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  "body": "\n## Statement\n\nEvery [[sphere packing|HF32CBN5MA]] of equal balls in $\\R^3$ has upper [[packing density|4P5PSWADJD]] at most\n$$\\frac{\\pi}{\\sqrt{18}} = \\frac{\\pi}{3\\sqrt 2} \\approx 0.7405.$$\nThe face-centred cubic packing attains this density.\n\n## Remarks\n\nJohannes Kepler stated the claim in 1611.\nGauss proved it for lattice packings in 1831.\nThomas Hales announced a proof in 1998, completed with Samuel Ferguson.\nIt reduces the problem to the minimization of a function over finitely many local configurations, and checks these by computer with linear programming bounds.\nThe Flyspeck project, led by Hales, completed a formal proof checked by the proof assistants HOL Light and Isabelle in 2014.\nThe bound is attained by infinitely many packings.\nAny stacking of hexagonal layers, each layer resting in the hollows of the one below, has density $\\pi/\\sqrt{18}$.\nThe face-centred cubic and the hexagonal close packings are two of them.\n\n## Explanations\n\n### L1 human\n\nThe usual way grocers stack oranges, in layers where each orange sits in a dip of the layer below, fills about $74\\%$ of space.\nKepler guessed that no arrangement of equal balls fills more.\nThis took almost four hundred years to prove, and the proof needed extensive computer checks.\n\n### L2 human\n\nThe Kepler conjecture is the three-dimensional case of the sphere packing problem; the two-dimensional case is [[Thue's theorem|3P246RHXMF]].\nUnlike in the plane, the densest local cluster of balls around one ball does not extend to a packing of space, so local arguments alone fail.\nHales's proof uses a carefully chosen partition of space into cells around the balls and a weighted volume score that handles this.\nThe other dimensions where the answer is known are $8$ and $24$, by the work of [[Viazovska and her coauthors|YJ48S89BZA]].\n",
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