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PAPER is the stuff of science. It fills laboratory notebooks, covers
desks with photocopied reprints, and pours forth from computers by the ream.
Look around any scientific office or laboratory and you will see paper put
to a variety of uses. Look around some, though, and you are likely to find
it put to a distinctly unscientific use: folded into small figures of birds,
fish, people and objects. They are examples of an art called origami, a
Japanese word meaning ‘folded paper’. Members of the scientific community
have become some of the most ardent devotees of this ancient art, which
has grown beyond its Japanese origins and taken sturdy root in the West.
A remarkable number of scientists, engineers, architects and the like have
been captivated by what is ordinarily viewed as a child’s pastime. Not content
with traditional designs, in recent years they have wrought dramatic changes
on the direction of the art. By applying the principles of mathematics and
geometry to origami design, modern ‘technical’ folders have taken the art
to undreamt-of heights of complexity and realism.

The rules of origami – one sheet of paper, no cuts – are daunting. It
would appear that only the simplest abstract shapes are feasible with such
rules. Yet over hundreds of years, by trial and error, two to three hundred
designs were developed. They were, for the most part, simple and stylised.
Complexity and realism – insects with legs, wings, and antennae – were not
possible until the development of specialised design methods in the latter
part of this century.

The design of an origami model may be broken down into two parts: folding
the ‘base’ and folding ‘details’. A base is a regular geometric shape that
has a structure similar to that of the subject, although superficially it
may appear to bear very little resemblance to the subject. The detail folds,
on the other hand, are those folds that transform the appearance of the
base into the final model. The design of a base must take into account the
entire sheet of paper: all the parts of a base are linked together and cannot
be altered without affecting the rest of the paper. Detail folds usually
affect only a small part of the paper: they turn a flap into a leg, a wing,
or a head. Converting a base into an animal using detail folds requires
tactical thinking; developing the base to begin with requires strategy.

By the mid-1960s, origami had spread around the world, and four bases
were in widespread use (Figure 1); in English-speaking countries, they are
called the Kite Base, Fish Base, Bird Base and Frog Base. (In addition,
there are two other shapes commonly called bases – the Preliminary Fold
and Waterbomb Base – that are precursors of the Bird and Frog bases.) It
is a testimony to the versatility of the four ‘classic bases’ that, although
they were known to the Japanese for more than a hundred years before origami
made it to the West, most of the hundreds of new designs produced until
then were based on one of these shapes.

The classic bases are not without their drawbacks, however. Major points
on a base get turned into major appendages of a final model. If the subject
has four legs, a head and a tail, it requires a base with six major points.
The Kite, Fish, Bird and Frog bases have, respectively, one, two, four and
five large points and one, two, one and four smaller points. A simple fish
has two large points (head and tail) and two small ones (pectoral fins),
which is why the Fish Base is so appropriate and so named. The average land-dwelling
vertebrate has five major points (four legs and a head), which pretty much
stipulates the Frog Base and rules out a tail. The point on the Frog Base
that is in a position to form a head is thick and difficult to work with,
however. One of the four points of a Bird Base would be easier. But to use
a Bird Base to fold a four-legged animal, you would have to represent two
of the legs (usually the rear legs) with a single point. In the 1950s and
1960s, there were a lot of three-legged origami animals hobbling around.

The first significant systematic design came in the 1960s with the development
of a folding style called ‘box pleating’, so dubbed because the paper is
initially pleated, and intermediate stages resemble a chain of boxes. Box
pleating has resulted in some of the most fantastic structures origami has
yet produced, including one of the earliest: Neal Elias’s ‘Llopio’s Moment
of Truth’, in which a bull, bullfighter and cape are all folded from a single
sheet of paper. One of the most complex box-pleated models is my ‘Black
Forest Cuckoo Clock’; it contains about 200 metres of creases in a model
40 centimetres high, and takes four to six hours to fold. Both of these
designs, like most box-pleated designs, however, depart from the traditional
square and are folded from rectangles (with proportions of 3:1 and 10:1,
respectively).

