  A striking result of McDuff and Schlenk asserts that in determining when a
four-dimensional symplectic ellipsoid can be symplectically embedded into a
four-dimensional symplectic ball, the answer is governed by an "infinite
staircase" determined by the odd-index Fibonacci numbers and the Golden Mean.
Here we study embeddings of one four-dimensional symplectic ellipsoid into
another, and we show that if the target is rational, then the infinite
staircase phenomenon found by McDuff and Schlenk is quite rare. Specifically,
in the rational case, there is an infinite staircase in precisely three cases
-- when the target has "eccentricity" 1, 2, or 3/2; in all other cases the
answer is given by the classical volume obstruction except on finitely many
compact intervals on which it is linear. This verifies in the special case of
ellipsoids a conjecture by Holm, Mandini, Pires, and the author.
