How Many Balls Can you Shake into a Can?
Malcolm E. Lines · 2021
Take a large known quantity of equal-sized balls and pour them into a cube-shaped can. Shake them up gently and measure the level in the container. Repeat the procedure and measure again, ... , and again. You quickly find that this level is remarkably stable if the shaking is performed in a thorough fashion, even though it seems inconceivable that the exact arrangement of balls in the can could be absolutely identical each time. Nevertheless, the effect is well known in real life; not only grocers, but their customers too, believe that the volume of a box containing one pound of coffee is well-defined. Although neither the grocer nor his customer may know how to calculate it, they feel sure that the first mathematician they meet on the street surely does. Their confidence in the abilities of those pursuing mathematical interests is commendable, but unfortunately it is misplaced. You see, no mathematician on earth knows how to do it either! What we are looking for is the best way of packing spheres into a specified volume; we talk about finding the maximum ‘packing fraction’ which is defined as that fraction of the ‘can’ which is ultimately taken up by the balls. If this fraction is denoted by f, then the inevitable spaces between the balls must account for the rest of the volume fraction 1 — f such that the sum of the two parts (occupied and unoccupied) add up to one. What we are therefore saying is that no-one has yet been able to calculate the number f for the densest possible arrangement of spheres in three-dimensional space, a situation which we refer to as ‘dense random packing’. Still more distressingly, we do not even have a convenient mathematical description of what random packing really is. Nevertheless, it seems quite clear from experiment that the number f exists, is reproducible with considerable accuracy, and is about f = 0.64.