Box pleating brought complexity to rectangles, but the square – the
traditional starting shape – remained relatively undeveloped, yielding only
a handful of complex designs. That situation changed in the 1970s, when
three Americans and a Japanese, working independently, hit upon a set of
techniques and symmetries suitable for folding complex models from squares.
They were John Montroll, an engineer and mathematics lecturer; Peter Engel,
a science writer and architect; Jun Maekawa, a nuclear physicist; and myself,
a laser physicist. The techniques we developed have come to be called ‘technical’
folding: origami composed of equal parts art and engineering.

The basic principles of our work were quite simple. In the four classic
bases, the same shape appears in multiples of two, four, eight and sixteen.
Technical folding simply expands upon that pattern.

This reappearing shape is an isosceles right-angled trianglewith two
creases in it; Figure 2 shows how it appears in eachbase in successively
smaller sizes. Two of the basic trianglescan be assembled into a square,
yielding the Kite Base. Fourgive the Fish Base. Eight give the Bird Base.
Sixteen givethe Frog Base. The pattern is clear. We could easily go to 32by
folding the four corners of the square to the centre and making a Bird Base
from the result. In this case, we would end up with what has become known
as a Blintzed Bird Base, which has already been used for a few complex designs.

The crease pattern acts as a map showing how the paper should collapse
to form a base. As in box-pleated bases, the base may be formed from the
creased square by collapsing the crease pattern on folds in alternate directions.
Every source of radial creases on the square becomes a point of the base;
therefore, more appendages may be achieved by creating more radial clusters
of creases. The Blintzed Bird Base, with its nine major points, was nowhere
near the limits of what was possible. The crease pattern for my ‘Sea Urchin’,
which incorporates 128 copies of this triangle, contains 25 equal-length
points.

This triangle is still not the most fundamental unit. It is composed
of three smaller triangles: two identical scalene right triangles (with
sides in the ratio of 1:1+t2) and one isosceles right triangle (with sides
in the ratio of 1:1) that is a smaller copy of the original. We will call
them type A and B triangles, respectively. These two triangles appear over
and over in different sizes throughout the creased patterns in Figures 3and
2. They are the true building blocks of technical folding.

These two triangles have some interesting properties. They can each
be dissected into two or four smaller copies of themselves (Figure 4); each
triangle can also be dissected into two triangles of type A and one of type
B. By selectively applying these dissections to simple crease patterns,
it is possible to get more complicated creased patterns, yielding more and
more points.

Rather than breaking up a square into smaller and smaller triangles,
we can assemble A and B triangles into larger and larger geometric patterns.
For example, two type As andtwo type Bs can be assembled into a 1:t2 rectangle.
Fourof these rectangles can be assembled into another 1:t2 rectangle with
two axes of symmetry (Figure 5). By thesemeans, we can create higher-order
building blocks withwhich to generate bases. By combining ever larger assemblies
of the basic modules, we can create ever morecomplicated bases, leading
to ever more complex models. Aswe explore the different combinations of
triangles, we candevelop ‘libraries’ of higher-order crease patterns; the1:t2
rectangle is an example of one. The problem ofdesigning an origami base
thereby reduces to that of ’tiling’a square with A and B triangles (or higher-order
combinationsof such) so that we get a radial pattern of creases for eachappendage
of the model.

Each point, or equivalently, radial cluster, takes up a certain amount
of paper on the square that depends on the length of the point; so, for
example, getting a crease pattern with eight radial clusters is still not
necessarily a guarantee of folding a spider, or its base. The points on
the base that represent a spider’s eight legs must be roughly equal lengths
and must emanate from the appropriate place on the body. (It would not do
to have six coming from the body and two from the head.) Additional design
rules can ensure that the points are the same length and fall in the right
places.

