  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. Attempts to maximize the
strength-to-weight ratio with purely repulsive spheres have proceeded by
removing spheres from already stable crystalline structures. This results in a
modestly low density and a strength-to-weight ratio much worse than can be
achieved with tensile materials. Here, we demonstrate the existence of a
packing of hard spheres that has asymptotically zero density and yet maintains
finite strength, thus achieving an unbounded strength-to-weight ratio. This
construction, which we term Dionysian, is the diametric opposite to the
Apollonian sphere packing which completely and stably fills space. We create
tools to evaluate the 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.
