I am not sure if this counts, because maybe it is physics more than mathematics, but the density of packing tetrahedra in space was experimentally investigated by Jaoshvili, Esakia, Porrati, and Chaikin, Experiments on the random packing of tetrahedral dice, Phys. Rev. Lett. 104 (2010), 185501. I believe that at the time they did their experiments, the empirically observed packing density was slightly higher than what mathematicians had been able to construct explicitly.