“Unbuilt Washington” is the National Building Museum’s best chance at drawing blockbuster crowds in years. Devoted to the might-have-beens in the wastebasket of Washington’s design history, it begins with variations on the basic city plan laid out by Pierre L’Enfant in 1791 and ends with a spectacular sculptural bridge the museum would like to construct in the enormous atrium of the Pension Building.

But it is far more than a catalogue of lost opportunities and terrifying near misses. It is also a compendium of Washington’s architectural phobias and obsessions, its neurotic compulsion to grandeur and countervailing fear of anything too elegant, too bold or too French. It is, in the best sense, profoundly disorienting, an exhibition that makes you laugh at the absurdity of a pyramid-shaped Lincoln Memorial or a new White House built on the scale of Versailles. You laugh, and then you wonder why you laugh. And in most cases, the chain of questions leads back to fundamental and often arbitrary assumptions about what the architecture of democracy should look like.

Washington residents will view many of these unrealized plans with profound and proprietary relief.

Residents of Capitol Hill will be glad that long-standing proposals for an East Mall, balancing the National Mall, running through the historic residential neighborhood all the way to the Anacostia River, were never carried out. The Capitol Hill Expressway, a plan for two elevated highways along the Mall, must have seemed like an obvious solution to an obvious traffic problem when it was proposed in 1946, but the effect would have been to turn core of Washington’s downtown into a dispiriting dead zone.

In both cases, better ideas about urbanism — and local resistance — won out in the end. But how to account for a seemingly risible proposal for an ornate, faux-medieval Memorial Bridge, with turrets and towers, instead of the relatively modest, low-arched Memorial Bridge we have now? Why does an 1875 design for a Victorian Gothic Library of Congress seem so strange?

We might call this the pyramid question: The pyramid has architectural pedigree even more ancient than the Greek temple, yet no major pyramid has been built in Washington. Why? Not because it’s never been proposed. Benjamin Latrobe designed a monument with a pyramid-shaped top and simple columns as a mausoleum for George Washington in 1799-1800, but it was never built. Nor was the 1837 colossal pyramid-shaped Washington monument proposed by Peter Force, who seemed to be channeling the terrifying utopian grandeur of the visionary French architect Etienne-Louis Boullee.

Nor were any of the various pyramid schemes, or the round ziggurat, proposed by John Russell Pope (architect of the Jefferson Memorial) as a monument to Lincoln in 1911-12. In each case, the form seems desperately, even comically out of place, and yet as curator Martin Moeller argues, a pyramid is no more or less arbitrary as a Lincoln memorial than the rectangular classical temple (designed by Henry Bacon) that was ultimately finished in 1922.

As the exhibition makes abundantly clear, the idea that Washington could only be a rigidly neo-classical city of white marble and columns is relatively recent. Buildings such as the Smithsonian Castle remain as echoes of a rich argument about style that only ossified into dogmatic classicism in the last century. The city we ended up with seems inevitable, the only possible Washington that can and should exist, but it was achieved by fiat, with grand disregard for the multiplicity of architectural styles and the diversity of cultural inheritances upon which this country is built. The Mall, which was still very much a work in progress in the 1930s, is a classic example of architectural groupthink, an aggressive manifestation of the fashionable City Beautiful movement that stressed the positive value of harmony and order, dramatic vistas and generic architecture in the classical and beaux arts idioms.

But it came at enormous cost. Parkland on the Mall was denuded of trees and paths, and landscape was give rigid geometric order. Government power was expressed in dramatically framed views and an orderly march of white palaces along the north and south sides of the lawn. Some of the prize pieces in the “Unbuilt Washington” exhibition remind us what could have been if the Mall had been treated as an urban amenity rather than a national symbol.

Designs by Robert Mills (the first architect of the Washington Monument) and Andrew Jackson Downing remind us that the Mall was once devoted to a diversity of uses, including pleasure gardens. Downing’s serpentine paths, evergreen grove and small lake look a lot more inviting than the shadeless greensward we have today. Benjamin Latrobe’s plans for a university campus (reminiscent of Jefferson’s University of Virginia design) where the Washington Monument now stands are also intriguing.

It’s hard to imagine, but a worthy thought exercise: What would America be like if this land hosted a functioning academic institution, questioning all aspects of American life, rather than a monument that serves only to express one idea, the greatness of the Founding Father?

Several of the unbuilt designs are intended as critiques of Washington rather than serious proposals for new construction. In 1984, the Museum of Modern Art commissioned Leon Krier, an architect and planner born in Luxembourg, to rethink the basic design of the city’s central core. He proposed reflooding the west end of the Mall, creating a Venice-like waterway, with the Washington Monument jutting into the lagoon on a large stone plinth. That part of his notorious design seems like pure flippancy. But his plans for filling in the neighborhoods near the Mall, now devoted to large federal buildings that empty out at night, with mixed-use buildings and housing make a lot of sense.

