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  "body": "\n## Statement\n\nLet $\\gamma \\colon [a, b] \\to \\R^3$ be a [[regular|06JMR2XWXW]] [[closed curve|0D032NSZKY]] of class $C^2$ whose derivatives up to order $2$ also agree at $a$ and $b$.\nThen the [[total curvature|QP9DY5K8QW]] of $\\gamma$ satisfies\n$$\\int_a^b \\kappa(t) \\, \\norm{\\gamma'(t)} \\, dt \\ge 2\\pi.$$\nEquality holds if and only if $\\gamma$ is a simple closed curve that lies in a plane and bounds a [[convex|0K06X9QXDX]] region of that plane.\n\n## Remarks\n\nThe unit tangent $T = \\gamma' / \\norm{\\gamma'}$ is a closed curve on the unit sphere, and its length equals the total curvature.\nFor each unit vector $v$, the integral of $\\ip{T}{v}$ over a period is $\\ip{\\gamma(b) - \\gamma(a)}{v} = 0$.\nSo $T$ is not contained in any open hemisphere, and a closed curve on the sphere with this property has length at least $2\\pi$.\nExample: a circle has total curvature exactly $2\\pi$, and an ellipse too, since both are plane convex curves.\nA figure-eight curve in the plane has total curvature greater than $2\\pi$, since it is not simple.\nThe theorem holds in the same form in $\\R^n$ for each $n \\ge 2$.\nWerner Fenchel proved it in 1929.\n\n## Explanations\n\n### L1 human\n\nAny closed loop in space must turn by at least one full turn in all.\nOnly flat convex loops, such as a circle or an ellipse, manage with exactly one full turn.\nEvery other loop wastes some turning.\n\n### L2 human\n\nThe proof turns a statement about curves in space into a statement about curves on the sphere, through the tangent indicatrix $T$.\nThis trick, the Gauss map of a curve, is the one-dimensional model of the Gauss map of a surface.\nThe [[Fáry–Milnor theorem|B1Z1C8AHVD]] sharpens Fenchel's bound for knotted curves, from $2\\pi$ to more than $4\\pi$.\n",
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