THE NEW HEADQUARTERS FOR Barry Diller's InterActiveCorp stick up from the low-rise terrain of Manhattan's West Chelsea neighborhood like Space Mountain at Disneyland. The 10-story asymmetrical protuberance has outer walls that veer every which way, a typical design for architect Frank Gehry. But the building's showstopper is a facade that looks like sails billowed by the wind. Gehry, famous for his complex compositions in titanium and stainless steel, had never before designed a major building in glass, and he was shocked to learn how difficult it would be to soften and mold the material around the contours of the building. Each of the 2,541 pieces of glass would have to be heated to 1,148 degrees Fahrenheit, then cooled and shaped. It was physically possible, but the sheer size of the project made it seem inconceivable. "We didn't think we could do it," Gehry says. "We were going to abandon it."

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Take a look at a major city skyline. Buildings have become more complicated, like engineering riddles that seem to defy both physics and common sense. For leading architects, every commission is an invention, an intricate one-of-a-kind experiment. Gehry, of course, is the leader of the starchitects, conjuring from his computers shapes so modern and complex they're practically baroque. But Gehry and his fellow designers don't actually build what they dream up – that's not their job. They rely on a new breed of contractor, the kind that can translate an architect's lofty vision into a physical structure. For the InterActive HQ, Gehry turned to a longtime collaborator, the Connecticut-based branch of Permasteelisa, the Italian curtain-wall couturier. The firm has spent nearly a decade figuring out how to fabricate the sort of brainteaser building "envelopes" that have made Gehry a household name. Permasteelisa has quietly erected the facades of some of the most significant buildings in the US, from Thom Mayne's Federal Building in San Francisco to Yoshio Taniguchi's MoMA in Manhattan. Permasteelisa also engineered towers for Skidmore, Owings & Merrill, Norman Foster, and the new partnership of Cook+Fox. "They've become the go-to people when you have a difficult job," says architect Richard Cook, for whom the company is fabricating the shell of One Bryant Park, a conspicuously asymmetrical 55-story tower in Manhattan. "We can be thinking whatever we want, but if you can't build it, what's the point?"

Much has been written about how Gehry conjures complex shapes from his computer, or how Mayne relies on data processors to create exteriors that ripple and shimmer in response to fresh air and daylight. But little has been said about how these digital models become buildings or about the surprising ingenuity of the teams of engineers and craftsmen more technically adept than the architects whose designs they execute. Companies like Permasteelisa must figure out how to mass-produce custom high-end architectural components – the only way to turn fantastical renderings into structures that developers can afford to finance. And because no part of a project is more crucial to an architect's reputation than the curtain wall – the glass, metal, or masonry skin that is the public face of a building – the specialized contractors who fabricate and hang the material can make or break an architect's career. Where other contractors are quick to push back on the feasibility of an architect's design, the engineers at Permasteelisa rarely say it can't be done.

For the InterActiveCorp building, Alberto De Gobbi, a 44-year-old civil engineer and the president of Permasteelisa's US operation, started with a simple principle: Glass bends – a lot – before it breaks. "We said, 'Why don't we take the natural, physical capability of the glass and see how much we can push it?'" He demonstrates by holding up a sheet of paper with his left hand and gently pushing one of its corners with his right. In the end, Permasteelisa gave each 14-foot panel of glass a 4-inch curve. That may not sound significant, but it was enough to achieve the startling effect Gehry had in mind. "This is the first time in the world that this has been done to such an extreme," De Gobbi says.

Permasteelisa learned an early lesson about dealing with complexity from Gehry himself. In 1992, the not-yet-famous architect designed a 260-ton steel-mesh fish for the Barcelona Olympics. Gehry recalls Permasteelisa's founder, Massimo Colomban, saying, "I can't do it. I can't get it. I can't build it." The architect advised him to invest in a 3-D modeling technology developed for aircraft design. Columban bought the $100,000 software and the outsize workstation needed to run it. "Two weeks later," Gehry says, "Massimo called me and said, 'Perfetto!'" With the software, the company's engineers could finally draw up Gehry's ideas, something they couldn't do with traditional 2-D CAD drawings. Later, they carried out the architect's designs by hiring about a dozen mountain climbers to weave 90,000 feet of gold-hued metal onto the 135-foot-high fish-shaped frame.

Since then, Permasteelisa has collaborated with Gehry on a dozen projects. Currently, the firm is working with him on both the InterActiveCorp headquarters and a 75-story hospital-school-condo tower, known as the Beekman, planned for lower Manhattan. A few years ago, the firm hung the gleaming stainless steel skin that covers the Gehry-designed Disney Concert Hall in Los Angeles. Before that, it helped Gehry build the cafeteria at the New York headquarters of Wired's parent company, Condé Nast.

Permasteelisa succeeds by breaking hugely complicated projects into more parts, not fewer. Then, even a topological nightmare like the Disney Concert Hall becomes manageable. "We broke it down into elements," De Gobbi says. "We're talking about 700,000 parts. The stainless steel sheets were pretty easy to fabricate once you had the design of each individual piece."

A stroll through the firm's 300,000-square-foot factory in Windsor, Connecticut – strategically located midway between New York and Boston – is an education in the prefab nature of skyscraper construction. It's a full-on assembly line for making one-off buildings. Thousands of "units" – the sophisticated assemblages of glass, coatings, and sealants surrounded by a crust of precisely extruded and trimmed metal – are shipped to building sites, where they're pieced together into huge exteriors. But something more interesting is out back. Test sections of recognizable facades line the property: There's the Hearst Tower, 7 World Trade Center, and a chunk of the new Louis Vuitton store. ("We dress the building; we do haute couture," De Gobbi says.) It's like visiting the New York set on a Hollywood back lot – except instead of creating illusion, Permasteelisa builds reality.