Cyan = normal disk
Yellow = rattler (Z<3)
Red = overlapping disk
φ_J ≈ 0.84 (random close pack)
Hard disks pack randomly until the jamming point φ_J ≈ 0.84 (2D random close packing), where every disk is mechanically trapped. At jamming, pressure diverges as P ~ (φ_J − φ)^{−1} and the shear modulus jumps from 0 to finite. Rattlers (fewer than Z=3 contacts in 2D) are not in the force-bearing network. Press "Compress Step" to inflate disks and watch jamming onset.