Editor’s note: If you would like to fold any of the tessellations shown in this article, there are two resources you may use. First, if you have a cutter, you can download the SVG patterns directly from the article by right clicking on a PC or control clicking on a Mac. We have also compiled a PDF booklet with one crease pattern per page. You can download it here.
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Introduction
The Miura Ori is a well-known origami corrugation named after the astrophysicist Koryo Miura, who proposed it in 1985 as a way to pack solar panels for deployment in space.1,2 The crease pattern is made up of parallelograms, which when folded, lie completely flat. 3 (important - please read) The other important property is that the individual parallelograms forming the pattern remain rigid in the folding process and after. Since that time, origami designers have used the basic idea behind the Miura Ori to create many interesting and delightful variations.
The set of ideas presented in this article is yet more variations – but under a unifying framework, which I have called Sourdough Miura Ori. This name comes from my making sourdough bread at home, where I was pinching and pushing the dough together, and stretching and pulling it outward.
Now, let’s imagine that all these movements are applied to a flat sheet of square dough that has no thickness and infinite elasticity, so it does not break when you stretch it out. This special piece of dough has the Miura Ori crease pattern printed on it. Mathematically, these movements (or in more technical terms, operations) are covered in a branch of mathematics called topology. What we are effectively doing in this article, is applying topological transformation to the Miura Ori, without going into the mathematical details.
Note that some of the resulting models from these dough-pulling operations have already been created by other designers, such as Paul Jackson, Ray Schamp, Jun Mitani and Ekaterina Lukasheva, among others. However, the primary goal of this article is to present the framework and method, rather than the end results.
Stretching the Miura Ori
Variation 1:
Let’s pinch and push together all the Miura Ori zigzags at one end of the sheet so the distance between these creases reduces. This means the sheet will now look like the picture below. So far, so good. Interesting, but not THAT interesting.
Variation 2:
Let’s stretch the two non-parallel sides of the dough from Variation 1 all the way around, so that these edges joint together. We now get a Miura Ori in a circle.
In the drawing below, I have used 16 zigzags, but you can use any even number, and it would still work. The number of zigzags has to be even because they need to alternate between mountain (M) and valley (V) (see Endnote 3). If the number is odd, we will end up with a M next to an M, or a V next to a V, which just would not work.
The photo of this variation is taken from directly above the model, so it is not apparent that the model does not lie flat. In fact, it exhibits the same curved surface that results from folding alternating mountain-valley concentric circles.
You can see this better in the side view:
Also, as you may have noticed, I have cut out a circle from the middle. But this is not a necessary step, since all the creases could converge at the center of the circles (and form a peak or trough, per Variation 9 and some of the examples after). However, I apply this cut to most of the variations because it makes folding easier.
If you wish, you can also straighten all the circle segments, connecting the corners of the adjacent zigzags. So, rather than concentric circles, you will have concentric polygons. In the example above, the result will be a 16-gon, the crease pattern of which I will leave for you as an exercise.
Variation 3:
Now, let’s keep stretching the outside of the circle. The crease pattern below is what we would get. (Geometrically speaking, the circles are still concentric but the distance between each circle increases the farther away we get from the center).
Unlike the previous variation, this one seems to be fairly flat when not too compressed, and slightly curved when compressed, but to a lesser degree than Variation 2.
It is also possible to draw concentric circles with a different distance between each circle, similarly to if you pull the dough out, then push it back together, then pull it out a bit again and push it back together. I won’t do a drawing here, but you can probably get the picture.
Variation 4:
Now let’s evenly stretch both the left and the right sides of the circle from Variation 3. We end up with an ellipse that is symmetric along the vertical axis.
This variation is also reasonably flat from the side.
Variation 5:
Next, we stretch just the left-hand side of the circle in Variation 3. We will end up with an ellipse, but the center is off to one side.
And from the side, this variation looks reasonably flat but with a bit of an incline on the left side because of the wider zigzags.
