Mr. Trump marked the G.M. building at his preferred height with a pencil. Mr. Brosterman sawed off the top third, leaving Trump Tower — in the one-thirty-second-scale model, at least — the tallest in the neighborhood.
“Trump said, ‘Great,’ and left,” Mr. Brosterman recalled recently.
In real life, Trump Tower could not be elongated. But Trump Arithmetic found a way.
Though the tower was built with 58 floors, Mr. Trump later explained to The New York Times that because there was a soaring pink marble atrium and 19 commercial floors at the bottom, he could see no good reason not to list the first residential floor as the 30th floor. The pinnacle became the 68th — the height that appears in marketing materials, online search results and news articles to this day.
The Republican who would be president has been accused of exaggerating his own height by an inch. To suggestions that he has abnormally short fingers, he has responded by boasting, during a presidential debate, about other parts of his anatomy.
He has tried and failed to develop the world’s tallest building at least three times. And when he wants his buildings to seem bigger than they actually are, he enlarges them the same way he has thwarted long odds and defied reality throughout the election season: with sheer bluster.
As a review of his Manhattan building portfolio shows, Mr. Trump repeated his Trump Tower innovation at least seven more times.
The idea quickly caught on with other New York developers looking to sell condominiums of ever loftier height, status and, most important, price. By 1985, the year after Trump Tower opened, the developer Harry B. Macklowe had employed the same stratagem to turn his 67-story Metropolitan Tower into a 78-story skyscraper.
Mr. Macklowe’s team credited Mr. Trump for the idea. So did Mr. Trump.
“A lot of people have copied me,” he told The Times in 2003.
In this case, imitation is shrewd salesmanship. One57, the billionaires’ aerie on 57th Street that laid temporary claim to being the tallest residential tower in New York when it was completed in 2014, was said to top out at the $100 million 90th-floor penthouse. Actual floor count: 75.
“The higher your building, the better it is for your marketing purposes,” said Amir Korangy, the publisher of The Real Deal, a real estate publication based in New York. “Nobody’s trying to have the shortest building in the city, so any sort of edge you can get to add a floor here and there, you take it.”
The city’s Buildings Department does not object, so long as the floors are counted accurately in the building’s certificate of occupancy.
Hence the Trump SoHo on Spring Street, completed in 2010, where, according to the condominium offering plan filed with the state attorney general’s office, Mr. Trump skipped the 13th floor for superstition’s sake and a few more for marketing’s sake. There are 43 floors, but the elevators go up to 46.
Or take the Trump International Hotel and Tower, the hotel and residential building on Columbus Circle that was, pre-Trump, the 44-story Gulf & Western office building. Mr. Trump improved the structure so thoroughly that it managed to stretch into a 52-story tower, even though it stayed, strictly speaking, the same height.
Because new apartment buildings usually have lower ceilings than office buildings, Mr. Trump explained in 1994, the 583-foot building was about as tall as a conventional 60-story residential building.
Seen this way, measuring the converted tower at 52 floors was an act of altitudinal restraint.
“Depending on how you view it,” he said, “you could look at it as 60 stories.” (A spokeswoman for the Trump Organization did not respond to several requests for comment.)
Then there is the Trump World Tower on the East Side, built in 2001, which enraged antagonists as varied as Walter Cronkite (whose views it blocked) and the United Nations (whose height it dwarfed). At 90 stories and 900 feet — actually 70 and 843, according to Buildings Department records — the World Tower was once billed as the “tallest residential tower in the world,” until it was overtaken by a skyscraper in Dubai, prompting Mr. Trump to switch to a less easily fact-checked superlative: “most luxurious.”
He arrived at 90 floors by dividing the tower’s purported height of 900 feet by 10 feet, the equivalent of what he said was a standard residential floor. Inconveniently, the apartments were also advertised as having higher-than-usual ceilings.
“The two forms of hype kind of contradicted each other,” said Karrie Jacobs, an architecture writer who covered the building’s development.
Again, Mr. Trump said he had chosen the more modest math. “I could have gone higher than 90 stories,” he told The Times in 2003. “I chose 90 because I thought it was a good number.”
In time, creative numbering became almost as much of a building signature as the Trump name. Other tactics included counting underground parking lots and other below-grade floors in the total, Mr. Korangy said.
The concept originated with Mr. Trump, said Costas Kondylis, the architect who designed many of Mr. Trump’s buildings in New York.
“It was considered a novel idea,” he said.
But Mr. Korangy said research for a documentary he produced about Mr. Kondylis suggested that Mr. Trump had gotten the idea from Philip Johnson, the modernist architect and a frequent collaborator.
At some Trump buildings, it can be hard to detect any mathematical principle at work in the floor count.
Mr. Trump converted the former 31-story Hotel Delmonico into the Trump Park Avenue in 2004. Trump International Realty’s website measures the building at 32 stories in one place, but describes it as 35 stories in another.
And at Trump Place, the series of towers marching down Riverside Boulevard on the West Side, Mr. Trump sprinkled on a few extra floors here and there, modestly leaving some buildings at the height they were born with, but stretching the 31-floor 240 Riverside Boulevard into a comparatively extravagant 41 stories.
Few buyers were hoodwinked. At the Trump International, for instance, buyers had to sign a document explaining the difference between the marketing floors and the actual floors. Still, Mr. Trump has said, people are not exactly opposed to saying they live on, say, the 50th floor, even if it is really the 43rd.
“People are very happy,” he said. “They like to have apartments that have height, the psychology of it.”
Developers outside New York have used elastic math, too, but it is not clear if they were inspired by Mr. Trump.
As a practitioner of the art, one developer in Hong Kong, where the number eight is considered lucky, managed to eclipse even Mr. Trump: He rebranded the top floor of his 46-story building as the 88th floor.
For Mr. Trump, the quest to own New York’s tallest and most glamorous structures eventually led him back to the G.M. building, which he and a partner bought in 1998 for $878 million. (He later sold his stake.)
“His mission was to add the Trump touch to the white-marble-plated skyscraper,” according to the Trump Organization’s account of his ownership.
This meant splashing his name across the building in four-foot golden letters, then ornamenting its plaza, where the Apple store on Fifth Avenue now sits, with two “breathtaking” fountains and a “gorgeous” marble sitting area.
This time, however, the Trump touch did not involve the loss of the building’s top third.
The Trump Organization says the G.M. building is 50 stories high. For the record, the city agrees.
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