





Mathematical Paper Folding Exhibit
 Introduction
 Interview with Dr Robert Lang
 Single Cut Origami
 Curved Crease Origami
 Business Card Menger Sponge
 Links to Other Sites and Resources 
Mathematical Paper Folding:
An Interview with Robert Lang
By Margaret Wertheim
This interview was first published in
Cabinet, Issue 17, Spring 2005 
Robert Lang is a pioneer in the emerging field of
computational origami, a branch of mathematics that explores the formal
properties and potentialities of folded paper. Like the study of knots,
pioneered in the late nineteenth century, computational origami and
its practical offshoot origami sekkei or “technical folding”
turn out to have a surprising range of applications to real world
problems; from working out how to fold up stents so they can be threaded
into arteries, to designing thinfilm telescopes that are packed into
the hold of a space shuttle. Lang is the inventor of the TreeMaker
computer program, which allows him to design and calculate crease
patterns for a wide range of origami models—including intricate
insects, crustaceans, and amphibians. He has been one of the very
few Western columnists for the Japan Origami Academic Society and
is the author of eight books, including Origami Design Secrets: Mathematical
Methods for an Ancient Art. Lang received a doctorate in physics from
Caltech and spent twenty years as a laser physicist before becoming
a fulltime paper folder. 

Origami model
of a tree frog
by Robert Lang. 
MW: You spent two
decades working as a physicist. What made you decide to give that
up and become a professional origamist?
RL: I’ve been interested in origami my entire life. In fact,
my interest in physics started sometime in college but I’ve
been folding since I was a child. Through the years I was a professional
physicist, I found that the tools I had learned doing physics and
engineering—the mathematics and the approach to breaking down
problems and studying the underlying theory—could be applied
to origami as well. So that helped me to develop my origami art
to a fairly high level. It just grew, and the number of different
facets I was involved in—the art, the science, the underlying
mathematics, the applications to technology—eventually got
to a point that I felt I could occupy myself fulltime doing origami,
consulting, lecturing, and making art. I made that change three
years ago and never looked back.
MW: Can you explain what the term “technical folding”
means?
RL: It’s used to apply to origami that’s complex enough
that it probably wasn’t discovered by accident. The word technical
really means techniques; you’re using specific techniques
for designing specific features. And while it had its roots maybe
as early as the 1970s, it really blossomed in the 1990s when various
origami artists developed mathematical and geometrical principles
for folding—they developed an understanding of how the crease
patterns turn into geometric shapes in the folded object. So if
they needed to make an object that has five parts (arms, legs, wings
and whatever), they could start to figure out what type of geometric
shapes to put into the crease pattern that would assemble into those
parts. In effect they developed a set of building blocks for origami.
MW: And does this enable you to do things that weren’t possible
with traditional origami techniques?
RL: Pretty much. The traditional origami designs were generally
simplified and abstract, and when people tried to do subjects that
had very complicated shapes—the caseinpoint being insects
and arthropods which have lots of long, skinny legs—they found
that traditional folding styles weren’t able to give them
the features they wanted. And even if they got an insect with, say,
six legs, they had to be sort of stumpy little points because they
didn’t know how to get really long, skinny points. What technical
folding allowed us to do was to create all the features with all
the dimensions that we really wanted to capture.
MW: So what creatures have you been able to develop with these techniques?
RL: The things that drove me to develop my techniques were cervids,
horned animals—deer, elk, moose, antelope, and the like. Whitetailed
deer, moose, and elk all have different branching patterns in their
antlers and I wanted to be able to make each species. That required
one to specify the lengths of the points, the numbers of points
and how they’re connected to each other with a great degree
of precision. But another class of subjects that this worked really
well for was insects and the broader class of arthropods—everything
from crabs and lobsters to scorpions and spiders. 

