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  "body": "\n## Statement\n\nLet $P \\subseteq \\R^3$ be a three-dimensional [[convex polytope|300K6CB1KH]].\nLet $v$ be a [[vertex|R7V3S9V891]] of $P$.\nFor a face $F$ of $P$ that contains $v$, let $\\alpha_F(v)$ be the interior angle of the polygon $F$ at $v$.\nThe *angular defect* at $v$ is\n$$\\delta(v) = 2\\pi - \\sum_F \\alpha_F(v),$$\nwhere the sum runs over the faces $F$ of $P$ that contain $v$.\nThen\n$$\\sum_{v} \\delta(v) = 4\\pi,$$\nwhere the sum runs over all vertices of $P$.\n\n## Remarks\n\nLet $P$ have $V$ vertices, $E$ edges and $F$ faces.\nA face with $k$ sides has angle sum $(k - 2)\\pi$, and the numbers of sides of all faces add up to $2E$.\nSo all face angles of $P$ add up to $2\\pi E - 2\\pi F$.\nHence $\\sum_v \\delta(v) = 2\\pi V - 2\\pi E + 2\\pi F = 2\\pi (V - E + F)$, and [[Euler's polyhedron formula|09HT3P35YP]] $V - E + F = 2$ gives $4\\pi$.\nThe same count shows that for a polyhedral surface of [[Euler characteristic|RNDDWDAFJ5]] $\\chi$ the total defect is $2\\pi\\chi$.\n\nExample: the cube has $8$ vertices, each with three right angles, so each defect is $2\\pi - 3\\pi/2 = \\pi/2$, and the total is $8 \\cdot \\pi/2 = 4\\pi$.\nThe regular tetrahedron has $4$ vertices with three angles of $\\pi/3$ each, so each defect is $\\pi$, and the total is again $4\\pi$.\n\nConvexity enters only through Euler's formula: the statement holds for every polyhedron whose surface is a topological sphere, where a defect may be negative at a non-convex vertex.\n\n## Explanations\n\n### L1 human\n\nAt each corner of a convex solid, the face angles add up to less than a full turn; the missing amount is the defect.\nIf you add the missing amounts over all corners, you always get $720^\\circ$, two full turns, whatever the solid.\nA cube has eight corners each missing $90^\\circ$, and $8 \\cdot 90^\\circ = 720^\\circ$.\n\n### L2 human\n\nThe theorem is the discrete form of the [[Gauss–Bonnet theorem|ESGEHKPD60]] for the sphere: the defect at a vertex plays the role of the total [[Gaussian curvature|SQVQ5DQT6W]] concentrated there.\nIt is equivalent to Euler's formula, given the angle count in the proof.\nIt shows that a convex polyhedron has at least one vertex with defect at least $4\\pi/V$, and it constrains the possible vertex configurations of Platonic and Archimedean solids.\nIt also explains why the total of the cone angles is fixed in [[Alexandrov's theorem on polyhedra|ZCDWXVMDFN]].\n",
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  "name": "Descartes's theorem on total angular defect",
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