The 1995 idea for a “National Sofa” on the White House grounds, proposed by Jim Allegro and Doug Michels, is also more conceptual art project than genuine architectural plan. In response to the closing of Pennsylvania Avenue — the first of many hysterical security measures since the Oklahoma City bombing and the attacks of Sept. 11, 2001 — Allegro and Michel envisioned a wide, arching “sofa” in front of the White House, with a giant video screen giving visitors an insider’s peek into the executive mansion. A nation of couch potatoes would have its own national symbol. But the interposition of a screen between visitors and the real White House was also a striking and prescient commentary on how cable news was degrading American political life, converting politics to entertainment.

Among the most striking objects on display is a magazine illustration, from 1905, titled “How Uncle Sam Is Spending Hundreds of Millions of Dollars to Make Washington the Most Beautiful Capital in the World.” The first part of that long-winded caption sounds like the beginning of a typical anti-government rant. The second part reveals how far we’ve come since the glory days of American self-confidence and ambition. Once upon a time, spending millions of dollars on making the nation’s capital beautiful was a good thing.

Disillusionment with government and with architecture has changed all that. Washington is now a city of small dreams, which makes it rather sad to revisit some of the better ideas that will probably never be realized. The exhibition includes Frank Gehry’s model for an addition to the Corcoran, and seeing it up close makes it clear that it could have been spectacular. An early design for a curvaceous Kennedy Center, including a central lobby space that would have been much more congenial and social than what got built, leaves one wishing we hadn’t been left with such an isolated and ugly arts center. The very real possibility of removing the Whitehurst Freeway in the 1980s was a tragic missed opportunity. And the Washington Channel Bridge, a 1966 plan to link the Southwest waterfront to East Potomac Park with a bridge of connected restaurant and shopping pavilions, might have saved that neighborhood from decades of decline only now being reversed.

It’s tempting to argue that the positive side of the city’s innate architectural conservatism is seen in some of the more wretched “might-have-beens,” but usually it’s all about money. In the 1920s, the Masons, who owned the land where the Washington Hilton now sits, wanted to build an enormous temple on the site, and they had the connections necessary to get a variance from the District’s height ordinance. Harvey Wiley Corbett’s designs would have made Albert Speer blush at the ego and overscale grandeur of the project. Several projects like this, which would have radically altered our sense of Washington, were thwarted only by economic downturns or wars that shifted attention away from building projects. The only hope that we won’t be saddled with the badly designed and ill-considered Vietnam Veterans Memorial visitors center, which will eat up public land near Maya Lin’s entirely self-sufficient memorial, now rests on the vagaries of fundraising.

The exhibition features two projects that shouldn’t be left to languish at the conceptual stage. A model of Morphosis Architects’ 2011 plan for reuse of the Smithsonian’s Arts and Industries Building — with deliciously amorphous internal bubbles of space that could be used as auditoriums — should be greenlighted. It’s far more innovative and exciting than any of the half-baked and halfhearted efforts to put a traditional museum or visitor’s center in the now empty Victorian brick building. And architect Jim Eyre’s proposal for a “tensegrity bridge” in the enormous open atrium of the National Building Museum would be a delightful folly. The web-like bridge seems to stay up without any rational support for its structure, and though it is conceived as a temporary installation, it might spark meaningful debate about one of this city’s strangest architectural tics: We are a city of rivers that is determined to build ugly bridges (the one exception is Memorial Bridge).

We are also a city entering in a new age of symbolism. The American century has passed and the United States, riven by political dysfunction, debilitated by cultural infantilism and saddled with unsustainable debt, is passing into an age of decline. Paradoxically, what’s bad for the country might actually be good for the city, allowing Washingtonians to reclaim their town from an outdated design ethos devoted to imperial grandeur, and rebuild it in more livable, sustainable and welcoming ways.

As the dissonance grows between the decrepit reality of our political life and the remains of our City Beautiful idealism, the District might seize the opportunity to think its way into a more pleasant senescence. The Mall doesn’t really make sense anymore, as a symbol or an urban feature. Rebuild the old parks and pleasure grounds, plant trees and retire the memorials that have become meaningless. Reclaim the waterfront, including the Navy Yard and Fort McNair (which would make an elegant academic campus). Eliminate parking on the Capitolgrounds and near the Reflecting Pool, no matter how loud the VIPs squawk. Rather than representation in Congress, pursue tax-free status for the District, and use the local wealth boom to focus on housing, culture and night life.

Given jurisdiction battles and the city’s second-class status relative to the federal government and the states, none of this is unlikely to happen any time soon. But as historians have long argued, Washington is the speculative city par excellence. The material on display in this fascinating exhibition suggests that there is something fundamentally American at work in the restless spirit of redesign, refashioning and remaking of the urban landscape. As we look for new forms that express our diminished role in the world, a more modest sense of self, one can imagine a more arcadian city, and new and exciting idylls among the ruins of passing splendor.

“Unbuilt Washington”

runs through May 28 at the National Building Museum. For information on admission charges and opening hours visit www.nbm.org.