Vision, vision, vision, it’s everywhere. Can’t move for it. Architects are living in one of those all-too-brief moments in which the world seems to be swimming with fat wallets — cities, Middle Eastern oil states, capitalist dictatorships — with the means and the egos to indulge in fantastical visions.Not in Britain, naturally. We prefer to get our visionary fantasies in the sale aisle at Matalan. No, it’s in China, of course, and Dubai, but also in culturally adventurous continental Europe, and even in the once architecturally cautious America, that experimentation is flourishing.

Next week a major exhibition, Future City: Experiment and Utopia in Architecture 1956-2006, opens at the Barbican. This vast survey of the avant-garde since the Second World War has been thrillingly designed by the modern-day experimentalists Foreign Office Architects as a labyrinthine city within what is the last old-school utopian complex built in Britain. Almost all the (living) architects in the show are building, and on a scale: FOA are co-designing the 2012 Olympic Park, if the shindig’s accountants allow them; Coop Himmelblau are realising their Sixties fantasy Cloud as a show complex for BMW in Munich; America’s king of crazy shapes Thom Mayne last year won architecture’s highest honour, the Pritzker Prize.

We can chuckle at the models’ fashions in the Smithsons’ House of the Future, the Austin Powers-style inflatable cells Haus-Rucker-Co thought of to expand Manhattan. But these dreams are coming true. There’s a market for Utopias these days. And yet they all began with one man.

Constant Niewenhuys died in August, at the age of 85. There were few obituaries beyond his home country, the Netherlands. True, the man hadn’t exactly been front-page news for a decade or three. But still, this was the intellectual leader of the Provos, those pot-smoking anarchists whose artsy pranks in the 1960s ushered in the stereotype of liberal, libertarian Netherlands.

Constant co-founded the Situationiste Internationale, too, Jean-Luc Godard’s “children of Marx and Coca-Cola”, inspiration for every sulky counter-cultural movement from Beatniks through May 1968 and punk to the anti-globalisation protestors. The man was also a leading light of CoBrA, whose paintings — great childlike scrawls designed to put a bat up the nightdress of bourgeois society — are today the kind more admired by art theoreticians than by anyone with eyes in their head. And he also happened to be the most influential architect since the war.

Of course you’ve never heard of him. The man didn’t lay a brick in his entire life. But his one great conceptual work, New Babylon, was so powerful a vision of the future, the true heir to great architectural fantasists on paper from Piranesi to Sant’Elia, that there are few architects since who don’t owe him an intellectual debt. New Babylon begat the swirling forms of Frank Gehry and Zaha Hadid, the technopop of Archigram and Cedric Price, the playful naivety of Will Alsop, even the pragmatic high-tech of Richard Rogers and Norman Foster, and certainly the provocations of Rem Koolhaas.

The exhibition’s story begins in 1956, in Paris, with Constant. In that year he met Guy Debord, soon to be the leader of the Situationists. They had a vision of what urbanism and society should be. But first they had a vision of what urbanism and society should not be: not the bourgeois, class-riven city of old, nor the corrupted modernist Utopias rising in the banlieue, nor the new consumerist cities of America that would come to dominate the West.

These cities were prisons for the soul. But by subverting them with “guerrilla” acts, both benign and violent, even riotous, a more authentic, alive city might be unearthed to free the individual, they said. “Beneath the pavement, the beach”, went their slogan.

If Debord gave Situationism a language, Constant, its artist and architect, gave it form. New Babylon was a city not of sin but of pleasure. Constant imagined a future in which humans, freed into permanent leisure by technology, would endlessly have fun.

Unlike the dystopian futurist visions of Metropolis or Things to Come, this freedom and leisure was communist, shared by all, not just an elite. Constant sketched it, painted it, lithographed it, built models of it — all on show at the Barbican — in furious labyrinthine swirls that thumbed their noses at the grids of the Modernists. The Situationists loved instead the organic bulges of Frank Lloyd Wright, the curls of the Baroque.

New Babylon went on public show in a small gallery in Essen in 1960. Debord excommunicated Constant from the Situationists, in part because of the very act of making concrete a postrevolutionary world that Debord believed should remain blank until revolution had occurred.

Constant never believed New Babylon was anything other than his personal interpretation of the “feeling” of Utopia. But he hoped that architects and artists could create physical manifestations of Utopia that might accelerate its arrival. The split cut to the heart of 1960s revolutionary politics and the role of architecture within it.

Does social change happen through total revolution or change from within? Can architecture change society? Ten years later such questions were as dated as rock’n’roll. The failure of May 1968, stirred by Situationism, was a starting pistol for the retreat from modernist Utopianism. Architecture ditched politics in favour of aesthetics.

In the real world, a Babylon of sorts multiplied, only one ruled by Heat magazine, rather than free, post-revolutionary citizens. Constant returned to the canvas in 1974. Twenty years later Guy Debord blew his brains out.

Future City: Experiment and Utopia in Architecture 1956-2006 is at the Barbican Art Gallery, EC2 (020-7638 8891), from June 15