The hexagon honeycomb …is the most efficient.

Regular hexagons provide the least-perimeter way to divide the plane into unit areas.

The recent reports on the disappearance of bee populations have omitted mention of their role in the longest-standing open problem in mathematics, dating from the first millennium B.C.E. and solved in the nick of time (actually with a year to spare; see Scientific American) for the second millennium in 1999. The ancient conjecture, now a theorem, says that the hexagonal honeycomb of Figure 1 provides the most efficient (least-perimeter) way to divide the plane into unit areas. A proof was announced by Thomas Halesin 1999, the same Hales who the year before had proved the 1611 Kepler sphere-packing conjecture, a mere 387 years old. This in turn was on the heels of the celebrated 1995 proof by Princeton mathematicianAndrew Wiles of Fermat’s 1637 “Last Theorem,” similarly a mere 358 years old.

2012-03-03-hexagons.jpgFigure 1. Regular hexagons provide the least-perimeter way to divide the plane into unit areas. Image from wikipedia.org.

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