Circle packing is a classical topic in discrete geometry. As Axel Thue and László Fejes Tóth showed, if you want to fit as many identical circular coins on a table as possible (all sitting side by side, no piling up or overlapping), the best you can achieve is for around 90.7% of the table to be covered. This is done by arranging the coins along a hexagonal lattice.
That’s fine, but we can pose the same problem, using coins which are not circular. Now here is an interesting question: which shape is the worst packer?
Then the answer is conjectured to be the smoothed octagon, with a maximum packing density of around 90.2%.
The smoothing is done by rounding off each corner with a hyperpola which is tangent to the two meeting sides, and which asymptotically approaches the two sides beyond those.
[Image from Wikipedia]