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  "body": "\n## Summary\n\nAn $m$-rectifiable set is a subset of $\\R^n$ covered by countably many Lipschitz images of $\\R^m$.\nThe cover may miss a set of $m$-dimensional [[Hausdorff measure|Q1TGZK8RN6]] zero.\nSuch sets are the measure-theoretic version of $m$-dimensional surfaces, and they may have corners, cusps and countably many pieces.\nThey are the basic objects of geometric measure theory, where they replace smooth submanifolds in problems such as the Plateau problem.\nEvery set of finite $m$-dimensional Hausdorff measure splits into a rectifiable part and a purely unrectifiable part, and the two parts behave very differently.\n\n## Explanations\n\n### L1 human\n\nThink of a set that is built from countably many bent and crumpled copies of a flat $m$-dimensional piece.\nIt may look rough, but at almost every point it has an approximate tangent plane, much as a smooth surface does.\nIn the plane, a polygon, a circle, and a countable union of line segments are all $1$-rectifiable.\nA filled square is not $1$-rectifiable, because countably many curves cover only a set of area zero.\nSome fractal dust sets of dimension one, such as the four-corner Cantor set, are not rectifiable at all: they meet every curve in a set of length zero.\n",
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