The origins of the cortical connectome’s complex topology and region-specific topography remain unclear, motivating a search for underlying physical principles. A neural-field-theory model that prioritizes connections between regions that facilitate the excitation of the resonant geometric eigenmodes of the cortex and reproduces topological and topographical features of diffusion-MRI and tract-tracing connectomes across five mammalian species and multiple spatial scales, indicating a universal geometric basis for connectome architecture.