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  "body": "\n## Description\n\nThe $E_8$ lattice is the [[lattice|WN9V7WSJKF]] in the [[Euclidean space|0XPR1E85W0]] $\\R^8$ given by\n$$E_8 = \\set{x \\in \\Z^8 \\cup (\\Z + \\tfrac12)^8 : x_1 + x_2 + \\dots + x_8 \\in 2\\Z}.$$\nSo its points have all coordinates integers or all coordinates half-integers, with even coordinate sum.\n\n## Remarks\n\nIt is closed under addition and negation, so it is a subgroup of $\\R^8$, and it contains $8$ independent vectors.\nIt is even: $x \\cdot x \\in 2\\Z$ for all $x \\in E_8$.\nIt is unimodular: its covolume is $1$.\nUp to isometry, it is the unique even unimodular lattice in $\\R^8$.\nIts shortest nonzero vectors have length $\\sqrt 2$, and there are $240$ of them.\nThey are the $112$ vectors with two coordinates $\\pm 1$ and six coordinates $0$.\nThe other $128$ are the vectors with all coordinates $\\pm\\tfrac12$ and an even number of minus signs.\nThese $240$ vectors form the root system of type $E_8$.\nBalls of radius $\\sqrt 2/2$ centred at the points of $E_8$ form the densest [[sphere packing|HF32CBN5MA]] in $\\R^8$, with density $\\pi^4/384$, as proved by Viazovska.\nEach ball touches $240$ others, which is the [[kissing number|B408FZYXF9]] in dimension $8$.\n\n## Explanations\n\n### L1 human\n\nThe $E_8$ lattice is a grid of points in eight dimensions.\nTake all points whose coordinates are either all whole numbers or all halves of odd numbers, and keep those whose coordinates add up to an even number.\nThis grid is extremely symmetric, and placing equal balls at its points gives the tightest possible packing of balls in eight dimensions.\n",
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