I will present a simple-minded view of interleaving: we want two diagrams to fit together inside a bigger diagram. I will show how this idea can be used to draw pictures that prove a number of persistence results. Surprisingly (to us!) carefully working out the theory underlying these ideas led us to discover that interleaving distance and Gromov-Hausdorff distance are both special cases of a richer theory. Joint work with Vin de Silva and Jonathan Scott.