My son uses a lot of LEGOs that we have to clean up every night before he goes to bed. The box we use for them is a little on the small side, so we often find that the LEGOs are stacked too high for the lid to close.
He suggested that he stick the LEGOs together into long "towers" before he puts them away to make them fit better, and I said that was a good idea but now I'm not really sure. On the one hand, stacking bricks together makes them have zero space in between them, eliminating air gaps in the container. However, while the "chunks" themselves have no wasted space inside them, it seems like they don't pack as tightly against each other when thrown into the box (of course, if carefully sized for the exact dimensions of the box and placed there in an organized way, that would be optimal, but we don't have time for that). What I'm wondering is whether stacking the bricks into towers or cubes actually might make them pack worse.
To simplify this real-world question for the theoretical domain, let's assume:
- Only one type of Lego brick -- let's say the classic 4x2 brick
- An infinitely large box, to eliminate edge-of-box effects
- A packing based on dropping the pieces in at random orientations from above the box
Then the question is: What packs the tightest?:
- Individual 4x2 bricks
- Towers of 4x2 bricks, of height N
- Rectangular solids of bricks, of dimension XYZ
If we restrict to either single or tall stacked 4x2 bricks then individual bricks pack denser. The best result we get for stacks of two bricks as they have the most spherical shape. You can find the answer in an investigation using long cylinders here. In figures 5 and 11 you see the packing density for cylinders with different aspect ratios. The density in infinitely tall vessels you find at $L/R+L/H=0$. The density is the ratio of occupied vs total space. These results are approximately also true for Lego bricks.