Variation 6:
What if we stretch out the circle from Variation 2 (but using only 12 zigzags rather than 16) in six different places at once, so you get a 6-lobed blob?
And what if we really stretch out the points so that the shape becomes more “starry”? Well, the answer is that it will still work. And, in fact, the result is interesting, with its many undulating curved surfaces.
This stretching was applied to five regions on the circle, forming a wonky 5-pointed star with rounded points and troughs.
(A quick note on drawing for this pattern: I am not using mathematics to shift the concentric-circle Miura Ori into a concentric rounded-stars/blob Miura Ori, but rather drawing it empirically using Inkscape. In the drawing of zigzags, you might need to make minor adjustments so that they don’t cut across the concentric blobs/stars in order to touch the rounded points. The way I did it, was simply to move the sharp point of the zigzag such that it touches a place near the rounded point that has the same tangent as the straight line, as in the drawing below, from the sharp blue corner, to the sharp red corner.
This method of shifting the corners of the zigzags is not very mathematical and will probably annoy purists. But since there is no stretchable paper with zero thickness in real life, I am happy to make small practical adjustments to get things to work).
Variation 7:
What if we straighten the lines of the six-lobed blob in Variation 6, so that all the lobes become pointy and we get something like a wonky straight-lined star?
This pattern is foldable. But if we want a flat surface for each polygon in the folded model in order to maintain rigidity (and also to maintain uniformity for aesthetic reasons), it is not going to work because of the numerous concave quadrilaterals (quadrilaterals with one or more internal angles over 180 degrees), such as the two highlighted below.
Fortunately, this is a problem easily solved by adding extra creases. First, we connect the points of the star to the center points. This method will take care of all the concave quadrilaterals located along these new lines, splitting them up and turning them all into triangles.
Next, we look at the remaining quadrilaterals (the ones that lie along the “troughs” of the concentric stars). As you can see, some of them are concave (red) but some are convex (blue). But all in the innermost ring, are convex.
If a quadrilateral is not concave, there is actually no need to add any extra crease and divide the shape up into triangles. However, if you do put those extra creases into all the rings except for the innermost one, irrespective of whether the quadrilateral is concave or convex, it will give the design better “compressibility” (that is, you can push the perimeter of the paper into the center better). And this is the result.
But what are the MV assignment of these straight lines from the peaks/troughs of the star to the center that have been added? The first thing to note is that when you move from one concentric star to the next, the segment switches from M to V, or V to M, that is, they alternate. So, if you can work out the MV assignment of the creases on outermost ring, the rest will follow.4
But how do we do that? Well, it appears the MV assignment depends on whether the crease starts at the point of the star, or the trough of the star. And then, it depends on the MV assignment of the zigzag that has its sharp point touching that line. The rules to follow, from observation, are listed in the table below:5
| Line from the point of the star to the center | Line from the trough of the star to the center | |
| Zigzag with points touching the line, and the zigzag is mountain | The outermost crease is mountain | The outermost crease is valley |
| Zigzag with points touching the line, and the zigzag is valley | The outermost crease is valley | The outermost crease is mountain |
There is likely to be a mathematical proof of why these rules work, but I will have to leave it for someone more mathematically capable than I am.
This pattern can be observed again in Variation 8 below.
Variation 8:
In the previous variation, we had only one zigzag for each peak-trough section of the 6-pointed star. Could we possibly add more zigzags into each of the peak-trough section? It looks like the answer is yes. As long as the total number of zigzags within the entire design is even (per the note in Variation 2), we can add as many zigzags into each segment as we like.
Here is a crease pattern where we have two zigzags for each peak-trough section of a regular 5-pointed star.
And below is the crease pattern if we add one more zigzag in the middle of the two zigzags from before.
You could also choose not to have the same number of zigzags in each peak-trough section within the star. As long as you follow all the rules listed in Endnote 3! and Variation 7, your design can be folded.
