Mechanically stable sphere packings at arbitrarily low densities

Lightweight materials can be formed by creating mechanically rigid structures with a combination of compressive and tensile forces. By considering purely compressive forces in sphere packings, we determine the limits on creating low density rigid systems. An Apollonian packing proves that a rigid packing can completely fill space, but proof for the existence of a lowest density rigid packing was unknown. The previously known lowest density packings are constructed by diluting simple crystals, but we present a new construction based on rigid bridges. This new construction not only demonstrates that lower density packings can be achieved, but it can be used to create rigid packings with densities arbitrarily close to zero. We demonstrate the rigidity of these low density packings using both established and novel procedures and we explore the properties of these configurations to gain a deeper understanding of the limits of rigidity in repulsive systems. Such constructions not only lay an old puzzle to rest, but enable the development of new lightweight materials.