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  "body": "\n## Statement\n\n1. Every [[sphere packing|HF32CBN5MA]] of equal balls in $\\R^8$ has upper [[packing density|4P5PSWADJD]] at most $\\pi^4/384 \\approx 0.2537$. This is the density of the lattice packing given by the [[E8 lattice|2CXH65EKY7]].\n2. Every sphere packing of equal balls in $\\R^{24}$ has upper packing density at most $\\pi^{12}/12! \\approx 0.00193$. This is the density of the lattice packing given by the [[Leech lattice|XETV3657DG]].\n\n## Remarks\n\nMaryna Viazovska proved the case of dimension $8$ in 2016.\nHenry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko and Viazovska proved the case of dimension $24$ soon after.\nBoth proofs use the linear programming bound of Cohn and Elkies.\nIf $f \\colon \\R^n \\to \\R$ is a suitable integrable function with $f(0) = \\hat f(0) \u003e 0$, $f(x) \\le 0$ for $\\abs{x} \\ge r$, and $\\hat f \\ge 0$, then every packing of balls of radius $r/2$ has density at most $\\operatorname{vol}(B(0, r/2))$.\nViazovska built such a function for $r = \\sqrt 2$ in dimension $8$ from modular forms and quasimodular forms, and the second paper did the same in dimension $24$.\nThe density values follow from the lattices.\n$E_8$ has covolume $1$ and minimal length $\\sqrt 2$, so the balls have radius $\\sqrt 2/2$.\nThe Leech lattice has covolume $1$ and minimal length $2$, so the balls have radius $1$.\n\n## Explanations\n\n### L1 human\n\nIn most dimensions nobody knows the best way to pack equal balls.\nDimensions $8$ and $24$ are special: there are two extremely symmetric arrangements there, and these are now proved to be the best.\nThe proof is short compared with the proof in dimension $3$, but it uses deep tools from number theory.\n\n### L2 human\n\nThe theorem completes the list of dimensions $1$, $2$, $3$, $8$, $24$ in which the sphere packing problem is solved.\nThe Cohn–Elkies bound is a Fourier analytic version of the [[Delsarte bound|7RPEDHAT7R]] for codes.\nNumerical computations by Cohn and Elkies suggested that it is sharp in dimensions $2$, $8$ and $24$.\nThe work was to find the exact auxiliary function in dimensions $8$ and $24$.\nViazovska received the Fields Medal in 2022, partly for this work.\n",
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