Infinitesimal affine geometry of metric spaces endowed with a dilatation structure

We study algebraic and geometric properties of metric spaces endowed with ndilatation structures, which are emergent during the passage through smaller nand smaller scales. In the limit we obtain a generalization of metric affine ngeometry, endowed with a noncommutative vector addition operation and with a nmodified version of ratio of three collinear points. This is the geometry of..