Variation 9:
What if we rotate the original Miura Ori crease pattern by 90 degrees and stretch it into a circle?
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There are other crease pattern variations that I have not included here, such as nested circles of different sizes that are not concentric or stacking: Variation 1 with an upside-down version of itself vertically (so we get an hourglass perimeter) or horizontally (so we get a parallelogram) or combinations of the variations above, such as turning ellipses into blobs or stars. What I have presented here is not an exhaustive list, and you can experiment to make up your own variations.
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One related note I would like to add is a computer program called “Freeform Origami,” created by Tomohiro Tachi. On Freeform Origami, you start with a crease pattern, and the program simulates the resulting 3D model. You can then stretch and push the 3D model, one point at a time, and the crease pattern is modified accordingly.
(This is obviously a bit different from what we are doing here, since we are stretching and pushing the starting crease pattern rather than the resulting 3D model.)
The program is free, and you can download it from Tomohiro’s webpage. I would fully recommend playing with it to see what you can come up with.
(Unfortunately, the program can be a bit fiddly to run. I have two Windows computers at home, both running Windows 10, and I have never managed to get it to work on either machine).
Stretching Other Miura Ori Related Crease Patterns
Now the next question – can we apply the same pinching and stretching processes using other Miura Ori related crease patterns as a starting point?
Let’s try it with a reflected Miura Ori pattern with the crease pattern below (for which the MV assignment rules 1 and 3 in Endnote 3 still apply and mirrored zigzags have the same MV assignment).
Similarly to Variation 2, this one does not lie flat (see picture on left, below). But unlike Variation 2, which has two opposing peaks and troughs, this one seems to be able to be pushed into having four alternating peaks and troughs (see picture on right right).
And an elliptical version of the same:
All the other variations, such as stretching into blobs and stars from the regular Miura Ori earlier (variations 6 to 9) should be foldable as well. But I have not tested them.
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Now, what about turning everything into curves? Well, it is possible to round out the zigzags so they are curved. In my upcoming book with Ilan Garibi, we look at how to create tessellations with curves using straight-lined tessellations as a base. Without going too much into the content of the book, the design below started as an off-center radial Miura Ori, with increasing distance between each circle, stretched into an ellipse, with the zigzags replaced by Bezier curves. The “rings” of ellipses have also been removed as hard creases since they are not necessary (and in fact their removal makes the design look much more fluid and aesthetically pleasing). This photo is taken looking straight down at the model and not at an angle.
And here is the next question: Are there other patterns that these operations can be applied to beyond the Miura Ori and its variations? The answer is yes. It seems if the pattern is primarily linear (basically made of molecules that can be laid out next to each other in parallel), then this framework should be applicable. But the drawing, and associated mathematics, will become quite complicated.
Below is the curved version of the Ron Resch Triangle Tessellation, which was used as a base.
Applying the method of Variation 2, with seven rotational symmetries (rather than the 16 in the Miura Ori above) gave me this pattern below (with thanks to Stuart Errol Anderson, who drew the base pattern for me).
The straight-lined triangles on the outermost ring in the crease pattern, between the curved triangles, are simply locks to keep structure together.
As you can see, this is a very interesting pattern. The two photos have the same crease pattern and the same MV assignment, but the pattern can yield two different outcomes, similarly to when you turn a woolen beanie inside out — except that these two models have different paper tension and topology, and you cannot flip them in and out like fabric because of the locks along the edge of the paper.
Applying the method of Variation 8 using the straight-line version of the Resch design6 gives me the pattern below and interesting front and back views (with thanks to Gömöri András, who came up with a base pattern based on my incomplete description of what I wanted to do, which I then expanded via trial and error).
These two crease patterns are also stretchable into ovals (and possibly n-sided blobs and stars as well), although I have only tried folding one of these variations.
