🏓A container “full” of ping-pong balls can still be about 36% empty

“Full” sounds like it should mean 100% occupied. With spheres, it doesn't.

When identical balls settle into a dense but jumbled arrangement, empty pockets remain between their curved surfaces. Using a packing benchmark of about 64%: 64% ball + 36% empty space = 100% container volume.

So a container can look completely full of balls while more than one-third of its geometric volume is still air.

The 36% isn't a separate experimental measurement. It is simply the complement of the packing model: 100% − 64% = 36%.

Change the arrangement and the empty fraction changes too.

The densest possible packing of identical spheres reaches approximately 74.05% occupied space. That leaves approximately 25.95% empty.

So there is no universal rule saying spheres must leave 36% empty. The arrangement matters.

In ordinary language, a box can be full when no more objects can conveniently be added.

In geometry, we can separately ask what fraction of its volume is actually occupied by solid material. Those are different meanings of “full.”

Approximate / model-dependent. The arithmetic from 64% to 36% is exact. The underlying 64% packing assumption is approximate and depends on the packing protocol and definition used.

Dense random packing of equal spheres leaves roughly 36% void space.

• Torquato, Truskett & Debenedetti

  Is Random Close Packing of Spheres Well Defined? Physical Review Letters 84, 2064 (2000)(opens external site)

  The ≈64% packing model and its limitations; 36% is derived from it

• Thomas C. Hales

  A proof of the Kepler conjecture. Annals of Mathematics 162 (2005)(opens external site)

  ≈74.05% densest packing of identical spheres

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