## Crates of Bottles

February 5, 2013

The other day I was looking at a crate of 12 water bottles, and looked at the wasted space between them. Why not use a honeycomb? or even other bottle shapes?

Well, let us have a mathematical look at bottle packing. A round bottle occupies a surface (at the bottom of the box) of $S=\pi r^2$ (for some radius $r$) in a square of $4r^2$. The occupancy is therefore