If 4% of the distance between cell centres is wax, how much of the total surface is wax? Further down on this page, you'll get a chance to submit your answer. The surface of the honeycomb is tightly covered by hexagonal "nuts" like the one shown on the right (not quite to scale), in which the yellow and red represent wax and the blue line marks half the distance between cell centres.

If you wish to cover the entire plane with identical circular discs (e.g., pennies) in the densest possible way, you cannot do better than the bees. While this may seem "obvious", it was proved only in the late nineteenth century by the Norwegian mathematician Axel Thue. What is obvious, is that the planar "kissing number" is 6: exactly six pennies can be laid around a single central one; what is not too difficult to prove, is that the honeycomb presents the most efficient lattice (i.e., regular) packing of pennies in a plane. The hard part is to show that this cannot be improved by some clever irregular scheme. If you restrict yourself to a finite part of the plane, the problem does not become easier, but much harder.

If you wish two pack the entire three dimensional space with identical spherical balls, you also have an "obvious" choice: the standard packing of oranges in a crate. That this is the most efficient way was conjectured by Johannes Kepler (of astronomy fame) around 1611 in an article about snowflakes, but a complete proof was only recently announced by Thomas Hales at MIT. Here, even the kissing question (pictured on the left) is difficult: in a famous 1694 dispute, Sir Isaac Newton contended that only 12 balls could touch a central one, while his colleague David Gregory thought it might be 13. After two centuries this was finally decided in favour of Sir Isaac. The lattice version of Kepler's Conjecture had been settled by Gauss in the early 1800's.

What's in it for you? Apart from its obvious application to solid state physics (packing atoms), it turns out that Sphere Packing is one of the corner stones of Coding Theory, the science of unscrambling garbled digital messages. Every conceivable "word" of your message is surrounded by a (possibly higher dimensional) "ball" of distorted versions, and these balls must be tightly packed for efficient operation.

The Pacific Institute for the Mathematical Sciences (PIMS) is a non-profit organisation supported by five universities of Western Canada and dedicated to the promotion of mathematical research -- but it also has a programme of education and public awareness.

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