Isostaticity of Constraints in Jammed Systems of Soft Frictionless Platonic Solids

The average number of constraints per particle $< C_{total} >$ in mechanically stable systems of Platonic solids (except cubes) approaches the isostatic limit at the jamming point ($< C_{total} > \rightarrow 12$), though average number of contacts are hypostatic. By introducing angular alignment metrics to classify the degree of constraint imposed by each contact, constraints are shown to arise as a direct result of local orientational order reflected in edge-face and face-face alignment angle distributions. With approximately one face-face contact per particle at jamming chain-like face-face clusters with finite extent form in these systems.