It's a sunny afternoon in Rome and high above the Via Guido Reni in warm, marshmallow-soft air, swallows dot and dash under a sky that might have been painted by Tintoretto, or the young Turner. The birds' trajectories possess a perfect metaphorical quality: four or five hundred feet beneath their febrile wingtips lies the MAXXI museum of contemporary art and architecture, whose form is a lavish, supersized riddle of sinuously overlapping voids and projections.
Outside the entrance, the huge plaster skeleton designed in the 1980s by the Italian artist Gino de Dominicis is laid out in droll extremis. And 20 metres beyond it, sitting in a clutter of chairs at the edge of the museum's geometrically faceted plaza, MAXXI's architect, Zaha Hadid, is discussing the contents of a panini with her minder, Roger "Three iPhones" Howie.
In London, at this very moment, builders are scrambling over the manta ray structure of Hadid's aquatics building on the site of the 2012 Olympic Games site; in Portland Place, a Royal Institute of British Architects judging panel is preparing to fly to Rome to decide if MAXXI is worthy of the coveted 2010 Stirling Prize, to be announced in early October; and, in architects' offices all over Britain, hundreds of designers are opening copies of Architects Journal or Building Design to find that the Baghdad-born Hadid has just been commissioned to design the new headquarters of the Central Bank of Iraq.
Hadid puts the panini back on its plate on the chair next to her. "I can't eat this," she says. "Where's Patrik? He won't talk to me, for some reason." Hadid glances towards the skeleton, wondering where her key collaborator, Patrik Schumacher, has gone.
He's on the other side of the plaza, behind her, walking to and fro enigmatically, like Jacques Tati's Monsieur Hulot. Perhaps he's musing about the architectural artworks that Hadid drew and painted in the 1970s, while studying at London's exclusive Architectural Association. Hadid's experimental art turned modernist forms into occult spatial patterns, and lines of force, that often resembled Technicolor tracer-fire.
Yet her proto-superstar status was confirmed not by these precocious successes, but by two high-profile failures: Hadid's competition-winning design for the Peak Club building in Hong Kong was spiked in 1982; and 11 years later, her Cardiff Bay Opera House proposal was deemed too challenging by both the city and Whitehall mandarins. She found these episodes almost crushing. "They couldn't deal with me because I was a woman," she said of the Cardiff debacle. "And because I was foreign, and because the design was foreign." Yet, as a British citizen with a CBE, she remains utterly anglophile: "My work developed totally because I was in England, and at the AA – a British situation. They allowed me to do what I wanted. People supported my ideas."
Decades on, the media has mythologised her as a diva who sometimes uses a megaphone in her Clerkenwell office, and who hobnobs with A-list friends such as Karl Lagerfeld. She encourages it, playfully: today her toenails are the colour of Greek lemons, to match the chic canary-yellow coat that she will wear at the fashionable Dal Bolognese restaurant in three hours' time.
Hadid means "iron" in Arabic and, with 400 staff working on massive building and urban regeneration projects in the US, Europe, the Middle East and Far East, she has plenty of that particular quality. "I think about architecture all the time," she says. "That's the problem. But I've always been like that. I dream it sometimes. If I'm drawing, I find that relaxing. But one doesn't have much time. The travelling is awful – but tiredness can also make your thinking more intense."
She has few consuming outside interests. "I like music," she says. "Country, hip-hop, R&B, sometimes classical. And I like art. It's inspiring. It might not guide you through life, but the way you react to it can be applied to other things, whether you're a lawyer or a writer, or whatever. It's this hybridising of different things which is really interesting."
And nothing is more interesting to her than cities: "Cities are a landscape, and people use them in different ways. One change has a ricochet effect on other things. Look at Bogotá: there are great things happening there, unexpected things. And Beirut is very interesting because there used to be so much complexity in the urban fabric. Now it's lost its charm. There has been regeneration, but it's not random and complex any more. London needs an overview, and then closer focus on smaller areas. But there are some really exciting things happening."
