Here we provide the solutions found by our method and presented in our paper. The descriptions of the solutions for the three-dimensional problems are available in the following format.

s
m
a1 b1 c1 x1 y1 z1 t1 u1 v1
a2 b2 c2 x2 y2 z2 t2 u2 v2
...
am bm cm xm ym zm tm um vm

The first line has the length s of the side of the container. The container is a cube centered at the origin. The second line contains the number m of ellipsoids packed. Then, each one of the next m lines contains the lengths of the semi-axes (a, b, c), the center (x, y, z), and the rotations angles (t, u, v) of an ellipsoid. We have adopted a left-handed coordinate system and the ellipsoids were rotated through an angle t about the x-axis, through an angle u about the y-axis, and through an angle v about the z-axis, in this order.

Packing identical ellipsoids

In this set of experiments, we considered the problem of packing ellipsoids with semi-axis lengths (1, 0.75, 0.5) within a cube with side length 30. We considered the strategy of packing $n$ ellipsoids at a time, for $n$ $∈ {1,2,3,4,5}$ . We minimized the sum of the middle heights (i.e., the sum of the z-coordinate of the centers) of the ellipsoids. The hyperrectangle side length factor $η$ and the hyperplane height factor $γ$ parameters vary from 4 to 10. Select the number of ellipsoids packed at a time $n$ $=$ and the values of the parameters $η$ $=$ and $γ$ $=$ to see the corresponding solution.

Number of ellipsoids packed:
Solution description

Minimization of other heights

Considering the packing of ellipsoids with semi-axis lengths (1, 0.75, 0.5) within a cube with side length 30, we also minimized other three different types of height: lower, upper, and random. The ellipsoids were packed one at a time. Select the type of height , and the values of the parameters $η$ $=$ and $γ$ $=$ to see the corresponding solution.

Number of ellipsoids packed:
Solution description

Packing ellipsoids with random semi-axis lengths

In this experiment, we considered the problem of packing ellipsoids with uniformly random semi-axis lengths on the interval [0.1, 1] within a cube with side length 30. We used the hyperrectangle side length factor $η = 10$ and hyperplane height factor $γ = 6$ . The solution found has 23860 ellipsoids.

Packing more than a million ellipsoids

We also considered the problem of packing ellipsoids with semi-axis lengths (1, 0.75, 0.5) within a cube with side length 140. We used the hyperrectangle side length factor $η = 10$ and hyperplane height factor $γ = 4$ . The solution found has 1126474 ellipsoids.