  Economic activities favor mutual geographical proximity and concentrate
spatially to form cities. In a world of diminishing transport costs, however,
the advantage of physical proximity is fading, and the role of cities in the
economy may be declining. To provide insights into the long-run evolution of
cities, we analyzed Japan's census data over the 1970--2015 period. We found
that fewer and larger cities thrived at the national scale, suggesting an
eventual mono-centric economy with a single megacity; simultaneously, each
larger city flattened out at the local scale, suggesting an eventual extinction
of cities. We interpret this multi-scale phenomenon as an instance of pattern
formation by self-organization, which is widely studied in mathematics and
biology. However, cities' dynamics are distinct from mathematical or biological
mechanisms because they are governed by economic interactions mediated by
transport costs between locations. Our results call for the synthesis of
knowledge in mathematics, biology, and economics to open the door for a general
pattern formation theory that is applicable to socioeconomic phenomena.
