{
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  "slug": "angular-defect",
  "title": "Angular defect",
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  "lead": "In geometry, the angular defect or simply defect (also called deficit or deficiency) is the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the excess. Classically, the defect arises in two contexts: in the Euclidean plane, angles about a point add up to 360°, while interior angles in a triangle add up to 180°. However, on a convex polyhedron, the angles of the faces meeting at a vertex add up to less than 360° (a defect), while the angles at some vertices of a nonconvex polyhedron may add up to more than 360° (an excess). In addition, the angles in a hyperbolic triangle add up to less than 180° (a defect), while those on a spherical triangle add up to more than 180° (an excess). In modern terms, the defect at a vertex is a discrete version of the curvature of the polyhedral surface concentrated at that point. Negative defect indicates that the vertex resembles a saddle point (negative curvature), whereas positive defect indicates that the vertex resembles a local maximum or minimum (positive curvature). The Gauss–Bonnet theorem gives the total curvature as 2\\pi times the Euler characteristic \\chi = 2 , so for a convex polyhedron the sum of the defects is 4\\pi , while a toroidal polyhedron has \\chi = 0 and total defect zero.",
  "body": "In geometry, the angular defect or simply defect (also called deficit or deficiency) is the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the Euclidean plane would. The opposite notion is the excess. Classically, the defect arises in two contexts: in the Euclidean plane, angles about a point add up to 360°, while interior angles in a triangle add up to 180°. However, on a convex polyhedron, the angles of the faces meeting at a vertex add up to less than 360° (a defect), while the angles at some vertices of a nonconvex polyhedron may add up to more than 360° (an excess). In addition, the angles in a hyperbolic triangle add up to less than 180° (a defect), while those on a spherical triangle add up to more than 180° (an excess). In modern terms, the defect at a vertex is a discrete version of the curvature of the polyhedral surface concentrated at that point. Negative defect indicates that the vertex resembles a saddle point (negative curvature), whereas positive defect indicates that the vertex resembles a local maximum or minimum (positive curvature). The Gauss–Bonnet theorem gives the total curvature as 2\\pi times the Euler characteristic \\chi = 2 , so for a convex polyhedron the sum of the defects is 4\\pi , while a toroidal polyhedron has \\chi = 0 and total defect zero. ==Examples== The defect of any of the vertices of a regular dodecahedron (in which three regular pentagons meet at each vertex) is 36°, or π/5 radians, or 1/10 of a circle. Each of the angles measures 108°; three of these meet at each vertex, so the defect is 360° − (108° + 108° + 108°) = 36°. The same procedure can be followed for the other Platonic solids: ==Positive defects on non-convex figures== It is tempting to think that every non-convex polyhedron must have some vertices whose defect is negative, but this need not be the case if the Euler characteristic is positive (a topological sphere). A counterexample is provided by a cube where one face is replaced by a square pyramid: this elongated square pyramid is convex and the defects at each vertex are each positive. Now consider the same cube where the square pyramid goes into the cube: this is concave, but the defects remain the same, and so are all positive. Two counterexamples that are self-intersecting polyhedra are the small stellated dodecahedron and the great stellated dodecahedron, with twelve and twenty convex points respectively, all with positive defects. ==References== ===Notes=== ===Bibliography=== Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton (2008), Pages 220–225. ==External links==",
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    "title": "Angular defect",
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