  Recently, McDuff and Schlenk determined the function c_{EB}(a) whose value at
a is the infimum of the size of a 4-ball into which the ellipsoid E(1,a)
symplectically embeds (here, a >= 1 is the ratio of the area of the large axis
to that of the smaller axis of the ellipsoid). In this paper we look at
embeddings into four-dimensional cubes instead, and determine the function
c_{EC}(a) whose value at a is the infimum of the size of a 4-cube C^{4}(A) =
D^{2}(A) times D^{2}(A) into which the ellipsoid E(1,a) symplectically embeds
(where D^{2}(A) denotes the disc in mathbb{R}^{2} of area A). As in the case of
embeddings into balls, the structure of the graph of c_{EC}(a) is very rich:
for a less than the square sigma^2 of the silver ratio sigma := 1+sqrt(2), the
function c_{EC}(a) turns out to be piecewise linear, with an infinite staircase
converging to (sigma^2, sqrt(sigma^2/2)). This staircase is determined by Pell
numbers. On the interval [sigma^2,7+1/32], the function c_{EC}(a) coincides
with the volume constraint sqrt(a/2) except on seven disjoint intervals, where
c is piecewise linear. Finally, for a >= 7+1/32, the functions c_{EC}(a) and
sqrt(a/2) are equal.
  For the proof, we first translate the embedding problem E(1,a)-->C^{4}(A) to
a certain ball packing problem of the ball B^{4}(2A). This embedding problem is
then solved by adapting the method from McDuff-Schlenk, which finds all
exceptional spheres in blow-ups of the complex projective plane that provide an
embedding obstruction.
  We also prove that the ellipsoid E(1,a) symplectically embeds into the cube
C^{4}(A) if and only if E(1,a) symplectically embeds into the elllipsoid
E(A,2A). Our embedding function c_{EC}(a) thus also describes the smallest
dilate of E(1,2) into which E(1,a) symplectically embeds.
