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Claims about the universality of human concepts have been predominantly assessed through linguistic similarity across languages and cultures. However, words are effective as communication devices because they compress rich experiential variation into shared conventions, potentially obscuring hidden individual and cultural differences in how concepts are mentally represented. Here, we analyse 2.6 billion human-made sketches of common concepts from 236 countries and territories to examine conceptual structure through people's visual imagination. Consistent with recent work on image-based cognition, we find that single concepts unfold into multiple distinct visual exemplars, revealing latent information about similarities and differences in conceptual structure across cultures. This variation is strongest for concepts involving haptic interaction, suggesting that visual imagery reflects variation in embodied experience as much as conventional definitions. Comparing embedding models of sketches with word embedding models across languages, we find that their geometries diverge, with visual representations preserving rich semantic and cultural structure that language models compress. Cross-cultural similarities derived from sketches align 45% more closely with established cultural distances than do text-based measures. Together, these results suggest that patterns of human conceptual universality may depend critically on the modality through which concepts are measured, with large-scale sketching providing a direct, high-resolution probe of conceptual diversity across embodied and cultural dimensions of thought.