We introduce a new type of growth diagram, arising from the geometry of the affine Grassmannian for GL_m. These affine growth diagrams are in bijection with the components of the polygon space for a sequence of minuscule weights. Unlike Fomin growth diagrams, they are infinite periodic on a staircase shape, and each vertex is labeled by a dominant weight of GL_m. Letting m go to infinity, a dominant weight can be viewed as a pair of partitions, and we recover the RSK correspondence and Fomin growth diagrams within affine growth diagrams. The main combinatorial tool used in the proofs is the n-hive of Knutson--Tao--Woodward. The local growth rule satisfied by the diagrams previously appeared in van Leeuwen's work on Littelmann paths, so our results can be viewed as a geometric interpretation of this combinatorial rule. Speaker(s): Tair Akhmejanov (Cornell)