But perhaps the most audacious part of Seville's grand plan is an architectural phantasmagoria called the Metropol Parasol. The ambition is to redefine what that most treasured of European public spaces — the square or piazza — should be, how it works, and how it serves the urban arts. The site is the only one large enough within the boundaries of the old city that happened to be available, the Plaza de la Encarnation. Until 1973, it was a covered market; after that it was torn down, and the bare land became a parking lot. With car mania in full flood, the next idea was to build an underground garage. When the earth was being excavated, parts of Roman Seville — a colony called Hispalis — were discovered. Work stopped, and the garage was abandoned.

In 2003, the city opened an international competition for a plan to both preserve the Roman relics and revitalize the life of the square. The winning design, by the German architect Jürgen Mayer H., suggests a “War of the Worlds” touchdown by immense, straddling semivegetal beings. But the term parasol is a clue to the concept: a series of undulating, almost amorphous canopies rising nearly 92 feet above the square, supported by six fat, circular stalks, veined like sinewy old tree trunks. (When I was there, only two concrete stumps had risen.) During the course of the day, the sun and the shadows will constantly move and refract across the surfaces, changing the atmosphere of the public spaces and suggesting that the structure itself is ephemeral rather than permanent. It's a building and yet it's not, making more conventional structures seem analog in the age of digital.

Looking at the computer projections, there is something biological about the outer surfaces: They resemble the kind of organic lattice you see, for example, in the membranes inside a lung. Just how dramatic this will be in actuality is impossible to tell — the Parasol won't be finished until late 2008, and digital images cannot, obviously, convey the physical experience that remains, at the moment, in the minds of the team that created it. But whether intended or not, whether acknowledged or not, I was reminded of the organic, plastic shape-shifting style of Antoni Gaudí — particularly of his playful, sinuous Park Guell in Barcelona, which also juggles different levels. If Gaudí — who, decades before a computer existed, deployed an instinctive ability to think in three dimensions — had had access to the right software, he might well have come up with something like the Metropol Parasol.

A great city square makes tangible a platonic ideal of the intermingling of peoples; locals and visitors are at one in a space that pleases the eye in every direction, rests weary feet, and distills the character of the city to the details of gesture, voice, and style. In a real sense, the whole world flows through it and is caught, from dawn until late at night, in the best of moods. It is also where the barriers between indoors and outdoors are porous: We like the freedom to move into the café and then out to the tables, to sit in a torpor under a tree or take the rays as they enrich the bloom of a jug of sangria. But this is strictly life conducted at one level, on an even keel without access to other perspectives. The Parasol opens up four levels, or stages, for the street life to migrate within the square — some large enough for, say, a chamber music performance; some shaped for café, bar, and restaurant; some for kids at play; some for shopping and marketing. With each level open on all sides but also giving selective shelter, the attractions are flexible enough to adapt to all the Andalusian seasons and the climate.

One can see why the concept won over the city managers. Before Carillo sent me off to look at the works in the Plaza de la Encarnacion, I asked him who he thought, from his perspective as a history professor and urban czar, were the best city planners. He didn't hesitate.

"The Arabs," he said.

Valencia is a Mediterranean city but doesn't feel like one. Unlike Barcelona, whose whole disposition within a broad amphitheater is canted toward the sea — where the city is, in effect, living in a bowl that focuses the Mediterranean light on the wide boulevards — Valencia has an old city center that is one-and-a-half miles from the coast and, like any once walled city, looks inward rather than outward. When I had digested my paella, I walked off to the project that has tipped the city's gravity further toward the sea: the City of Arts and Sciences. The science and the art are powerfully apparent in the buildings themselves — they are an eye-popping sight as you approach, following the old bed of a diverted river, resembling, from a distance, a sculpture park.

Rick Lew / Conde Nast Traveler Floating visions: Calatrava's Planetarium, left, and Science Museum, right, are set amid wide, shallow pools to catch and reflect Valencia's magical brand of Mediterranean light, called La Clara.

Santiago Calatrava, the hometown boy, has four buildings: the Palace of the Arts Reina Sofia, containing four theaters with a total of more than 4,000 seats; a planetarium and IMAX cinema called L'Hemisferic; the Science Museum; and a car park complex called L'Umbracle. Beyond these, somewhat out on a limb, is Europe's largest aquarium, L'Oceanographic, the last work by the renowned Spanish architect Felix Candela, who died in 1997, well before it was completed.

The Palace of the Arts, wrapped in a swooping, sharp-ended white cowl, looks like a gigantic bicycle racer's helmet fallen to earth behind palm trees. Beyond the Palace of the Arts and a soaring road bridge finished in a lapis-like mineral blue, another agent of drama is revealed: water. Two of the buildings, L'Hemisferic and the Science Museum, are placed amid three wide and shallow pools. I wasn't prepared for this — nothing I had seen or read about the site had suggested the effect of the water on the appearance and even the substance of the buildings. Valencia is known for a distinctive quality of light, called La Clara. It was near the end of winter when I was there, and the sky was arctic blue. I was reminded of Toronto's lake-reflected light, which has the same clarity — and yet it was different. Certainly the Mediterranean, although out of view, was collaborating.

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