Packing is a way to place objects of the same kind so that they touch in some specified way, often inside a container with specified properties. The objects to be packed may be polyhedra, polygons, spheres, ellipsoids, hyperspheres, or any other type of shape, and the number of dimensions involved may range from two upwards.

 

The fraction of a space filled by a given collection of objects is called the packing density. The densest packing of circles in the plane is the hexagonal lattice of the bee's honeycomb, which has a packing density of 0.9069 ... In 1611, Johannes Kepler proposed that hexagonal, or face-centered cubic, packing is also the densest possible way to arrange spheres in three dimensions – an assertion known as Kepler's conjecture. Currently, the worst known convex packer in two-dimensions is the smoothed octagon, with a packing density of about 0.902. Stanislaw Ulam conjectured that the sphere was the worst packing object in three-dimensional space.

 

Slothouber-Graatsma puzzle

The Slothouber-Graatsma puzzle is a packing puzzle in which six 1 × 2 × 2 blocks and three 1 × 1 × 1 blocks must be fitted together to make a 3 × 3 × 3 cube. There is only one solution. A similar but much more difficult puzzle, named after its inventor, John Conway, calls for packing three 1 × 1 × 3 blocks, one 1 × 2 × 2 block, one 2 × 2 × 2 block, and thirteen 1 × 2 × 4 blocks into a 5 × 5 × 5 box.

 

Sphere packing

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A long-standing puzzle was to ask: what's the most efficient way to stack spheres of the same size so that you leave the smallest amount of space between them. For more on the most efficient way to pack spheres, see Kepler's conjecture, and the Cannonball Problem.