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  "body": "\n## Statement\n\nLet $P$ and $Q$ be two simple [[polygons|8VEFWD3EJB]] in the Euclidean plane with the same area.\nThen $P$ can be cut into finitely many polygonal pieces that can be moved by [[isometries|0J1N3WTKM4]] to form $Q$, with disjoint interiors.\n\n## Remarks\n\nThe proof cuts a polygon into triangles.\nEach triangle can be cut and reassembled into a rectangle, and each rectangle into a rectangle with one side of length $1$.\nStacking these rectangles gives a rectangle of width $1$ and height equal to the area.\nSo $P$ and $Q$ can both be cut into the same rectangle, and the two dissections combine, since the relation \"can be cut into\" is transitive.\nThe analogue in dimension $3$ is false: this is the negative answer to [[Hilbert's third problem|81035XQ5HE]].\n\n## Explanations\n\n### L1 human\n\nTake any two polygons with the same area, for example a square and a triangle.\nYou can always cut the first into finitely many pieces and reassemble them, like a jigsaw puzzle, into the second.\nSo in the plane, area is the only thing that matters for cutting and rearranging.\n\n### L2 human\n\nThe theorem justifies the old method of finding areas by cutting and rearranging.\nAn example is the proof that a parallelogram has the area of a rectangle with the same base and height.\nIt shows that in the plane, equal area is the same as scissors congruence.\nIn space there is a second invariant, the [[Dehn invariant|S0TGRX3ARS]], and equal volume is not enough.\n",
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