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Using a newly assembled data set of US patents, we show that innovation activity is less concentrated in high-density locations than commonly believed. Yet, inventions based on atypical combinations of knowledge are indeed more prevalent in high-density urban centres. To interpret this relation, we propose that informal interactions in densely populated areas help knowledge flows between distant fields, but are less relevant for flows between close fields. We build a model of innovation in a spatial economy that endogenously generates the pattern observed in the data: specialised clusters emerge in low-density areas, whereas high-density locations diversify and produce unconventional ideas.