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Timeless truths
Folds and creases: Clues for architects, biologists and mathematicians

Some people don't even think this exists,' says Erik Demaine, turning in his hands an elaborately folded paper structure. The intricately pleated sail-like form swooshes gracefully in a compound curve and certainly looks real enough, if decidedly tricky to make.

Demaine, an assistant professor of computer science at the Massachusetts Institute of Technology, is the leading theoretician in the emerging field of origami mathematics, the formal study of what can be done with a folded sheet of paper. He believes the form he is holding is a hyperbolic paraboloid, a shape well known to mathematicians ' or something very close to that ' but he wants to be able to prove this conjecture. 'It's not easy to do,' he says.

Over the past few years he has published a series of landmark results about the theory of folded structures, including solutions to the longstanding 'single-cut' problem and the 'carpenter's rule' problem. These days he is applying insights he has gleaned from his studies of wrinkling and crinkling and hinging to questions in architecture, robotics and molecular biology.

As a child, he and his father, Martin, a goldsmith and glass artist who home-schooled his son as a single parent, travelled around the US, settling somewhere new every six to 12 months.

At 12, after Erik had become intensely interested first in computer games, then in computer programming and finally in mathematics, he persuaded the administrators of Dalhousie University in Halifax, Nova Scotia, to let him take classes in maths and computer science. His father sat in as an auditor. Demaine received his doctorate at 20 and at the same age became the youngest professor ever at MIT. In 2003 he was granted a MacArthur 'genius' fellowship.

Today, at 23, he has published more than 100 academic papers in fields as diverse as computational geometry, combinatorial game theory, data structures and graph theory.

'He loves working with other people,' says Jo-seph 'Rourke, a mathematician and computer scientist at Smith College who has been collaborating with Demaine since he was 16. 'He has a very broad understanding of a whole range of topics and he often brings in ideas that at first seem off the wall but really help to enrich what you are doing.'

Yet for all Demaine's smarts, the pleated form in front of him is not giving up its secrets easily. The perplexing question is whether its concertina-like structure can be derived by purely mathematical transformations of a flat sheet, or whether the sheet must be stretched in places to take on this complex shape.

As Demaine explains, stretching will warp the intrinsic flatness, thereby destroying the underlying geometry. If that were the case then, mathematically speaking, it will not exist. 'But if it doesn't exist mathematically, then something else is going on and it would be nice to know what that is,' he says.

Demaine's office is littered with these models. On the windowsill is a collection of glass vases and sculptures made by Demaine and his father, who is now a researcher in his son's lab.

Aside from the mathematical value of the hyperbolic forms, Demaine has also taught courses in the school of architecture and imagines being able to computationally generate a scaffolding of these shapes over which a flexible skin could be draped.

Demaine is primarily a theoretician. 'I love the idea of timeless truths,' he remarked.

Though Martin Demaine trained as a glass artist, when his son developed a fascination for computing and mathematics he happily read the books and attended lectures with him. 'I don't really think of them as such different activities,' he said of this switch from art-making to mathematical theorising.

Today, the father and son have written 43-papers together. Meanwhile, Martin, who has just been appointed artist-in-residence in the computer science department, also runs the MIT glass blowing workshop, where one of his students is his son.

Among the topics the two have researched together is the 'single cut' problem, whose roots go back to ancient China and to magic tricks. Before Houdini became an escape artiste he had a career as a magician and supposedly performed a trick in which he folded a piece of paper, then cut across the creases to 'magically' create a five-pointed star. Other examples of single cut magic are sprinkled through historical literature. The question that arises is, what sorts of shapes can you make this way' In 1998 the two Demaines, working with Anna Lubiw at the University of Waterloo, Ontario, proved that you could effectively make any shape just with folding and a single cut ' a star, a swan or a unicorn.

You can even create multiple shapes with a single snip of the scissors ' two stars, 10 stars or 50 stars if you like. One set of shapes that can be produced this way is the letters of the alphabet.

And since Demaine's proof shows that you can get as many shapes as you want, 'in theory you could produce the complete works of Shakespeare with a single cut,' said Robert Lang, a former laser physicist and professional folder who is collaborating with Erik on a major origami math project.

Understanding what you can do with paper is a two-dimensional problem, but Demaine also works with the one-dimensional analogue, or what are known as linkages. A linkage is a set of line segments hinged together like the classic carpenter's rule. It sounds simpler, but Lang noted that the one-dimensional case is often much harder to understand and analyse than the two-dimensional case.

The major part of Demaine's doctoral thesis was a solution to the so-called 'carpenter's rule problem,' which asks a question about how linkages can be unfolded. Put simply: imagine a carpenter's rule arranged on a table in a complicated pattern. Is it always possible to unfold the rule, or are there patterns that cannot be opened out, that are in what mathematicians call a 'locked' state'

Robert Connelly, a mathematician at Cornell who worked with Demaine on the solution, noted by phone that the problem was a good deal subtler than it initially sounded. At first mathematicians thought all linkages could be unfolded, but during the 1990s they discovered a number of very clever arrangements that looked impossible to unfold. 'Many people thought a lot of these were locked,' Connelly said.

But he and Demaine, along with Guenter Rote of the Free University of Berlin, proved that all linkage arrangements could be unfolded. It turns out that the problem of folding and unfolding linkages is applicable to one of the major scientific questions of our time: how do proteins fold up' Proteins are made up of long strings of amino acids, and as the strings are produced inside a cell by the ribosome they fold up into complicated shapes.

It is this shape that largely determines the biochemical function of each protein. Molecular biologists and pharmaceutical companies are extremely interested in understanding how protein folding occurs, in part because they like to design specialised proteins for use as drugs.

Recently Demaine has been working on the question of how protein folding occurs. 'We think they fold by keeping their backbones as linkages,' he said. He and 'Rourke, along with Stefan Langerman at the Free University of Brussels, Belgium, have created a computer model of this process and will report their results soon in a paper for the journal Algorithmica. 'Rourke said their model proved that protein linkages could not become locked. If their main assumptions hold up, he said, this result can help pharmaceutical companies to radically speed up the time it takes to find useful proteins.

Ideally, molecular biologists would like to be able to predict from the chemical structure of a protein what shape it will fold into. 'If you could predict that,' Demaine said, 'then you wouldn't have to do all the hard work of synthesising and crystallising the protein to find out what it does.'

Demaine hopes to solve the protein-folding problem completely. 'I'm an optimist,' he said. 'I believe it can be done in my lifetime.' (NYTNS)

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