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  "body": "\n## Summary\n\nThe Ehrhart polynomial of a [[lattice polytope|F39N9NBFPZ]] $P$ counts the integer points in the dilates $tP$ for positive integers $t$.\nEhrhart's theorem says that this count is a polynomial in $t$ of degree $\\dim P$.\nIts leading coefficient is the volume of $P$ when $P$ is full-dimensional, and its constant term is $1$.\nIt is a discrete analogue of volume, and it satisfies [[Ehrhart–Macdonald reciprocity|ZDBER9FJ6T]].\n\n## Explanations\n\n### L1 human\n\nBlow up a lattice polygon by a factor $t$ and count the grid points inside it, including those on the boundary.\nFor the unit square the count is $(t+1)^2$: a $t$ by $t$ square holds $t+1$ points along each side.\nFor every lattice polytope the count follows a polynomial formula in $t$ like this one.\nFor a full-dimensional polytope in $\\R^n$ and large $t$, the count is close to the volume times $t^n$, as one expects from area.\n",
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