Urbino is a thick exhibition of hills shared by walkers, Italian-talkers and residential Fiats. Surrounding the two grand piazzas is a hive of narrow cobblestone paths steeped in the shade of ancient buildings. This is my fourth four-week residence, and still I discover new nooks in this city no bigger than Lexington, Va. (inside Urbino’s high, ancient walls at least).

The layers of Time did crazy things to the design, as did the slopes of hills upon a hill.

But somewhere deep in its soul, Urbino had an idea, a pitch-perfect ideal based on mathematics and a hallucination of perfect civic design. There is a painting of this, a long, narrow work called “Città Ideale,” the ideal city. The painting (there are other versions in Berlin and Baltimore) is mounted in the museum of the duke’s palace. No one is sure who painted it. But we were told, in the Palazzo Ducale tour the class got last Thursday, that art critics are sure it’s an important depiction of the Renaissance idea of utopia, the perfect city.

But these were Humanists. Where are the people? Ah, we’re told, this is not Man but the Work of Man. But, wait. They were great observers of Nature. What about nature – why not a single geranium or shrub in all this marble? Nature is represented, they say, in that little plant puffed from a window ledge, and two hints of mountains in the background.

But what about politics, the new realism of power that made the cunning Caesare Borga (after he conquered Urbino for his father Pope Alexander VI) the model of Machiavelli’s Prince? Well, there’s no constitutional “balance of power.” But the painting is. . .in “dialogue.”

We’re supposed to talk back to it, the Enlightenment vs. the Renaissance. Now vs. then.

I don’t know. The older sights of Urbino leave me sort of speechless (especially in my Italian). The iPhone panorama I took inside the Palazzo courtyard looks like “Città Ideale,” without people, nature or the politics of Mitch McConnell. There’s no dialog with the present.

But there are clues to the past. Another painting in the museum is an obscure work of Pietro della Francesca that is like the “Città Ideale” in look, shape, and. . . obscurity.  It’s called “The Flagellation of Christ.” (Both might be by the same painter.) In an oddly serene moment in a Renaissance setting, an un-bloodied, un-bent Christ is being scourged in the background while three figures in the foreground could be discussing silk prices on the Rialto. Our guide says there are something like 46 official interpretations of what the painting means (and another 63 unofficial, he jokes).

I am told that a loggia near our classroom was the basis for the setting where Christ is being whipped. And I wonder if a column nearby is the model for the column Christ is tied to. The one in Urbino is topped by St. George slaying his dragon; in the painting, it looks like a Roman emperor. One of the interpretations is that this is about the tension between Christian and pagan cultures. Another is that it is about Christendom threatened by the Islamic conquest of Byzantium in the East, which happened at the time of the painting. Another, that it is about the new geometry of perspective, with various implied circles intersecting at the crown of Christ’s head.

What I notice in the painting and in the ideal Città is that there are no ads. No cars, and no ads. That seems ideal enough to me right now, thinking of America from over here. Italy seems closer to these paintings than to any five minutes off I-75 outside Atlanta or a shopping mall in McLean, Va.