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 1 and aerogels 2 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. Attempts to maximize the strength-to-weight ratio with purely repulsive spheres have proceeded by removing spheres from already stable crystalline structures 3, 4 . This results in a modestly low density and a strength-to-weight ratio much worse than can be achieved with tensile materials. However, the lowest density stable sphere packing is unknown 4, 5 . Here, we provide such a packing and demonstrate that it has asymptotically zero density and yet maintains finite strength, thus achieving an unbounded strength-to-weight ratio. This construction, which we call Dionysian, is the diametric opposite to the Apollonian sphere packing 6 which completely and stably 7 fills space. The Dionysian structure is inspired by Böröczky bridge packings 8, 9 which are unstable zero density sphere packings in which every individual sphere is constrained. We create tools to evaluate stability and strength of compressive sphere packings. Using these we find that our structures have asymptotically finite 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.

Dionysian hard sphere packings are mechanically stable at vanishingly low densities