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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 panels 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 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?
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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 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.
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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
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 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 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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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.
22
Saadya Sternberg
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.
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.
Collapsible Origami
23
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.