In this elliptical version, I had to take out most of the creases for the locks in the crease pattern because the locking creases’ positions and angles need to shift once you stretch the crease pattern out from a circle. New flat-foldable locking creases can be put in, but I simply folded them via angle bisection on the paper when I made the model.
And just like the original circle version, the elliptical version can also be made inside-out.
Final Notes
Other designers have made radial versions of linear tessellations, for example, Ben Parker and Fernando Sierra, with their radial Yoshimura pattern, and Ekaterina Lukasheva, with her various spiral designs. You can find these on their respective social media pages.
As you can see, this framework is conceptually very simple, and can give you many variations of a base pattern. It should also be codeable, if you are so inclined. I hope you will have a go at applying these methods and come up with your own design and have fun in the process!
Endnotes
1. The design has existed in fashion and napkin folding, made using fabric, before then.[back]
2. The Space Flyer Unit, a spacecraft launched by Japan in March 1995, deployed a 2D Array with a Miura Ori configuration. [back]
3. I am a little lazy with the drawings and did not put mountain and valley folds in this article. The reason is because they are very fiddly to do, particularly if I have to cut parts of a circle or ellipse into segments to change the line format to distinguish the two later on. When I score the pattern on a cutter, mountain-valley assignments are irrelevant, and I do not need to specify them on the drawing. However, to understand which creases are mountain and which are valley in the crease patterns in this article, there are three general rules:
- Each zigzag has the same mountain-value (MV) assignment for the entire zigzag, no matter how many turns there are. There is no switching of MV assignment along the zigzag.
- Adjacent zigzags must have alternating MV assignment.
- The straight crease that comes OUT of a sharp point of the zigzag has the same mountain-valley (MV) assignment as the MV assignment of the zigzag, and the crease that goes INTO the sharp point of the zigzag has the opposite MV assignment.
These rules apply to ALL the variations from 1 to 9 and should help you understand the MV assignment of each crease. There are some complications, though, once we get to Variation 7, and we will go through how to deal with them. [back to introduction] [back to Variation 2] [back to Variation 8]
4. We could also start with the innermost ring. However, since there is no need to break up the quadrilaterals in the innermost ring into triangles, we will focus only on the outermost ring. [back]
5. In my experiments, I have tried folding the other way. But what that seems to do is unravel the zigzags completely. [back]
6. It is possible to do this version using the curved version of the Resch design. However, it is much harder to fold and aesthetically not that interesting. [back]
Bibliography
Demaine, Erik and Martin Demaine. “History of Curved Origami Sculpture.” http://erikdemaine.org/curved/history/.
Jackson, Paul. Complete Pleats. London: Laurence King Publishing, 2015.
Kasahara, Kunihiko. Extreme Origami. New York: Sterling Publishing Co., 2003.
Lang, Robert. Twists, Tilings and Tessellations – Mathematical Methods for Geometric Origami. Abingdon: Taylor and Francis, 2018.
Lukasheva, Ekaterina. Instagram. https://www.instagram.com/ekaterina.lukasheva/.
Mitani, Jun. Curved-Folding Origami Design. Boca Raton: CRC Press, 2019.
Parker, Ben. Instagram. https://www.instagram.com/benparkerstudio/.
Resch, Ron. Ron Resch Official Website. http://www.ronresch.org/ronresch/gallery/paper-folding-origami/.
Sallas, Joan. Folded Beauty. Chungju: Jong Ie Nara, 2018.
Schamp, Ray. Flickr. https://www.flickr.com/photos/miura-ori/.
Sierra, Fernando. Flickr. https://www.flickr.com/photos/elelvis/.
Tachi, Tomohiro. “Freeform Variations of Origami.” Journal of Geometry and Graphics 14, no. 2 (2010).
Tachi, Tomohiro. TT’s Page (website) https://tsg.ne.jp/TT/index.html.
Comments
Absolutely beautiful work! I love seeing others work though a systematic approach to test all variations. As shown here, it leads to tangents that can offer totally new models and processes. Thank you for sharing!