Collapsible Origami 3a3m1n

SYMMETRY: CULTURE AND SCIENCE

SYMMETRY ISSUES IN COLLAPSIBLE ORIGAMI Saadya Sternberg

Abstract:‘Collapsible Origami’ is the field which invents and investigates folded forms made from flat uncut sheets that can be squeezed into smaller 2D or 3D shapes and be re-expanded, repeatably and reliably, in a single fluid movement. In this paper I consider the effects of a fold-pattern’s symmetry and asymmetry on its collapsibility. Symmetry tends to distribute compressive/tensile forces more evenly in a sheet: for many purposes that is a good thing, but with excess equilibrium there is no clear channel of release for these forces. So, introducing a single small asymmetry often aids in compaction. That said, folding and unfolding are rarely the SOLE purpose of collapsible origami. Often the sheet also has to unfold into a pretty shape, or compress into a specific shape, or do something interesting as it unfolds, or work using rigid hinged parts like steel (unlike paper which can flex), or compress only partway and resist compression thereafter, or fold along curved lines which have a distinct logic of their own, etc. For each of these varied requirements pattern and symmetry—but also the breaking of symmetry—are relevant. I illustrate this theme by considering a few novel patterns of my own, as well as some famous cases.

Keywords: origami, collapsible, symmetry, design, biomimicry.

compressible, deployable,

1. BACKGROUND Origami, the art of making things by folding flat sheets without cutting, is sometimes confused with pop-up art, which is about making things that unfold into interesting shapes and then close back flat—but starting from discrete elements, not a single continuous sheet. In the former, the ultimate folded shape is smaller than the initial one (typically a square);

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in the latter, the resultant popped-up shape is larger. But there is also an overlap between these two fields: collapsible origami, in which the form that opens and closes is itself made from a single uncut sheet. The sheet is scored or pre-folded typically with a pattern of alternating mountain and valley folds, and it then opens and shuts in a systematic, repeatable way in a single fluid movement. Probably the simplest and oldest such shape is the paper fan, but collapsible origami is also to be found in familiar objects like umbrellas and tents, as well as in ultramodern deployable structures such as unfolding satellite s and medical stents—all situations where it is necessary to deliver a continuous surface in a compact form to where or when it is needed and then have it unfurl or unfold into a larger one, smoothly and reversibly. This field is currently undergoing an explosion of interest. Aspects of it have been in existence for decades, and possibly for centuries; but it has recently begun to coalesce and to draw the attention of serious paperfold artists as well as the physicists and engineers who study deployable structures. Indeed the field is a natural point of between art and science: and symmetry issues, I hope to show, are key to both. I will focus primarily on the mechanical effect which adding asymmetry and symmetry to a pattern has on collapsibility. To that end I will introduce some collapsible origami designs of my own, alongside the discussion of more familiar cases.

2. CIRCULAR PAPER FAN We may begin with one of the oldest of collapsible forms: the paper fan, which is a corrugation of rectilinear parallel equidistant pleats on a sheet that is glued or held manually together at the bottom and opened and closed at the top. Although along its lines of collapse the fan tends to open and close somewhat uncontrollably, it is otherwise a remarkably stable shape; and even of this simple form it is almost worth asking (and easy enough to answer for oneself) why this is so. Which aspects of the pattern affect collapsibility: the parallelism, the equidistance, the rectilinearity, or the closure at the bottom and opening at the top? If you vary any of these how does that affect form, function, stability, beauty?

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Take this same pattern, drawn on a sheet the length of which is at least 2pi times its height. Such a fan can be opened all the way round, and then two of its edges may be glued underneath to form a complete circular fan. Can this shape be compressed? It can, if you break from the plane of the circle, raising the center and squeezing the circumference into a sort of cylinder. What about while keeping to the plane of the circle? –No, that will not work. The following variation, however, will allow planar compression of a circular fan. Instead of having the parallel folds of the fan be orthogonal to the long edge of the rectangle, make these parallel folds slightly skewed. Glue the edges as before. This will make a circular paper fan with a small hole at its center (the hole being larger the greater the angle of skew). And now the shape will compress laterally, just by pressure from the sides. The hole at the center will wind up and simultaneously the fan’s corrugations will close as they do normally in a spreadable fan. In fact, with suitable paper the shape will also spring back open. [Invented by the author, 2009.]