Some of these rules are almost obvious: the cluster that becomes the
head should be at the opposite end of the paper from the cluster that becomes
the tail; the forelegs should be closer to the head than the tail; and so
forth. In general, points that are close together on the base should be
close togetheron the square. The length constraints are less obvious, butstill
easy to deal with. A spot on the square that is the tip ofa 2-centimetre
point on the base uses the surrounding 2-centimetres of paper on the square.
This means, for example, that two clusters of creases that are to yield
2-centimetrepoints cannot be any closer to each other than 4 centimetres(the
sum of the two lengths). This property may be summarised by saying that
two points on the square cannot becloser to each other than the length of
the shortest pathbetween them on the base.

Therefore, in designing a base, we must arrange combinations of type
A and type B triangles on a square so that we get a radial cluster of creases
for every appendage of the subject, located and spaced according to the
location and lengths of the subject’s appendages. While it may be necessary
to develop a unique base for a particular subject, it is also possible to
use a single base for many different subjects. For a subject with six major
appendages, two of the 1:t2 rectangles can be combined with two Bird Bases
to give the crease pattern in Figure 6, which yielded an alligator for Peter
Engel, a shark for John Montroll, a kangaroo for Jun Maekawa, and for myself,
Albert, the Experimental Rat.

As the technology to design origami models has improved, the subject
matter has shifted towards the more challenging end of the spectrum. As
is often the case in the sciences, we find technology in search of a problem
to solve. The ability to fold multipointed creations cries out for a multipointed
subject. Insects, once considered all but impossible, are now commonplace.
Legs are no longer a sign of a realistic arthropod; mandibles are. The ultimate
challenge to a designer was once thought to be a lobster, with its eight
long, skinny legs, two split claws, antennae and segmented tail. In 1970,
origami lobsters didn’t exist. In 1990, you can choose your favourite species.

The engineering approach to folding brought origami into the modern
age. We may ask what origami can bring to science. Origami has always enjoyed
an interest among recreation mathematics, and achieved a prominent appearance
in Martin Gardner’s ‘Mathematical Games’ column in Scientific American magazine
in 1960. It made it to the world of engineering in 1969, when Jon Myer,
a scientist at Hughes Research Laboratories, published an article showing
how origami could be used to simulate optical systems. Just in the past
year, Alan Huang, a researcher at AT&T Bell Laboratories, has developed
a technique called ‘computational origami’ to aid in the design of optical
computers. It is interesting to speculate on the opposite influence. Could
a computer be used to design origami figures? The prospect is not outrageous.
Most of the design techniques that have been developed can be formalised
as algorithms: to get these N legs, follow these M steps. Computerised ‘expert
systems’ are already diagnosing illness and locating oil fields; surely
a flapping bird is no more complicated.

In fact, it has been done. In 1971, Arthur Appel, a researcher at IBM,
programmed one of the company’s System 360 computers to print out simple
geometric configurations of folded paper. Ninety per cent were considered
unsuccessful (one wonders what qualifies as a ‘successful’ geometric configuration),
but it raises an interesting question: Coulda computer some day design a
model deemed superiorto that designed by a person? Technical folders havereduced
much of the process of design to algorithms, whichnaturally lend themselves
to computer implementation. Indoing so, the folders may have sown the seeds
of theirown obsolescence.

However, while a computer could conceivably design a base far more complex
than could a mortal (one could envision a paper version of Rodin’s Gates
of Hell), folding it would be another matter entirely. Paper has a finite
thickness and tears easily. Already, the most advanced designs of technical
folders strain the tensile properties of the paper to its limit. The sea
urchin, which has 25 points, is easily generalised to 36, 49 or more points
were it not for the limitations of the paper. A computer could conceivably
design a Gates of Hell complete with every figure, but the model would not
bedazzle; it would merely rip.

There is another reason why origami design will likely remain a human
domain. A computer may be programmed for efficient design, because ‘efficiency’
is an attribute that may be quantified. The quality of ‘artistry’ is not.
Engineering may have made it easier, and science may have opened up new
vistas, but origami is still an art. Throughout the design of a model, there
are artistic decisions to be made, and the folder, technical or not, remains
firmly in the loop.

Robert Lang is a laser physicist at NASA’s Jet Propulsion Laboratory
in Pasadena, California.