Urban sprawl reshapes cities, yet its quantitative laws remain elusive. Analyzing built-up expansion in 19 cities (1985–2015) with tools from surface growth physics in radial geometry, we reveal anisotropic, branchlike growth and a piecewise linear scaling between area and population. We uncover a robust local roughness exponent ${\alpha }_{\mathrm{loc}}\approx 0.54$, coexisting with variable $\beta$ and $z$. This unusual coexistence of universal and variable exponents offers a rare empirical test bed for nonequilibrium growth and an empirical basis for modeling urban sprawl.

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• Complex systems
• Patterns in complex systems
• Physics & society
• Roughness
• Surface growth