Figure 1. Collapsible Fan Disk, invented by Saadya, 2009

Why does this work, when it doesn’t with the orthogonal pattern? In the ordinary and more symmetrical shape the tensile forces are in dynamic equilibrium: closing the fan edges in one direction increases the tension in the other. This holds the shape in balance, but it also keeps all the corrugations pointing straight toward the center, where they interfere with each other if compression is attempted. With the skewed-angle circular fan, there is (a) a hole at the center instead of a point, like the

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hole at the center of a vortex, so there is room for winding; (b) the corrugations come in at an angle and so don’t run into each other; and (c) the upper parts of the corrugations have a single direction to bend in as the shape compresses (again, rather like the spiral arms of a galaxy). All this happened by taking a pattern that was too symmetric and introducing a single asymmetry into it. The symmetry lends the shape rigidity and helps it resist collapse; the slight skew opens a channel by which these compressive and tensile forces can be released; there is only one such channel, so the released forces do not interfere with each other. A fine article by Taketoshi Nojima (Nojima, 2002) contains a collection of images of known collapsible origami forms. This collapsible disk is not in it, although others, made from disks instead of from an initial rectangle, are. One observes that many of the forms shown there involve slightly skewed angles with respect to a symmetrical (typically radial) grid.

3. MIURA MAP FOLD The skewed-angle fan disk may be collapsible but it is not ‘rigid origami’—it would not work if the parts were hinged steel or plywood instead of paper. Obviously, since plywood can’t curl up at the center; a little less obviously, because even an ordinary fan can’t be made of hinged plywood parts. A regular fan has its outer extremity oriented in the plane of the circle, and the compressed sheets at its center oriented perpendicular to that plane, facing the holder. And plywood cannot twist. I turn next to a pattern that is rigid origami. It is probably the instance of collapsible-origami best-known to engineers, made famous by Prof. Koryo Miura, who wanted a reliable way of deploying continuous satellite s in space, and of course a compact way of transporting them. The pattern also solves an old nagging problem here on earth of how to get a large paper map to fold up properly along its creases: the Miura-ori or ‘Miura map fold’, as it is called, can only fold up one way, and it does so in a single continuous action.

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Figure 2: Miura Map-Fold

Pictured here is the Miura pattern with its recognizable zig-zags. It is easy enough to fold. On first acquaintance it is astonishing how reliably and simply this form opens and closes, by pulling and pushing at just two opposite corners. Apart from reliability, the size of the opened form relative to the initially-compressed shape is particularly impressive, compared to other collapsible-origami patterns. In the present context, we must ask two questions: (1) What happens when you add symmetry to the pattern? (2) What happens when you add asymmetry? If you make the pattern more symmetrical, it becomes an orthogonal grid. It was Miura himself who first asked why an orthogonal fold pattern does not a smoothly collapse and expand, whereas a skewed-angle pattern does. Paraphrasing Professor Miura: (a) the slightly skewed angle takes the shape out of dynamic equilibrium, and gives each unit a preferential initial direction to collapse in—in both the X and Y dimensions. (b) Change in each unit induces a certain amount of change in each of its neighbors, in the correct direction. And (c) the zig-zag pattern imparts some springiness to the overall shape in its extended form, so it is never quite flat; an orthogonal grid does not have a comparable springiness and so lies flatter, increasing the likelihood of a hinge turning the wrong way (Miura, 1993).

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But you can also make the pattern less symmetrical, preserving the signs for the folds relative to the nodes but varying the angles of the zigzags. Figure 3 shows some simple progressive deformations of the Miura-ori. As each pattern becomes less and less symmetrical, it is reasonable to inquire: can this sheet still be collapsed so that it is flat when folded? Does it collapse as rigid origami? How does the altered symmetry affect the mechanics of collapse/expansion? And how does it affect its beauty?

