The District is dotted with holdouts, properties that look like they no longer belong because the city has grown up around them, without them.

There's the little shop on Massachusetts Avenue east of the convention center: a former 1890 rowhouse, now a Le Pain Quotidien, dwarfed on both sides by modern high-rises (or what counts for such in D.C.).

Two blocks away, there's a shuttered AV repair shop on K Street, its defunct signage in Chinese characters. On both sides, a dozen stories of new high-end apartments loom over the little twinned homes, giving the block a glassy skyline that looks like it's missing a tooth.

Across town, there's the odd little gas station at the tip of the Watergate complex, occupying a sliver of the mega-block that was surely coveted for other purposes when the Watergate was built in the 1960s.

These properties — holdouts to the development around them — juxtapose Washington's past with what it's becoming today. Each has its own backstory. And all attest to the challenge of a city struggling to remake itself in pursuit of denser, taller, more modern.

Austin Spriggs, the man who owned the Massachusetts Avenue home, famously rebuffed developers who offered him nearly $3 million to buy his mid-block parcel to make way for the large new developments rising in the area during the 2000s construction boom. The redevelopment eventually happened anyway, around him. Then that little rowhouse sold in 2011 in foreclosure, worth a fraction of what it had been when the possibilities for the land underneath were so much larger than those 1,796 square feet.

"It has to be a person of great perseverance," says Leah Brooks, a George Washington University professor who shares my fascination with these properties and the people who refused to sell them, or were determined to save them. "For the gas station guy, I think if he’d gotten a share of the Watergate building instead, he’d be better off."

There is something romantic about the idea of a holdout, a David to the big developer's Goliath, a protagonist for whom home matters more than money, a solitary survivor. In the Pixar movie "Up," the holdout is the hero. In the real-life Seattle version of the story that reportedly inspired the film's premise, an elderly woman who refused to sell her home became — along with her home itself — a city icon. "I don't want to move," Edith Macefield told the Seattle P-I in 2007 before her bungalow was enveloped by new development. "I don't need the money. Money doesn't mean anything."

In practice, though, modern cities grow out of older ones in large part through the unglamorous process of parcel assembly — of fitting together the once-smaller pieces of the city, "Tetris"-like. And while the result often produces fantastically bizarre neighbors, cities can't evolve if every property owner is such a hero.

"When you walk around downtown D.C., downtown D.C. used to be entirely rowhouses," Brooks says. "So somebody had to assemble them."

The area just east of Mount Vernon Square — where the Spriggs home and AV repair shop are located — looked like this in the 1892 plat book of the city:

If many of those properties weren't combined, it would be hard for Washington — or any other city of its vintage — to accommodate over time its growing population, or its changing economy, or the ways we live now (with underground parking garages and apartments made possible thanks to elevators).

Just off that map, on the north side of K Street at Sixth Street NW, sits the modern Association of American Medical Colleges building, a block-size office cobbled together from many smaller lots — and a few restored historic properties. Here, the project's architect, Shalom Baranes, describes how the site came together and why it looks today as if it emerged from two different eras a century apart:

"I feel like they add a fourth dimension — of time," Baranes says of the older buildings incorporated into the newer one. "You walk down the street and you realize this wasn’t all created yesterday. It was created across a series of decades."

An economist like Brooks looks at holdouts and sees impediments to the necessary process of urban evolution.

In Los Angeles, she and Byron Lutz, a researcher with the Federal Reserve Board of Governors, looked at more than 2 million parcels of land between 1999 and 2011. They calculated that parcels meant for assembly sell at a 15 to 40 percent premium relative to land not destined for larger projects. The scale of that premium startled them, and it implies that holdouts are driving up prices or preventing projects to a degree that Brooks and Lutz argue hinders the city's reinvention.

"I feel like a real economist grouch in some ways with this paper," Brooks jokes, acknowledging how the lonely holdout can become a folk hero. But an economist argues that new development — particularly if it boosts the housing supply — creates the greater good. "Economists are like, 'Take the money, everyone else is better off!' I think it’s really easy to empathize with the person who owns their house right now. It’s hard to empathize with the 60 people who could be living in apartments where that house is, at a slightly lower price [than would be possible without new development]. You don’t see them. They’re not out picketing."

To buy this argument, you have to believe, as economists do, that increasing housing supply lowers housing prices. And that is a hard sell when a woman is nudged to leave her home so that a new apartment can accommodate 60 well-paid professionals. If that development happens without her, though, the woman gets none of the windfall from the rising value of the land underneath her — and all of those gains go instead to a developer. That's not an equitable outcome either, Brooks argues.

When holdouts sell, yes, developers win, but so do cities at large, Brooks argues. They're able to add housing, or to boost the agglomeration benefits that kick in when businesses cluster tightly together. When new construction is blocked in existing neighborhoods, it may also migrate instead to the edge of town, consuming undeveloped land there and harming the environment.

"In the biggest, biggest, biggest picture," Brooks says of holdouts, "we think it inhibits economic growth."

It is a "tragedy of the anti-commons," she and Lutz argue, in which fragmented ownership gets in the way of the greatest social good. The alternatives, though, are unappealing, too. Eminent domain — forcibly taking property from people — isn't a fair answer. Nor would people want to give the community at large a vote on whether to force one member to sell their property.

"In some ways, you can see why the problem is so bad," Brooks says, "because the solutions are so bad, too."

It's hard for economists to identify and count the developments that were never built because a lone property owner or two held out. But new development, for its part, often does find a way, hopscotching old buildings, hovering over them or hugging them close. And the resulting effect, for alert passersby, can be entrancing.

"That," Baranes says, "is what makes living in a city, I think, interesting for all of us."

Daron Taylor contributed to this report.