  The world's urban population is expected to grow fifty percent by the year
2050 and exceed six billion. The major challenges confronting cities, such as
sustainability, safety, and equality, will depend on the infrastructure
developed to accommodate the increase. Urban planners have long debated the
consequences of vertical expansion---the concentration of residents by
constructing tall buildings---over horizontal expansion---the dispersal of
residents by extending urban boundaries. Yet relatively little work has
predicted the vertical expansion of cities and quantified the likelihood and
therefore urgency of these consequences.
  We regard tall buildings as random exceedances over a threshold and use
extreme value theory to forecast the skyscrapers that will dominate the urban
skyline in 2050 if present trends continue. We predict forty-one thousand
skyscrapers will surpass 150 meters and 40 floors, an increase of eight percent
a year, far outpacing the expected urban population growth of two percent a
year. The typical tall skyscraper will not be noticeably taller, and the
tallest will likely exceed one thousand meters but not one mile. If a mile-high
skyscraper is constructed, it will hold fewer occupants than many of the
mile-highs currently designed. We predict roughly three-quarters the number of
floors of the Mile-High Tower, two-thirds of Next Tokyo's Sky Mile Tower, and
half the floors of Frank Lloyd Wright's The Illinois---three prominent plans
for a mile-high skyscraper. However, the relationship between floor and height
will vary across cities.