Figure 3. The Miura-ori and various simple permutations. (A) With sharper angles (which sometimes also are considered Miuras); (B) with increasingly shallower angles in each row; (C) with randomly varying angles in the zigzags; (D) with verticals no longer straight, but bending at the nodes.

I leave these questions for the reader to explore for himself. But I will note that I have used an asymmetric Miura pattern in an application, one of the flower-cards presented below. It is surprising, at any rate, how robust this pattern is to various kinds of deformation. Just what does one have to do to destabilize a Miura-ori?

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4. MIURA PREHISTORY A few words are in order on the history of this pattern. Various writers (e.g., Vincent, 2000; Mahadevan, 2005) have pointed out that something like the Miura-ori has long existed in nature. For instance, the leaf of the hornbeam plant has a zigzag pattern, and in opening its tip jumps forward in the way that is characteristic of the expansion of the Miura-ori. But not just in nature: also in art. On February 8, 2008, the origami and fabric historian Joan Sallas posted to the Origami List (the main origami forum on the internet) a reference to a painting made in 1535 by the Italian master Agnolo Bronzino, of the aristocratic Lucrezia Panchiatichi (Bronzino, 1535). In particular he pointed to Lucrezia’s collar, with its unidentified pattern of “curved folds”—part of Sallas’ argument for the antiquity in Europe of knowledge of curve-folding (Sallas, 2008). I immediately reconstructed the pattern in paper. (It is amusing to reverseengineer an object from a painting made of it 500 years ago.) This let me identify the basic pattern as fundamentally of the class of the Miura-ori, with its characteristic zig-zags. The curviness of the edges of the zig-zag results from the shape having been made from fabric rather than paper. This pattern nevertheless varies from the regular Miura in that there are two scales of the pattern, one at twice the size of the other, which alternate with each other. What is interesting is that having these two scales interferes with the complete collapse of each. (As the crease pattern in Figure 4 shows, a square is necessarily formed in the handoff between the two scales, and that square is bounded by folds of the same sign—a configuration that prevents collapse.) The net result is that the form functions like a spring, compressing partway but no further: exactly what one wants in a collar of this kind. This is perhaps a case where complicating the symmetry changes the mechanical properties of the sheet with respect to collapse and expansion. In any event, what it clearly shows is that not only was there knowledge of a Miura-like pattern in the 16th century, but this knowledge was quite advanced—more advanced in some respects than that which exists today.1

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Figure 4: Agnolo Bronzino,1535, and the 2-scaled Miura reconstructed by the author. The paper reconstruction uses only the folds visible in the painting and what they directly imply (e.g., a valley between two mountains). The resultant pattern resists full collapse.

5. MIURA SEQUENCES: RAY SCHAMP What is the connection between the accordion or fan pleat with which we began, and the Miura pattern? Ray Schamp, who is perhaps the leading figure exploring Miura-type folds artistically and theoretically today, thinks that the Miura is the second in an indefinite series of rigidly collapsible patterns, each formed from its predecessor by the addition of a bending line: the line itself being an intersection of a plane with the predecessor pattern in its 3D state (Schamp, 2006). Schamp’s work is an engagement of reflective symmetry at a deep level; it is almost a sort of optics. Yet I am unable to follow his ideas fully. Among other things I would like to know whether (a) there is always one successor to each state, or several; (b) it can be proven that the folds are always rigid origami and close flat; (c) no successors which are rigid and close flat can be added by other methods; (d) the series is infinite (e) what determines whether the successor fold is a zigzag or a crenelation or

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any other shape; and so on. –What I cannot dispute, however, is the beauty of his results.

Figure 4: Ray Schamp, 3 degrees of pleat. According to Schamp, a new flat collapsible fold can be generated from each predecessor state by a simple algorithm which can be applied indefinitely.