Hadid has a strong interest in Docklands and the Lea Valley that dates back more than two decades to research projects she led with her students at the Architectural Association. "Those projects were about how London could develop in the future," she recalls. "It was a series of drawings and sections about going eastwards, with linear cities in the estuary. They predicted the later development of Stratford and the Thames Gateway. It's very interesting how, through that kind of research, one could predict these changes in cities. People say I design architectural icons. If I design a building and it becomes an icon, that's ok. But cities are really the ultimate project for architects – to apply your mind, your ideas, to a big area, in one go."
Hadid is excited about London's multi-billion pound infrastructural projects – Crossrail, for example – and says they will totally transform places such as Greenwich. But she is scathing about the quality of new housing, which she says has often been carried out poorly in Britain. "I mean, luxury housing in London is like social housing in other European cities. It's been badly done – very schlocky. The government should make an example of doing good stuff. Doing Dickensian vernacular isn't enough."
Education is her other hot topic – "it's top of the list in all cultures" – and she thinks the state should be more committed to progressive approaches to the design of schools and colleges. "When we designed the Maggie's cancer care centre in Kirkcaldy, it was the modernist agenda – a place whose design made you feel good. It should be the same with the design of schools. Simple stuff, like nice light and good outdoor space."
"People are more aware, superficially, about more and more stuff," she muses. "It's a world of information and it affects the way we live. Good education is so important. We do need to look at the way people are taught. It not just about qualifications to get a job. It's about being educated. That's why the state is critical, in helping."
Hadid's remarks reflect the golden glow of her own early education – a perfect example of the hybrid conditions that so fascinates her. In 1950s Baghdad, as the secularised daughter of cosmopolitan, intellectual parents, she attended a Catholic school; there was further education in Switzerland, then a degree in maths at the American University in Beirut. And underlying it all was her first formative inspiration: visits to the remains of Sumerian cities in Iraq as a teenager. Hadid was particularly insulted by Simon Jenkins's suggestion, on a Radio 4 programme, that she was a fantasist whose architecture was more appropriate for the Iraqi desert. "You don't find this baffling? I find it baffling. The city of Ur wasn't a fantasy!"
And neither is the fiscal Ur that she will design for Iraq's Central Bank. She hasn't returned to Iraq since her family's departure in the wake of Saddam's Baa'thist takeover, and working in Iraq will clearly challenge her. She recalls "the beautiful suburbs and modernist housing which are now all abandoned. It's difficult to place myself in that context. It's been 30 years... you still have memories, as a child. It's very difficult, emotionally. My [late] parents aren't there. Our house is still there, but I'm told it's all wired up. When the Central Bank contacted me [their directors interviewed Hadid in Turkey] I really thought, at the time, that there was so much rebuilding and social structures to deal with that it wasn't just about one building."
But it is. One of the quickest ways for any country to signal internationally recognisable normality is to commission a big, and very striking, building by a world- famous architect. "The Bank are actually very proud of the fact that I'm an Iraqi. They feel very strongly about using my practice to design the building – it's symbolic."
The conversation has been interrupted by perhaps a dozen requests from strangers, for autographs or photo-opportunities, to which Hadid invariably assents. And now she is approached by her latest clients, the billionaire philanthrophists Eli and Edythe Broad, for whom she's designing a new museum in Michigan. The perfectly turned out Edythe Broad congratulates Hadid on MAXXI, her eyes and smile radiating a die-stamped, Forbes Rich List charm. Various pleasantries are exchanged, and the Broads retreat graciously.
Hadid glances again at the treacherous panini, and then spots Patrik Schumacher, who is still doing his Hulot routine. "Ask him to come here," she says to Roger Howie. "Tell him I want to talk to him." High overhead, the swallows flicker like ideas, or the tentative first marks in a drawing.