Collapsible Origami

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-patterns 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 symmetrybut also

the breaking of symmetryare 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, compressible, deployable,

biomimicry.

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 flatbut 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 panels and medical

stentsall 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 contact 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?

Collapsible Origami 3

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 fans 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 doesnt 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 dont 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

origamiit would not work if the parts were hinged steel or plywood

instead of paper. Obviously, since plywood cant curl up at the center; a

little less obviously, because even an ordinary fan cant 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 panels 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.


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 inin 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).

Saadya Sternberg 6

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?


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 Lucrezias collar, with its

unidentified pattern of curved foldspart 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 reverse-

engineer 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

signa 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 advancedmore advanced in some respects than that which

exists today.1

Saadya Sternberg 8

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).

Schamps 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


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

designing 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 diagonalwhich also makes

the resultant shape appear somewhat floral. As the two corners are

Saadya Sternberg 10

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


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 itwhich causes the surfaces to curve toothose 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 join to one surface, let alone become flat. The

dominant appearance of curved-fold origami is thus of one-layer thick

curving surfaces, joined 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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Lucrezias 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 dome-

like shapes such as the top of a persons head entirely through folding. I

have used this possibility before to good effect.

Figure 8: Ernestine, by Saadya, 2006


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 cant

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 otherand continue to slide it.

The mountain-valley folds grow steeper, and the disk begins to shrink.

Keep on sliding itthe disk gets smaller and smaller.

Figure 9: Concentric Winder, invented by Saadya, 2008

Saadya Sternberg 14

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 cones 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

themselveswhich is not to be ruled outthere 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.


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

Saadya Sternberg 16

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 added-

radius/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,


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 gastrulationwherein 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.

Saadya Sternberg 18

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.


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 arcto a pie-

wedge, 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 pie-

wedge; (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

balls 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.

Saadya Sternberg 20

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 essays

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


charm it holds for artists, geometers and engineers. There is still room

today for the simplest, most elementary of discoverieseven for

reinventing the wheel. Meanwhile there are technical and mathematical

challenges enough to satisfy any engineers 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 oneseeing 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, Sakyo-

ku, Kyoto, Japan, 2002.

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subject line History of Curved Origami Sculpture, Feb. 8, 19, 25, 2008.

Saadya Sternberg 22

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/miura-

ori/355100166/in/set-72157594478882139/.

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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.

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Procedures of the Institution of Mechanical Engineers, vol. 214, part C, 2000.

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434.

1 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 (Germany). 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.


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 mechanicswhat 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 cant 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.

Documents

Collapsible Origami