6. BLOOMING ORIGAMI I became interested in the Miura-ori when setting myself the challenge of deg pop-up cards using collapsible-origami flowers as elements. It turns out that the pattern not only collapses flat, but does so equally well when reflected symmetrically across one diagonal—which also makes the resultant shape appear somewhat ‘floral’. As the two corners are

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pulled out the ‘flower’ grows right before your eyes, and when the corners are pushed back the form shrinks reliably (also discussed in Sternberg, 2009).

Figure 5: Pop-up flower cards, an application of Collapsible Origami. All flowers are folded from uncut squares, each glued at two corners to the background card and activated by the card pulling and pushing at those points. (A) Purple flowers are Miura fold variants; yellows are variants of the Preliminary Fold. (B) Roses twist radially as they open. (C) Sunflowers rise, using a Miura principle, and then pivot forward, taking a bow. (D) ‘Bat-flower’ locks into a 3D shape as it opens.

But the Miura is by no means the only reasonable collapsible pattern or the only one that can serve the special aims of pop-up floral design. I have designed a flower based on a variant of the preliminary fold (dividing the square into nine rather than four square sections); a radial pattern that spins as it expands; a pattern that shifts its direction of

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movement as it opens; a pattern that locks into a 3D shape instead of (like most collapsible origami) becoming progressively flatter as it opens; and others. All these flowers collapse and expand solely by pulling and pushing at two of their corners. Each of them is an opportunity to rigorously explore a concept of Collapsible Origami by meeting the challenges of a specific artistic task.

7. CURVES I turn now to another area of interest in which the geometry and symmetry of the folds affects compressibility and collapse: curve folds. This is an exciting, relatively new field in origami, or, to be more accurate, a field in which there have been relatively few pioneers and real advances over the last 100 years but of which the public is finally now becoming aware. More crucially, it is today drawing the analysis and experimentation of first-rate paperfold and corrugation artists, people like Polly Verity, Philip Chapman-Bell and Fernando Sierra. 2 Perhaps the most important thing to know about curved folding is that when you put a curved crease in a sheet and begin to bend the paper along it—which causes the surfaces to curve too—those surfaces to either side of the curved crease can never be brought flush to each other. Geometry dictates that in order to have surfaces of a flat sheet be moved by folding from an open state to a fully-flush closed state, the hinge has to be a straight line, not a curved one (Fuchs and Tabachnikov, 1999]3. This fact has immediate mechanical and aesthetic consequences: patterns made of curved folds through single surfaces are always open. (Patterns of straight folds may likewise be left open but it is rare that they are forced to be such.) These open folds imbue a sheet with some compressibility, but this can never be taken continuously all the way to a state where two layers to one surface, let alone become flat. The dominant appearance of curved-fold origami is thus of one-layer thick curving surfaces, ed by sinuous open folds.4 Another way to say this is that when the fold-lines are straight, an investment of thought needs to be made (as we saw in the case of

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Lucrezia’s collar) so that a pattern that compresses part-way will not collapse all the way to flat. This investment does not need to be made for curved folds, which resist full collapse naturally.

Figure 6: Curve-Fold Tessellations, by Saadya, 2006

Curved folds impart a degree of springiness to a sheet because they compress the sheet along both its x and y axes. Some curve folds can be arranged in tessellations, and when this is done the compression can often be controlled so that the sheet maintains an average flatness and retains its original X/Y ratio (Sternberg, 2006). Alternatively, the compression can be applied variably, making it possible to form domelike shapes such as the top of a person’s head entirely through folding. I have used this possibility before to good effect.

Figure 8: Ernestine, by Saadya, 2006

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This same sculptural flexibility, however, also causes trouble for collapsible design: there are too many degrees of freedom in a curve tessellation, so that controlling or predicting how the paper will respond to a given mechanical motion becomes nearly impossible. Small changes in wrist-action result in great changes in shape. The sheet has been turned into a mass of independently-minded springs. It is nevertheless possible with some curve patterns to achieve controlled compression-expansion in a single fluid movement, and I want to turn now to a few such cases.

