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  "body": "\n## Summary\n\nThe total curvature of a curve is the integral of its [[curvature|7JQS6VKFEZ]] with respect to arc length.\nIt measures how much the direction of the curve turns in all, from start to end.\nFor closed curves it is bounded below by $2\\pi$, by Fenchel's theorem, and knotted curves need more than $4\\pi$, by the Fáry–Milnor theorem.\n\n## Explanations\n\n### L1 human\n\nWalk once around a closed path and add up all the turning you do, ignoring left and right.\nThat sum is the total curvature.\nWalking once around a circle you turn by a full turn, so the total curvature is $2\\pi$.\nA wiggly path makes you turn back and forth, and the total grows.\n",
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