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Packing Pennies in the Plane

An illustrated proof of Kepler's conjecture in 2D

by Bill Casselman

NOTE: This month's contribution contains several Java applets. They may not work on your particular computer, for any of various reasons. If you do not have Java enabled in your browser, for example, you will see only static images representing the animated applets. If you have trouble with viewing the applets even though Java is enabled, or if you want to print out this note, you should disable Java. If Java is enabled and you still have trouble viewing the applets, please let Bill Casselman know about it.

1. "Kepler's Conjecture"

This and the other image nearby are from Kepler's pamphlet on snowflakes. Contrary to what one might think at first. they are not of two dimensional objects, but rather an attempt to render on the page three dimensional packings of spheres.

In his book De nive sexangula (`On the six-sided snowflake') of 1611, Kepler asserted that the packing in three dimensions made familiar to us by fruit stands (called the face-centred cubic packing by crystallographers) was the tightest possible: Coaptatio fiet arctissima: ut nullo praetera ordine plures globuli in idem vas compingi queant.

He didn't elaborate much, and his statement lacks precision. It is almost certain that he had no idea that this assertion required rigorous proof. At any rate, this claim came to be known as Kepler's conjecture, and it turned out to be extremely difficult to verify.

Kepler quite likely would have thought that the analogous assertion about the hexagonal packing in 2D was even more obvious. However, it took about 300 years before it was proven, by the Norwegian mathematician Axel Thue. It is arguable that it took that long just to understand that such an `obvious' assertion required proof. It took another century before a proof of the much more difficult claim about 3D was found, by Tom Hales.

Both images are from photographs taken of the copy of the original edition of Kepler's pamphlet now located at the Thomas L. Fisher Library at the University of Toronto.

Kepler's assertions were possibly prompted by correspondence beginning in the year 1606 between him and the remarkable English mathematician Thomas Harriot. And Harriot's interest was perhaps prompted by a question his employer, Sir Walter Raleigh, had asked him much earlier about how to count the cannon balls in stacks on a ship. (Such were applied mathematics and the military-industrial complex in the XVI and XVII century.) Now Thue's and Hales's theorems have little to do with real world packings in a finite region. Optimal packings of finite regions are ridiculously difficult to ascertain rigourously, even in the simplest cases. Thue's and Hales' theorems are concerned instead with ideal packings throughout all of the plane and space. Hales' proof is one of the most complicated yet required by any theorem, in any branch of mathematics, and I will say little about it here. But Hales observed that, by combining in this small dimension ideas of Fejes-Toth and C. A. Rogers, one could arrive at an extremely elementary proof of Thue's theorem.

This note will follow Hales' suggested argument for the 2D conjecture, filling in a few minor gaps here and there, and relying almost exlusively on illustrations and a few animations to explain the reasoning.

Bill Casselman
University of British Columbia