8. CONCENTRIC WINDER Here is an interesting collapse-pattern that begins with a circular disk that has concentric mountain-valley circles drawn on it, and no straight folds. The pattern is not entirely symmetrical, as it requires a single radius cut. (This fact may offend origami purists; but while some asymmetry can’t be avoided those averse to cutting may start instead with a semi-circle and fold concentric semi-circles. It works just as well, though it does not reopen to as large a shape.) Slide one edge of the cut radius under the other—and continue to slide it. The mountain-valley folds grow steeper, and the disk begins to shrink. Keep on sliding it—the disk gets smaller and smaller.

Figure 9: Concentric Winder, invented by Saadya, 2008

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Mathematically, this sliding and shrinking could be carried on ad infinitum. But physics as usual gets in the way, here in the form of the thickness of the walls, which eventually constrains further movement. How does this work? If you started with a disk with a radius cut and no folds, you could slide one side underneath the other and form a shallow cone; and could continue to twirl the layer underneath indefinitely, making the angle at the cone’s apex more and more acute. Putting circular mountain and valley folds on the disk does not change the logic of any of this (it is equivalent to turning part of the cone over in space). But it does look more unusual. Many of the geometric traits and mechanical possibilities of concentric folds apply also to curve-folds more generally. However, this property of infinite compressibility seems, so far as I can see, to be unique among curve-folds to those which are specifically concentric. 5

10. THE ‘ALBERS EFFECT’ In the 1920s and 1930s, Josef Albers, the Bauhaus artist, taught a series of design workshops in some of which students were given the exercise of forming 3D objects entirely by paperfolding. Photographs survive of student projects at the Bauhaus from 1927-1928 (Wingler, 1969). All these photos are of great interest for the history of Collapsible Origami, as they suggest that, unless he or his students invented the shapes themselves—which is not to be ruled out—there may have been a tradition in which knowledge of collapsible patterns was transmitted. (Almost needless to say, one of the shapes shown is rather like the Miura-ori.). But one image is specially striking. It shows an uncut disk of paper on which concentric mountain-valley folds have been scored. The folded shape does not lay flat, but rather has undergone an extreme contortion and has settled into a saddle shape much like the one shown here.

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Figure 10: The ‘Albers effect’: an uncut disk with folded concentric mountain-valley folds deforms into a saddle-shape. The Concentric Winder be used to demonstrate the exact point at which this contorted shape emerges from a flat-on-average state.

Why does this happen? Once paper has been scored and folded concentrically it has been given a springiness; this pulls the edge circle and indeed each of the interior circles toward the center. In effect, then, the radius of each of these circles has been shrunk; but meanwhile its circumference has not. This excess circumference can be accommodated only if the circles twist out of the plane, settling into a saddle shape. It should be noted that saddle formation of this kind is by no means an exclusive property of concentric folds: the laws that C = 2!r and A = !r2 is just one of many that relate lines to lines and lines to areas in the plane, and when any of these expected relations for a flat surface are not met, a sheet will be forced to break from the plane, typically in origami either in the direction of a cone or in that of a saddle. (See Sharon, 2004, for a discussion of how a sheet deforms under differential growth.) Folders of other curve patterns and even of open-fold straight corrugations, which

