random close packing
Take a big jar and pour in thousands of identical ball bearings, then shake it down until they stop rattling and settle as tightly as they will go without any effort to line them up. They pack densely — each ball touches several others, no big gaps — yet there are no neat crystalline rows. That jammed, dense-but-disordered arrangement is random close packing: the densest way you can jam together equal hard spheres if you deny them the chance to form a regular crystal. It is the geometric model for the atomic structure of a metallic glass, where the 'balls' are metal atoms frozen in a liquid-like tangle.
The numbers make the contrast sharp. If you DO let spheres arrange into a crystal — the grocer's orange-pile stacking, face-centred cubic or hexagonal close packing — they fill 74 percent of space (packing fraction 0.74) with every atom having exactly 12 neighbours. Random close packing cannot reach that: jammed disordered spheres fill only about 64 percent of space (roughly 0.64), a value so robust it is often called the random-close-packing limit. The average coordination number is still around 12, but now it is an AVERAGE over a spread — some atoms have 11 neighbours, some 13 — and the local arrangements include five-fold, icosahedral clusters that a crystal can never contain. That extra 10 percent of empty space, distributed as many small irregular gaps, is closely tied to the free volume that lets a glass exist.
Random close packing matters because it explains the structure of metallic glasses without invoking any hidden crystal: pour metal atoms into a jar and freeze them fast, and you get essentially a frozen dense random packing. It also carries an honest health warning. Unlike the 74 percent of true close packing, which is an exact, provable geometric fact, the 64 percent figure is NOT a precisely defined mathematical constant — it depends on how you shake, how you define 'jammed', and how you measure it, and mathematicians have shown 'random close packing' is not perfectly well-defined. It is an excellent, widely used empirical model, but it is a model, not a theorem.
In the 1960s J. D. Bernal literally poured thousands of steel ball bearings into a rubber bladder, squeezed them into a jammed heap, then poured in paint to mark every contact point and counted. He found a packing fraction near 0.64 and an average of about 12 contacts per ball, with many non-crystalline five-fold clusters — a hands-on demonstration of the structure now used to model metallic glasses.
Jammed equal spheres fill about 64 percent of space, versus 74 percent for a crystal.
The 74 percent of crystalline close packing is an exact geometric fact; the roughly 64 percent of random close packing is not. It is protocol-dependent and, strictly, not a uniquely defined constant, so treat it as a robust empirical figure, not a theorem.