In joint work with Bestvina and Fujiwara we showed that the mapping class group embeds equivariantly and quasi-isometrically into a product of $\delta$-hyperbolic spaces. Hammenstädt improved this result by replacing the $\delta$-hyperbolic spaces with quasi-trees. We will present a different proof of Hammenstädt's result (also joint with Bestvina and Fujiwara). A key tool will be a new version of the Masur-Minsky distance formula that only measures the the "thick" distance between points in a curve complex. We will explain this formula and how it leads to the proof of the embedding theorem.