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often deform a sheet unevenly, will encounter a tendency to buckle from average flatness in the direction of a saddle commonly enough. But while saddle formation is not a unique feature of concentric circle patterns, it is especially elegant with them, for one can almost see the tension chasing around the symmetrical circles seeking some slight imperfection at which to break the symmetry. With less symmetrical patterns the points of the symmetry-breaking are given in advance. Now, among its other virtues, the Concentric Winder discussed above can also be used to demonstrate the exact phase at which this contortion or ‘Albers effect’ begins to set in. As you begin the unwinding, the object steadily maintains an average flatness: indeed the circular monotony is part of its hypnotic charm. The shape maintains this flatness both because when wound up, the multiple layers make it harder to bend, and because its furrows give it a resistance to bending (via the corrugation effect). But then it undergoes a phase-change. As it is unwound, the number of layers decreases and the corrugations partially flatten, so it is less able to resist the springiness of the creased paper. But then, long before the disk is entirely unwound, it starts to wobble, refusing to stay in the plane. Ultimately it settles into the shape shown above. Notice that this explanation is mechanical: the stage at which the buckling occurs depends on the qualities of the paper and its memory of how tightly it has been wound. But unless I am much mistaken there is also a mathematical component here. As you unwind it, the circumference of each circle increases along with its effective radius, but not at a constant rate. Initially the angles of the furrows are small, so most of the addition to furrow size is horizontal (the ratio of addedradius/added-circumference ratio is high). But once the angles of the furrows get past the 90 degree point, further widening of them has more of a vertical than a horizontal effect. In other words there comes a stage when you are adding much more circumference compared to radius. And there the pressure to break from the plane will be great. The twistiness of concentrically-folded circles or related shapes and its potential for making striking new forms has not gone unnoticed by artists over the past century. Others that I know of who have explored this theme artistically, following Albers, include Irene Schawinsky,

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Konohiko Kasahara, Thoki Yenn, Erik Demaine, Polly Verity and Fernando Sierra. The related field of Cone-Folding, which also uses concentric-folds but does not result in spontaneous twisting (there is no excess circumference: wedges are cut out) has been investigated by Ron Resch and David Huffman. Many of these concentric-circle sculptures are visually clean and mysterious, simultaneously geometric and unexpected. Some of them have made it into prestigious venues, such as the Museum of Modern Art in New York (Demaine, 2008)6.

10. ORGANIC FORM The distinct look and special geometry of concentric-folding (as a limiting case of curve folding) may be why, in the 80-plus years it has been played with, there have been few if any attempts to mix concentric and straight-line folds within a single pattern. But just this is what is practically dictated by the ‘vary the symmetry’ approach being advocated in this essay. Here is a pattern drawn on an uncut circle, with concentric folds on the lower half of the circle that are continued in the upper half as straights. The hand-off is mediated by a zigzag that reverses the direction of the folds. When collapsed, a surprising, elegant form results that has a large number of differentiated parts, as well as significant 3D bulging. Some of this derives from the forced-marriage of straight folds, which willingly accept compression, with concentric folds, which do not (or accept it only at infinity). This form still qualifies as ‘Collapsible Origami’ because the shape emerges from a single fluid movement following a unique path. I think of it as ‘organic’ not just because of the symbolism in its folded form of root, leaf and orb, but because the emergence of its shape is reminiscent (to me) of the astounding event in nature of gastrulation—wherein an apparently symmetrical sphere, through a process that also involves folding, in a single sudden movement becomes a body with its principal organs differentiated and in their correct spatial location.

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Figure 10: Organic Form, by Saadya, 2009. The pattern consists of half concentric, half straight parallel corrugations, which are mediated by a zigzag. W hen collapsed it bulges into a surprising shape.

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11. BALL FROM A DISK Having now, possibly for the first time, applied reverse-folds to a concentric pattern, let us take this idea further. In the Concentric Winder toy, each time you wind a circumference of the shrinking circle once, you are adding another small circle atop a pile of others, like a circular parking garage when you add another floor. This corresponds in the original large circle to a section of arc—to a piewedge, curved around into a circle. In theory you could peel back one of these layers, bending it along a diameter via reverse folds, so that it would stick out from the plane of the other circles; in fact you could peel layer after layer at the same diameter, all at slightly different angles, and then you would get the outline of a sphere. (Imagine the shape rotated in space). Simplifying this idea considerably, I came up a new way of making spheroid origami shapes by folding an uncut circle: (1) Draw concentric circles of mountain-valley folds; (2) draw zigzags to the concentric folds along equi-angled radii (any even number of radii will do), to form pie wedges; (3) reverse the direction of all folds in each alternating piewedge; (4) fold up the shape. The resultant form is not a hollow sphere, but a ball; indeed, this fold has the unusual property that the center of the circle becomes the center of the ball, the circumference of the circle becomes the surface of the ball, and the points in between in the ball have the same pair-wise ordering (closer or farther from the center) as they had in the circle. Pictured here is the formation of a ball of eight wedges. Indefinite even-numbers of wedges can be added from the same initial circle, adding to the accuracy of the outline of the ball, but decreasing the ball’s linear dimensions in direct proportion to the number of wedges. The neat thing is that as wedges are added and the shape closed, the curvature of all the concentric circles adjusts itself to accommodate the shorter radius of the shrinking sphere.

