In the May Transactions of the American Mathematical Society , he outlined progress in proving the hexagonal honeycomb conjecture and its variants, and he suggested a possible route to a proof. With curved sides, the complication is that a side that bulges out for one cell must bulge in for its immediate neighbor. Bulging out helps minimize the cell perimeter, while bulging in hurts. Hales proved that the advantage of bulging out is less than the disadvantage of bulging in. Therefore, straight-sided polygons work better than curved ones, and regular hexagons are truly best of all.
Bees have that aspect of their honeycomb structures down pat.
Two possible structures for the closed end of a honeycomb cell. T here's more to a honeycomb than a vertical, hexagonal grid, however. It actually consists of two layers of cells placed back to back. The cells themselves are tilted upward at an angle of about 13 from the horizontal-just enough to prevent stored honey from dripping out. The cells of the two layers are offset so the center of a chamber on one side is the corner of three adjacent cells on the other side.