Dionysian Hard Sphere Packings are Mechanically Stable at Vanishingly Low Densities

High strength-to-weight ratio materials can be constructed by either maximizing strength or minimizing weight. Tensegrity structures and aerogels take very different paths to achieving high strength-to-weight ratios but both rely on internal tensile forces. In the absence of tensile forces, removing material eventually destabilizes a structure. Nevertheless, we construct hard sphere packings that have asymptotically zero density and yet maintain finite strength, thus achieving an unbounded strength-to-weight ratio. These constructions, which we call Dionysian, have positive bulk and shear moduli and are linearly resistant to every applied deformation, both internal and external. By demonstrating that there is no lower bound on the density of stable structures, this work allows for the construction of arbitrarily lightweight high-strength materials.