  The shape of buildings plays a critical role in the energy efficiency,
lifestyles, land use and infrastructure systems of cities. Thus, as most of the
world's cities continue to grow and develop, understanding the interplay
between the characteristics of urban environments and the built form of cities
is essential to achieve local and global sustainability goals. Here, we compile
and analyze the most extensive data set of building shapes to date, covering
more than 4.8 million individual buildings across several major cities in North
America. We show that average building height increases systematically with
city size and follows theoretical predictions derived from urban scaling
theory. We also study the allometric relationship between surface area and
volume of buildings in terms of characteristic shape parameters. This allows us
to demonstrate that the reported trend towards higher (and more voluminous)
buildings effectively decreases the average surface-to-volume ratio, suggesting
potentially significant energy savings with growing city size. At the same
time, however, the surface-to-volume ratio increases in the downtown cores of
large cities, due to shape effects and specifically to the proliferation of
tall, needlelike buildings. Thus, the issue of changes in building shapes with
city size and associated energy management problem is highly heterogeneous. It
requires a systematic approach that includes the factors that drive the form of
built environments, entangling physical, infrastructural and socioeconomic
aspects of cities.
