Symplectic geometry is the geometry underlying Hamiltonian dynamics, and symplectic mappings arise as time-1-maps of Hamiltonian flows. In the last two decades, spectacular rigidity phenomena for symplectic mappings were discovered, that show that symplectic mappings are much more special than volume preserving mappings. At the same time, various methods were found to construct certain symplectic mappings. Despite this progress, it is still quite mysterious what a symplectic mapping is. Recently, in joint work with Dusa McDuff, we have calculated the function c(r) computing the radius of the smallest 4-dimensional ball into which the 4-dimensional ellipsoid E(1,r) symplectically embeds. The graph of this function is surprisingly rich, and is given in part in terms of Fibonacci numbers. Along the proof, many other number theoretic quantities, relations and identities appear in a rather mysterious way. We have also shown that for this problem, the invariant obtained by the Embedded Contact Homology of Hutchings and Taubes is a complete invariant. The goal of this project is to compute the corresponding function for similar symplectic embedding problems, such as embedding ellipsoids into ellipsoids and polydiscs. In this way, we hope to better understand these number theoretic aspects of the answer, and to see how strong the invariant from Embedded Contact Homology is in other problems.