Honeycomb theorem

Remarks

Thomas Hales proved the theorem in 1999. The proof reduces to a single inequality, the hexagonal isoperimetric inequality. It bounds the perimeter of each cell from below, in terms of its area and of how far the cell is from a hexagon. Curving an edge outward helps one cell and hurts its neighbour, and the inequality controls this trade. Before Hales, László Fejes Tóth had proved the result under the hypothesis that all cells are convex. Check of the constant: a regular hexagon of side $s$ has area $\frac{3\sqrt{3}}{2} s^2$, so area $1$ means $s^2 = \frac{2}{3\sqrt{3}}$. Each edge is shared by two cells, so the length per cell is $3s$, and $(3s)^2 = \frac{6}{\sqrt{3}} = 2\sqrt{3} = \sqrt{12}$, so $3s = \sqrt[4]{12} \approx 1.861$. For comparison, a tiling by unit squares has length $2$ per cell, which is larger.

Going deeper

For a student

The theorem is the two-dimensional case of the problem of partitions of least interface area. The three-dimensional analogue, the Kelvin problem, is open. Allowing curved edges is what makes the problem hard, since a cell can gain by bending its walls into its neighbours.