Paper folded model
of a fiddler crab by Robert Lang. Note the asymmetric front claws
– an effect
impossible to achieve with traditional origami techniques. 
MW: In addition to your own designs
you also do origami consulting. Whom do you consult for?
RL: Recently I was at the Jet Propulsion Laboratory. I’ve
also done work for the Lawrence Livermore Laboratory and a variety
of commercial companies that are doing product development in areas
like medical devices and packaging.
MW: What sort of packaging requires a professional origamist?
RL: Typically it’s a package that has some dual purpose. One
purpose would be when you have to enclose several different objects
and you want to use the same container for all of them so that it
has to fold between several different states. I don’t know
if they need a professional origamist, but someone who has been
folding for twenty years knows a lot of different structures. Usually
I manage to come up with something they haven’t seen before.
MW: You’ve also helped the Lawrence Livermore Lab develop
a spacebased telescope.
RL: The idea was to make a telescope with a 100meter aperture that
could be deployed in space, meaning the main lens of the telescope
would be approximately a football pitch across. The lens itself
would be a diffractive lens, a pattern of grooves formed on a thin,
plastic substrate, like ones used in overhead projectors. So now
they had the problem of a 100meter sheet of plastic that needs
to be taken into space and the only way we have to take things into
space is a rocket or a space shuttle, which are only a few meters
across. That pretty much stipulates some form of folding. They built
a fivemeter prototype based on a design I proposed that was very
successful in their tests.
MW: That suggests there is a general class of subjects to which
origami is applicable: something that needs to be folded up in order
to ship it or launch it, but later on, its end state has to be much
bigger.
RL: That’s a pretty good description. Whenever you have an
object that exists in a large state that is generally a surface,
something that’s roughly flat, and it also has to exist in
a much smaller state, usually for transportation, then origami plays
a role. In the space program, that shows up in things like folded
lenses, solar sails, various types of collapsible antennas, and
collapsible shrouds or shields. In the area of medicine, there are
various types of implants such as stents that go into the body and
which need to be put in through as small a hole as possible. Or
maybe it goes in arthroscopically through a vein. Again, folding
is a way of collapsing the structure down so it can be threaded
through an artery or inserted into an incision and expanded once
it’s close to its final resting place. 

A folded stent designed
to open collapsed arteries, designed by Kaoru Kobayashi  under professor
ZhongYou. 
MW: Haven’t origami techniques
also been used for working out how to pack airbags into steering
columns in cars?
RL: Actually, in that case it was for a computer simulation of an
airbag rather than the actual airbag. If you’re simulating
an airbag, you need to know where the creaselines form when the
airbag is collapsed. In this project, a German firm, EASi Engineering,
was developing a software tool for simulating airbags so that automotive
manufacturers could figure out whether a certain airbag design would
work without having to actually crash a bunch of Mercedes. They
came to me because I had published some papers about algorithms
for folding things flat and it turned out that one of these algorithms
was right for their needs.
MW: What is the technical problem regarding how to fold things flat?
RL: Mathematically the problem is very simple. Given a polyhedral
surface, can you construct creases so that when you fold on all
the creases, all of the faces of the polyhedra lie on the same plane.
That can be reduced to another problem: given an arbitrary polygon,
can you construct creases so that when you fold on all the creases,
all of the boundaries of the polygon lie along a single straight
line? That has a purely mathematical solution that is of interest
maybe only to mathematicians, yet it turns out to have these practical
applications as well.
MW: What other mathematical problems are technical folders interested
in?
RL: I think one of the most vibrant is the question of what distances
and shapes can be constructed just by folding alone, without doing
any measuring. That harkens back to an ancient problem in pure mathematics—what’s
called compass and straight edge construction. This goes back to
the early Greeks, who wondered what shapes and distances could be
constructed using just a compass and a straight edge for making
arcs and drawing straight lines. And there is an origami analogue
of that problem, which asks what distances and shapes you can construct
just by making folds in a sheet of paper without being allowed to
measure any distance with a ruler. It turns out that the field of
shapes you can construct with folding is richer than what you can
construct with compass and straight edge.
For example, just with folding you can solve the problem known as
trisecting the angle—this means dividing a given angle into
thirds. For 2000 years, people tried to find a way to do this with
a compass and unmarked ruler until finally in the late nineteenth
century mathematicians proved that it could not be done at all.
But in the 1980s, a French folder name Jacques Justin and a Japanese
folder name Tsune Abe independently showed how it could be done
with origami. Mathematically, trisecting an angle is the equivalent
of solving a cubic equation—an equation involving x to the
power of three. Using straight edge and compass you can only solve
equations with x to the power of 2. Once cubics were done with origami,
the question was naturally asked, “Well, can you solve higher
order equations with folding?” Just recently I have shown
that in fact origami can solve fifthorder equations—ones
involving x to the power of 5. That’s pretty interesting,
I think.
MW: Are there any practical applications for these constructions?
RL: Absolutely none that I can think of outside origami—it’s
appreciated purely for the mathematical beauty.
MW: One area in which I gather technical folding is proving useful
is one of the major problems in biology. We know that with proteins
often the most important thing about them is not the chemical composition,
per se, but the shape they eventually fold up to.
RL: There’s both relevance and differences here, because paper
folding is twodimensional and a protein is roughly a onedimensional
shape, a linear chain with a bunch of joints in the chain. Protein
folding is actually much more complicated than paper in that folds
can happen only at certain angles and there are bits that stick
together if you get them close. There are also other molecules jostling
around that can knock the protein about as it’s folding. But
the fundamental theory of folding is the same, and if you can develop
general concepts that apply across dimensions—from onedimensional
to twodimensional, and even higherdimensional problems—then
the results that you derive are going to be applicable to these
very fundamental issues like protein folding and biological activity.
MW: It reminds me of another branch of mathematics– knot theory.
In the late nineteenth century, mathematicians and physicists became
interested in how many different ways were there to tie a knot.
And it’s turned out in the late twentieth century that some
physicists believe knot theory might explain the nature of subatomic
particles. Mathematicians seems to have this way of taking what
seem to be unbelievably trivial things and developing from them
incredibly powerful abstract techniques. Do you think paper folding
may one day have some relevance to our understanding of fundamental
physics?
RL: Whenever you’re developing new mathematics, there’s
always that possibility. The hallmark of these sorts of surprise
applications is that they always turn out to have been a surprise.
There is a great example of this that is close to origami. In technical
origami when we’re designing complicated forms like manylegged
insects, we use a technique called “circle packing”
which basically asks the question how can you efficiently pack a
bunch of circles into various shaped containers. Now over the years
mathematicians have also studied how to pack spherical objects into
higherdimensional spaces and how close a packing you can get. Well,
it turned out that in 24 dimensions there is a particularly dense
packing. That sounds about as irrelevant an idea as you can get,
except it turns out that 24dimensional packing gives a very dense
compression algorithm for sending data. So using this 24dimensional
sphere packing result has become the basis for developing a very
efficient code for 24bit binary words. Now, who would have predicted
that?