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Figure 12: An uncut circle with a concentric fold pattern being folded into a ball. Invention by Saadya, 2009.

This too is an example of Collapsible Origami, at least in the sense that there is one path for the folds to take and one motion to create the form. Of course as the number of wedges grows the process becomes harder to control; for an eight-wedged sphere one needs the manual dexterity of an octopus to close the shape up all at once. The Winder and the Ball are two novel ways of folding up a circle ‘symmetrically’, that is, in ways that preserve the outline of the original shape while also transforming it. It is worth asking here too this essay’s basic questions: What are the symmetries or asymmetries which are crucial for their operation? In the case of the Winder, the shape needs to be a circle, the interior folds need all to be concentric, but the equidistance between the concentric folds adds an element of symmetry that is aesthetically pleasing yet serves no function in the compression. In the case of the circle folded into a ball, the initial shape needs to be a circle, the folds in the interior need not be equidistant, and they need not even be concentric so long as they allow an even compression to take place (e.g., long spirals might work too). The pattern is necessarily asymmetric top-bottom, but this fact becomes less noticeable the more concentric circles there are. And the wedges all need to be of an equal size.

12. CONCLUSION I have painted a small and very partial picture of this emerging field of Collapsible Origami, yet hopefully enough of one to show some of the

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charm it holds for artists, geometers and engineers. There is still room today for the simplest, most elementary of discoveries—even for reinventing the wheel. Meanwhile there are technical and mathematical challenges enough to satisfy any engineer’s appetite for complexity. Asymmetry and symmetry are key to the successful mechanical operation of these folds which open and collapse, sometimes in subtle and unexpected ways. Analysing a pattern, asking which of its aspects contributes to or impedes its function, adds to or detracts from its beauty and surprise, and then varying these one by one—seeing what happens— that is the essence of artistic experimentation, here as anywhere; but as this is origami it can also be done by anyone within reach of the nearest scrap of paper.

ACKNOWLEDGEMENTS I am indebted to the anonymous reviewer for many helpful suggestions.

REFERENCES Bronzino, Agnolo, “Lucrezia Panchiatichi”, painted ca. 1535. Image is in the public domain worldwide. Demaine, Erik, http://erikdemaine.org/curved/history , first posted 2008. Fuchs, D, and Tabachnikov, S, “More on Paperfolding”, American Mathematical Monthly”, vol 106, no 1, Jan 1999, pp. 27-35. Mahadevan, L. and Rica, S. “Self-Organized Origami”, Science 18 March 2005: Vol. 307. no. 5716, p. 1740. Miura, Koryo, “Map Fold a La Miura Style, Its Physical Characteristics and Application to the Space Science”, Research of Pattern Formation, ed. R. Takaki, KTK Scientific Publishers, pp. 77-90. Paper first presented at the First International Meeting of Origami Science and Technology, Ferrara, Italy, December 6-7, 1989. Nojima, Taketoshi, “Origami Modeling of Functional Structures based on Organic Patterns”, Dept. of Engineering Science, Graduate School of Kyoto University, Sakyoku, Kyoto, Japan, 2002. Sallas, Joan, series of postings to the Origami-List, an origami forum on the Internet, subject line “History of Curved Origami Sculpture”, Feb. 8, 19, 25, 2008.

