  This article introduces a new method to construct volume-filling symplectic
embeddings of 4-dimensional ellipsoids by employing polytope mutations in toric
and almost-toric varieties. The construction uniformly recovers the full
sequences for the Fibonacci Staircase of McDuff-Schlenk, the Pell Staircase of
Frenkel-Muller and the Cristofaro-Gardiner-Kleinman's Staircase, and adds new
infinite sequences of ellipsoid embeddings. In addition, we initiate the study
of symplectic tropical curves for almost-toric fibrations and emphasize the
connection to quiver combinatorics.
