Last year, physicists Jaoshvili, Esakia, Porrati and Chaikin published a paper in Physical Review Letters confirming that the D4-shaped tetrahedral die pack more densely than any other shape (yet tested).
“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.”
Applications of the research include the flow of granular materials like sand and gravel, the storage of information in digital memory, and determining optimal shapes for packaging consumer products.