Honeycomb theorem

Theorem

Let ${\displaystyle \mathrm{\Gamma }}$ be any system of smooth curves in ${\displaystyle {\mathbb{R}}^2}$, subdividing the plane into regions (connected components of the complement of ${\displaystyle \mathrm{\Gamma }}$), all of which are bounded and have unit area. Then, averaged over large disks in the plane, the average length of ${\displaystyle \mathrm{\Gamma }}$ per unit area is at least as large as for the hexagon tiling. The theorem applies even if the complement of ${\displaystyle \mathrm{\Gamma }}$ has additional components that are unbounded or whose area is not one; allowing these additional components cannot shorten ${\displaystyle \mathrm{\Gamma }}$. Formally, let ${\displaystyle B(0,r)}$ denote the disk of radius ${\displaystyle r}$ centered at the origin, let ${\displaystyle L_r}$ denote the total length of ${\displaystyle \mathrm{\Gamma }\cap B(0,r)}$, and let ${\displaystyle A_r}$ denote the total area of ${\displaystyle B(0,r)}$ covered by bounded unit-area components. (If these are the only components, then ${\displaystyle A_r=\pi r^2}$.) Then the theorem states that $${\displaystyle \underset{r\to \mathrm{\infty }}{lim sup}\frac{L_r}{A_r}\ge \sqrt[4]{12}.}$$ The value on the right hand side of the inequality is the limiting length per unit area of the hexagonal tiling.