More and more, the National Mall is living up to its moniker “America’s front yard”: patchy turf, puddles, and cracked sidewalks give it an air of foreclosure. The National Mall Design Competition, now under way, will surely produce ambitious proposals to mend the Mall, but getting them approved and funded could take years and is far from guaranteed.

Now on display through May at the National Building Museum, “Unbuilt Washington,” reminds us that the Washington Monument was a half-finished stump for decades, until money could be found to complete it. And even then it was not done according to the original design. And that Eliel and Eero Saarinen’s daring 1939 scheme for a Smithsonian art gallery—selected out of 400 entries—fell victim to politics, like so much else in our capital.

The might-have-been monuments and cityscapes on display are beguiling, often strange, and surprisingly varied (for a city that seems married to neoclassicism). If history had tracked just a degree or two from its eventual course, our postcards of the Lincoln Memorial would depict a gleaming ziggurat; Dupont Circle would be known for a huge tower complex by Frank Lloyd Wright; and the White House would sport two additional southern wings flanking a large conservatory (this last proposal was championed by First Lady Caroline Harrison in the 1890s).

Of all the lost opportunities included in the show, the one that curator Martin Moeller most wishes had been built is the Washington Channel Bridge, designed by Chloethiel Woodard Smith in 1966. Linking Southwest D.C. to the East Potomac Park spur of the Mall, this modernist answer to the Ponte Vecchio would have been lined with shops and restaurants that beckoned strolling pedestrians. Washington, finally, would have turned toward and not away from the water all around it.

Still, lucky escapes probably outnumber missed chances. Leon Beaver’s Second-Empire-on-steroids competition entry for the Library of Congress, and an amateur’s entry for the Capitol featuring an oversized, crudely drawn eagle are proof that the competition process does sort the wheat from the obvious chaff. And that, at least, should cheer National Mall Design Competition finalists and jurors.