting the folding methods from the simpler forms to more complex designs. Here is the crease pattern I developed in my previous tutorials:
Crease pattern for bowl
As far as paper choice, I find that heavier papers that wet-fold well work best for this style of folding. Elephant Hide paper is well-suited for these forms. I have also used Strathmore charcoal and pastel paper and Canson Mi-Teintes paper, but these papers are a bit more limited in their suitability. Other papers that work well for folding tessellations are probably going to work well for these forms too.
In general, the reference points for this type of model are extremely difficult to find by folding, and relatively small errors can make a big difference in the final shape. It is usually much easier to use a ruler to find the reference points for the ends of the straight folds and along the curved folds. I usually round the measurements to the nearest 0.5 mm or 1/32 inch. The straight valley folds are easiest to find first, and then the curved mountain folds can be filled in.
As with any crease pattern, the next step is to pre-crease along all the fold lines. For the straight folds, I typically score the folds with a scoring tool. This makes it much easier to make the folds look clean and neat, and it also speeds up the folding process.
Scoring straight lines with a ruler
For the curved folds, there are several approaches. One possibility is to score and then fold the curves. I sometimes cut a template from cardstock or another sturdy material to score those folds. This only requires a few reference marks on the paper and can make the curves more consistent. However, any imperfections in the template will be repeated in every gore. Also, any mistakes will be very difficult to fix because the scoring tool cuts into the paper.
Scoring curved folds with a cardstock template
Another approach is to fold the curves by hand. This generally requires measuring more reference points for each curve. Folding consistent curves is more challenging than folding straight lines, and folding the curves by hand is generally slower than using a template. However, especially for new designers, a big advantage of folding the curves by hand is that it is easier to tweak the curves as needed while collapsing the form and clean up any imperfections.
Folding curved folds freehand
Once all the folds are pre-creased, the next step is to transform the paper from a flat sheet into a tube. Since there is one more gore than is needed to go around the form, the first and last gore will overlap. Glue the front of the gore on one end of the paper to the back of the gore on the opposite end.
Gluing the paper into a tube
Often, the pre-creases need to be reinforced at this stage. For each mountain-valley crease pair, fold both creases and pinch along the folds. This makes it easier to collapse the final shape.
Reinforcing the pre-creased pleats
To collapse the form, I find it easiest to start with the base. This will use the mountain and valley folds that are already pre-creased. Squeeze the paper together using the pre-creased folds, and push down on the base to collapse it. It can take a little practice to collapse the base smoothly. Using tape to help hold the folds in place can make the collapse easier, and sometimes wet-folding is needed to get the base to stay in place.
Collapsing the base of the bowlThe base of the bowl after collapsing
Then, collapse the rest of the form along the pre-creased mountain and valley folds. Typically, wet-folding is required to lock the curves in place. While the paper is drying, either tape or clips can help hold the shape in place as needed. Each of these approaches brings its own advantages and disadvantages. Sometimes clips can dent the paper, and tape can tear the paper, especially if it is removed before the paper is completely dry. If the wet-folding does not hold the paper in place well enough, a small amount of glue will often help.
Collapsing the rest of the bowl
The finished form from the design looks like this, not too far from the original design:
This concept is an extension of the ideas I worked on this winter and early spring, particularly my Intersections series and my last test fold. Like those earlier pieces, I’m combining a curved form with a flat plane. The difference here is that instead of the flat plane being perfectly vertical, here the flat plane is along a diagonal. This requires a very different folding strategy.
I have several ideas of where I’d like to go with this folding strategy, but for now I’m still working out the engineering for it. This test piece was folded from a fairly simple-looking crease pattern, but actually folding it was surprisingly difficult. Even using Elephant Hide paper, which is a very tough paper, some of the internal layers started to tear. I want to try various tweaks to the proportions to see if I can find a version that collapses more easily. Also, in my current version the flat plane is essentially a circle, but it should be elongated into an ellipse. If I can find a way to cleanly fold an ellipse, that may remove some of the distortions to the tube.
This model is a practice piece, somewhat building on my Intersections series. It’s the first model with no curves that I’ve folded in quite a while. Since the top diagonal plane is intersecting a simple square box, it’s not too hard to figure out how to tuck the extra paper inside. However, that will change very quickly if I move to more complex shapes. I like the simplicity and sleekness of this form, which gives a very different aesthetic than my normal folding style. This piece is folded from one uncut rectangle of Elephant Hide paper.
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:
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.
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
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:
A couple of months ago I played with hand-marbling paper using oil paints. The paper is still an experiment, but I’m fairly happy with how it turned out. The oil tended to create ripples in the paper, which limits the complexity of what I can fold. This model is a fairly simple design to try out my new paper. This piece is folded from one uncut rectangle of Strathmore pastel paper.