Crease pattern for
fiddle crab model, showing the “circle packing” that entails
an approximate bestfit solution for how to arrange the creature’s
anatomical parts within the square sheet. 
MW: One thing that’s fascinating
about technical folding is that it’s both a physical and an
intellectual process. You have to have an analytical mind to design
the structures, but then you also need to have a great deal of practical
skill in terms of folding them as they are actually quite difficult
to make.
RL: I think of it like music. In fact there are a lot of analogies
between origami and music. You can compose both simple melodies
and symphonies with a lot of different instruments and themes moving
in and out. It’s the same with origami. There are simple beautiful
folds, in the same way that there are simple melodies, but if you’re
trying to do these very complicated structures you need practice
to get good at it.
MW: Is there a limit to the complexity of the models you can make
with origami?
RL: Mathematically there’s no limit. Theoretically, you can
take a finite sheet of paper and you can fold a star shape that
has an infinite, arbitrarily large perimeter —10,000 miles,
if you like. That shape’s points would have millions of layers
in them. So, that’s a problem you can do mathematically but
not in practice because, in the real world, paper has a finite thickness
and you’re limited in what you can do by the tensile properties
of the paper. In the last five to ten years, as people have designed
more complex figures, their ability to fold these figures has also
been enhanced by improvements in the field of papermaking. So you
can now get extremely thin, strong shapes that probably couldn’t
have been folded fifteen years ago.
MW: You’ve written a computer program, called TreeMaker that
will work out very complicated designs and calculate the crease
pattern. Yet traditional folders won’t even allow the use
of a ruler to make measurements. Is it somehow cheating to bring
a computer into the origami design process?
RL: There are people who get deeply nervous about the idea of using
a computer as a tool in design. The general consensus, I think,
is that when it comes to using a ruler, it’s fine if you’re
using it for designing the model as long as you don’t make
the person trying to follow the instructions use a ruler. Long before
I wrote my computer program, and even today, I still use a pencil
and paper to sketch out and calculate a draft of the creasepattern.
In that sense, the computer is no more of a tool than a pencil and
paper. I’ve done designs where I’ve intentionally asked
the folder to measure with a ruler just because it was a bit provocative,
but in my own folding I don’t consider a design to be finished
until I have a sequence that I can do with just with a square of
paper and my hands, not using any devices. 
Download
a pdf of Robert Lang's full size fiddle crab pattern. 
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