dimensional analysis = projective algebraic geometry

it's no big surprise that dimensional analysis is related to projective algebraic geometry, although you could have spent some time thinking about either or both of these subjects without particularly noticing it. both concern the study of "homogeneous" quantities, meaning quantities that transform under rescaling in certain simple ways. what might be more surprising though is that clarifying this relationship is linked with a unified development of good category-theoretic foundations for algebraic geometry and physics and logic based on the concept of "theory of a doctrine" occurring in the work of beck and lawvere.