Symplectic embeddings encode possible coordinate changes in phase space, the even-dimensional Euclidean space whose coordinates record the position and momentum of moving particles in a physical system. In 2012, McDuff and Schlenk analyzed symplectic embeddings of four-dimensional ellipsoids into 4-ball, and found the ellipsoid embedding function of the 4-ball contained a complex number-theoretic structure called an infinite staircase. Recently, Cristofaro-Gardiner--Holm--Mandini--Pires found a conjectural list of all rational convex toric domains in R^4 whose ellipsoid embedding function admits an infinite staircase. We will introduce recent work identifying six new infinite families of infinite staircases (which fall outside the scope of the conjecture of CGHMP), as well as extensive numerical evidence supporting the CGHMP conjecture in the case of one-point blowups of the complex projective plane.