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  "body": "\n## Description\n\nThe Leech lattice $\\Lambda_{24}$ is a [[lattice|WN9V7WSJKF]] in the [[Euclidean space|0XPR1E85W0]] $\\R^{24}$, discovered by John Leech.\nUp to isometry, it is the unique lattice $L \\subset \\R^{24}$ with these properties:\n\n1. $L$ is unimodular: it has a basis whose Gram matrix has determinant $1$;\n2. $L$ is even: $x \\cdot x$ is an even integer for every $x \\in L$;\n3. $L$ has no vector $x$ with $x \\cdot x = 2$.\n\nSo its shortest nonzero vectors have norm $x \\cdot x = 4$.\nThere are $196560$ of them.\n\n## Remarks\n\nThe Leech lattice can be constructed from the extended binary [[Golay code|SM1TJSAR75]].\nIts automorphism group $\\mathrm{Co}_0$ has order $8315553613086720000$, and its quotient by $\\set{\\pm 1}$ is the [[Conway group|366D4ETGQH]] $\\mathrm{Co}_1$.\nThe $196560$ shortest vectors show that the kissing number in dimension $24$ is at least $196560$, and in fact it equals $196560$.\nThe sphere packing obtained from the Leech lattice is the densest sphere packing in $\\R^{24}$; this was proved by Cohn, Kumar, Miller, Radchenko and Viazovska.\nEven unimodular lattices exist only in dimensions divisible by $8$.\nIn dimension $24$ there are $24$ of them up to isometry, and the Leech lattice is the only one without vectors of norm $2$.\n\n## Explanations\n\n### L1 human\n\nA lattice is a regular grid of points, like the integer points in the plane, but it can be tilted and stretched.\nThe Leech lattice is a very special grid in $24$ dimensions.\nIf you put equal balls at its points, each ball touches $196560$ others, the most possible in $24$ dimensions, and the packing is the densest possible there.\nIts symmetry group is huge and gave rise to several new finite simple groups.\n",
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