I have a 3D torus, say with radii $R$ (around the hole) and $r$ (thickness), where $R \gg r$. What is the densest packing, to fill a rectangular block with as many identical tori as possible?
Tori may link.
Origin of the problem: The torus is a design for bracelet, to be 3D printed. Actually, the torus is cut in one place to provide some flexibility. That will also allow them to be unlinked after printing. In this case, for the ratio of the radii we have $R : r \approx 25$.
Stacking them in columns, and then placing columns in a hexagonal grid is one option. But this leaves lots of space inside the tori unused.
I also looked at two other ideas (see images), based on linking.
The latter allows more to be packed next to each other, but seems vertically less efficient.
Any other ideas? Is any literature known on this problem?