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Schamp, Ray, photoset on Flickr.com: http://www.flickr.com/photos/miura-ori . For his ‘degrees of pleats’ theory see e.g. http://www.flickr.com/photos/miuraori/355100166/in/set-72157594478882139/. Sharon, Eran, et al., “Leaves, Flowers and Garbage Bags: Making Waves”, American Scientist, vol. 92, May-June 2004, pp. 254-261. Sternberg, Saadya, “Collapsible Origami: Symmetry and Asymmetry”, in Symmetry: Culture and Science, vol. 20, numbers 1-4, pp. 345-360 (2009). Sternberg, Saadya, “Curves and Flats", in Robert J. Lang, ed., Origami4; Wellesley: AK Peters, 2009. First presented at the Fourth International Conference on Origami in Science, Mathematics, and Education (4OSME), on September 8-10, 2006, at the California Institute of Technology, Pasadena, California, USA. Vincent, J F V, “Deployable structures in nature: potential for biomimicking”, Procedures of the Institution of Mechanical Engineers, vol. 214, part C, 2000. Wingler, Hans M., Bauhaus: Weimar, Dessau, Berlin, Chicago, MIT Press, 1969, p. 434.

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Though it seems I am the first to identify the fold pattern as a Miura-variant at alternating full and half-scales, and first to state the interesting mechanical properties of this pattern, and probably the first to reconstruct it in paper from the painting (since the two former points would have been immediately apparent to anyone who had done so), credit for noticing the pattern itself must surely go to the origami and fabric historian Joan Sallas, whose painstaking research into the history of folding in cloth and paper in Europe is worthy of all praise. Sallas reported his historical discoveries in a series of postings to the origami list (the main internet origami forum on the internet) in February 2008. After I identified the pattern as ‘Miuras’, he listed still earlier European exemplars of this fold: “The Miura pattern of alternating mountain & valley folds existed on cloth in Europe at least since the end of the 14th century, as seen on the headscarfs of the statue of Katharina Markgraefin of Baden of 1385 in the Cathedral of Basel (Switzerland), and the statue of Guda Goldstein in 1371, wife of Johann of Holzhausen, in the Cathedral of Frankfurt am Main (). The pattern of Lucretia Panachiatichi's clothing was a folding of North-Italian Renaissance development as seen on the just mentioned statues.” Joan Sallas, origami-list Internet postings, February 8, 19 and 25, 2008. 2

Polly Verity, http://www.polyscene.com ; Philip Chapman-Bell, http://origami.oschene.com ; Fernando Sierra, http://www.flickr.com/photos/elelvis. It honors me to have their respect as well. 3

I am indebted to the anonymous reviewer for this reference.

Collapsible Origami

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4

A curved crease can also be put through two layers of paper instead of one, or through a sheet folded over itself along a straight segment, and then there will be the option of bending one layer as a mountain fold while leaving the other intact. Indeed, as has been recognized, this sometimes weakly locks the surface into its curving shape (and is called by some in the origami world a “tension-fold”). Such a process will make it seem as though on one side of the sheet there is a closed fold: a seam line instead of a continuous open valley. 5

In 2006, while working on those spiral or vortex-like curve-folds (see e.g. Figure 6; discussed further in Sternberg, 2006) I was curious whether the visual similarity of individual spiral units to certain galaxies was superficial or whether the analogy went deeper. Vortexes after all are things that pull all material in them toward a center, whether in paper or in space. I was then in Boston, so I asked Brian Chan— known in the origami world as a champion of complex origami, but then also getting a degree from MIT in fluid mechanics—what a vortex is, as his field sees it. His answer: a perfect vortex is not, as is sometimes thought, a spiral that winds toward a center, but rather a series of concentric circles. So it is of special interest that the origami pattern that turns out to allow infinite compressibility of a sheet is a set of concentric circles— rather than the spirals as originally believed. And indeed, if you think of the points along the circumference of these circles, they are increasing in density with each winding (here not by compaction but by the addition of layers); and meanwhile the circles themselves are also being drawn closer to the center. I can’t conceive of any other curve pattern that would accomplish this in as orderly a fashion. 6

Erik Demaine, http://erikdemaine.org/curved/history, Posted in 2008. Much of the historical information here on Albers and 20th century concentric-circle folding (which Demaine calls “curve-folding”) derives from this source.