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We study algebraic and geometric properties of metric spaces endowed withdilatation structures, which are emergent during the passage through smallerand smaller scales. In the limit we obtain a generalization of metric affinegeometry, endowed with a noncommutative vector addition operation and with amodified version of ratio of three collinear points. This is the geometry ofnormed affine group spaces, a category which contains the ones of homogeneousgroups, Carnot groups or contractible groups. In this category group operationsare not fundamental, but derived objects, and the generalization of affinegeometry is not based on incidence relations.

Infinitesimal affine geometry of metric spaces endowed with a dilatation structure