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  "body": "\n## Summary\n\nThe packing density of a packing is the fraction of space covered by its bodies.\nIt is measured inside large balls and passed to the limit, since space is infinite.\nThe packing density of a body is the largest density of a packing of its congruent copies.\nFor equal balls this is the optimal [[sphere packing|HF32CBN5MA]] density, known exactly only in dimensions $1$, $2$, $3$, $8$ and $24$.\n\n## Explanations\n\n### L1 human\n\nFill a huge box with equal coins laid flat, as tightly as you can, and compute the share of the box that the coins cover.\nMaking the box larger and larger, this share settles near one number, the density of the arrangement.\nCoins in a square grid cover $\\pi/4 \\approx 78.5\\%$ of the plane.\nCoins in a hexagonal pattern cover $\\pi/\\sqrt{12} \\approx 90.7\\%$, and no arrangement does better.\n",
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