Writing in the May 3 issue of Physical Review Letters, a group of New York University and Virginia Tech physicists confirmed that there is no polyhedron that packs more densely than a D4-shaped tetrahedral die:

The revelation is the result of a series of experiments that involved pouring tetrahedral dice into containers, shaking them, and adding more dice until the containers were completely filled. After adding water to measure the open space between the dice, the researchers confirmed that the tetrahedrons fill roughly 76% of the available space in a large container. Similar experiments with spheres typically only fill containers to about 64% of the total volume.

Densest Dice Packing: Tetrahedral Dice Pack Tighter Than Any Other Shape

(Image: 4-sided dice, a Creative Commons Attribution (2.0) image from dicemanic’s photostream)