By MARGARET WERTHEIM
June 22, 2004
SANTA CRUZ, Calif. - On the mantel of a quiet suburban home
here stands a curious object resembling a small set of organ
pipes nestled into a neat, white case. At first glance it
does not seem possible that such a complex, curving form could
have been folded from a single sheet of paper, and yet it
The construction is one of an astonishing collection of paper
objects folded by Dr. David Huffman, a former professor of
computer science at the University of California, Santa Cruz,
and a pioneer in computational origami, an emerging field
with an improbable name but surprisingly practical applications.
Dr. Huffman died in 1999, but on a recent afternoon his daughter
Elise Huffman showed a visitor a sampling of her father's
enigmatic models. In contrast to traditional origami, where
all folds are straight, Dr. Huffman developed structures based
around curved folds, many calling to mind seedpods and seashells.
It is as if paper has been imbued with life.
In another innovative approach, Dr. Huffman explored structures
composed of repeating three-dimensional units - chains of
cubes and rhomboids, and complex tesselations of triangular,
pentagonal and star-shaped blocks. From the outside, one model
appears to be just a rolled-up sheet of paper, but looking
down the tube reveals a miniature spiral staircase. All this
has been achieved with no cuts or glue, the one classic origami
rule that Dr. Huffman seemed inclined to obey.
Derived from the Japanese ori, to fold, and gami, paper, origami
has come a long way from cute little birds and decorative
boxes. Mathematicians and scientists like Dr. Huffman have
begun mapping the laws that underlie folding, converting words
and concepts into algebraic rules. Computational origami,
also known as technical folding, or origami sekkei, draws
on fields that include computational geometry, number theory,
coding theory and linear algebra. This weekend, paper folders
from around the nation will gather at the Fashion Institute
of Technology in New York for the annual convention of Origami
USA. At an adjacent conference on origami and education, Dr.
Robert Lang, a leading computational origamist, will give
a talk on mathematics and its application to origami design,
including such real-world problems as folding airbags and
Dr. Lang, a laser physicist in Alamo, Calif., who trained
at the California Institute of Technology, gave up that career
18 months ago to become a full-time folder. "Some people
are peculiarly susceptible to the charms of origami,"
he said, "and somewhere along the way the ranks of the
infected were joined by mathematicians." Dr. Lang is
the author of a recent book on technical folding, "Origami
Design Secrets: Mathematical Methods for an Ancient Art."
Most computational origamists are driven by sheer curiosity
and the aesthetic pleasure of these structures, but their
work is also finding application in fields like astronomy
and protein folding, and even automobile safety. These days
when Dr. Lang is not inventing new models using a specialized
origami software package he has developed, he acts as an origami
consultant. He has helped a German manufacturer design folding
patterns for airbags and advised astronomers on how to fold
up a huge flat-screen lens for a telescope based in space.
Dr. Lang has been studying Dr. Huffman's models and research
notes, and is amazed at what he has found. Although Dr. Huffman
is a legend in the tiny world of origami sekkei, few people
have seen his work. During his life he published only one
paper on the subject. Dr. Huffman worked on his foldings from
the early 1970's, and over the years, said Dr. Lang, "he
anticipated a great deal of what other people have since rediscovered
or are only now discovering. At least half of what he did
is unlike anything I've seen."
One of Dr. Huffman's main interests was to calculate precisely
what structures could be folded to avoid putting strain on
the paper. Through his mathematics, he was trying to understand
"when you have multiple folds coming into a point, what
is the relationship of the angles so the paper won't stretch
or tear,'' said Dr. Michael Tanner, a former computer science
colleague of Dr. Huffman who is now provost and vice chancellor
for academic affairs at the University of Illinois in Chicago.
What fascinated him above all else, Dr. Tanner said, "was
how the mathematics could become manifest in the paper. You'd
think paper can't do that, but he'd say you just don't know
paper well enough."
One of Dr. Huffman's discoveries was the critical "pi
condition." This says that if you have a point, or vertex,
surrounded by four creases and you want the form to fold flat,
then opposite angles around the vertex must sum to 180 degrees
- or using the measure that mathematicians prefer, to pi radians.
Others have rediscovered that condition, Dr. Lang said, and
it has now generalized for more than four creases. In this
case, whatever the number of creases, all alternate angles
must sum to pi. How and under what conditions things can fold
flat is a major concern in computational origami.
Dr. Huffman's folding was a private activity. Professionally
he worked in the field of coding and information theory. As
a student at M.I.T. in the 1950's, he discovered a minimal
way of encoding information known as Huffman Codes, which
are now used to help compress MP3 music files and JPEG images.
Dr. Peter G. Neumann of the Computer Science Laboratory at
the SRI International (formerly the Stanford Research Institute)
said that in everything Dr. Huffman did, he was obsessed with
elegance and simplicity. "He had an ability to visualize
problems and to see things that nobody had seen before,"
Dr. Neumann said.
Like Mr. Resch, Dr. Huffman seemed innately attracted to elegant
forms. Before he took up paper folding, he was interested
in what are called "minimal surfaces," the shapes
that soap bubbles make. He carried this theme into origami,
experimenting with ways that pleated patterns of straight
folds can give rise to curving three-dimensional surfaces.
Dr. Erik Demaine of M.I.T.'s Laboratory for Computer Science,
who is now pursuing similar research, described Dr. Huffman's
work in this area as "awesome."
Finally, Dr. Huffman moved into studying models in which the
folds themselves were curved. "We know almost nothing
about curved creases," said Dr. Demaine, who is using
computer software to simulate the behavior of paper under
the influence of curving folds. Much of Dr. Huffman's research
was based on curves derived from conic sections, such as the
hyperbola and the ellipse.
His marriage of aesthetics and science has grown into a field
that goes well beyond paper. Dr. Tanner noted that his research
is relevant to real-world problems where you want to know
how sheets of material will behave under stress. Pressing
sheet metal for car bodies is one example. "Understanding
what's going to happen to the metal,'' which will stretch,
"is related to the question of how far it is from the
case of paper," which will not, Dr. Tanner said.
The mathematician G. H. Hardy wrote that "there is no
permanent place in the world for ugly mathematics." Dr.
Huffman, who gave concrete form to beautiful mathematical
relations, would no doubt have agreed. In a talk he gave at
U.C. Santa Cruz in 1979 to an audience of artists and scientists,
he noted that it was rare for the two groups to communicate
with one another.
"I don't claim to be an artist. I'm not even sure how
to define art," he said. "But I find it natural
that the elegant mathematical theorems associated with paper
surfaces should lead to visual elegance as well."