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  "body": "\n## Statement\n\nLet $\\Gamma \\subseteq \\R^2$ be a locally finite graph made of smooth curves, such that $\\R^2 \\setminus \\Gamma$ has infinitely many bounded connected components, all of area $1$.\nLet $C$ be the union of these bounded components.\nFor $r \u003e 0$ let $B_r$ be the disc of radius $r$ about the origin.\nThen\n$$\\limsup_{r \\to \\infty} \\frac{\\operatorname{perimeter}(C \\cap B_r)}{\\operatorname{area}(C \\cap B_r)} \\ge \\sqrt[4]{12},$$\nand the regular hexagonal tiling by hexagons of area $1$ attains equality.\n\n## Remarks\n\nThomas Hales proved the theorem in 1999.\nThe proof reduces to a single inequality, the hexagonal isoperimetric inequality.\nIt bounds the perimeter of each cell from below, in terms of its area and of how far the cell is from a hexagon.\nCurving an edge outward helps one cell and hurts its neighbour, and the inequality controls this trade.\nBefore Hales, László Fejes Tóth had proved the result under the hypothesis that all cells are convex.\nCheck of the constant: a [[regular hexagon|01KJJ9A400]] of side $s$ has area $\\frac{3\\sqrt{3}}{2} s^2$, so area $1$ means $s^2 = \\frac{2}{3\\sqrt{3}}$.\nEach 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 $3s = \\sqrt[4]{12} \\approx 1.861$.\nFor comparison, a [[tiling|GKJK293DXS]] by unit squares has length $2$ per cell, which is larger.\n\n## Explanations\n\n### L1 human\n\nBees build their combs from hexagonal cells.\nThe theorem says that hexagons are the best choice.\nAmong all ways to divide the plane into regions of equal area, the hexagonal grid uses the least total length of walls.\nThis was believed since antiquity and was proved only at the end of the twentieth century.\n\n### L2 human\n\nThe theorem is the two-dimensional case of the problem of partitions of least interface area.\nThe three-dimensional analogue, the Kelvin problem, is open.\nAllowing curved edges is what makes the problem hard, since a cell can gain by bending its walls into its neighbours.\n",
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