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- Angular_defect abstract "In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the plane would. The opposite notion is the excess.Classically the defect arises in two ways: the defect of a vertex of a polyhedron; the defect of a hyperbolic triangle;and the excess also arises in two ways: the excess of a toroidal polyhedron. the excess of a spherical triangle;In the plane, angles about a point add up to 360°, while interior angles in a triangle add up to 180° (equivalently, exterior angles add up to 360°). However, on a convex polyhedron the angles at a vertex on average add up to less than 360°, on a spherical triangle the interior angles always add up to more than 180° (the exterior angles add up to less than 360°), and the angles in a hyperbolic triangle always add up to less than 180° (the exterior angles add up to more than 360°).In modern terms, the defect at a vertex or over a triangle (with a minus) is precisely the curvature at that point or the total (integrated) over the triangle, as established by the Gauss–Bonnet theorem.".
- Angular_defect wikiPageID "673575".
- Angular_defect wikiPageLength "5981".
- Angular_defect wikiPageOutDegree "37".
- Angular_defect wikiPageRevisionID "707789253".
- Angular_defect wikiPageWikiLink Alexandrovs_uniqueness_theorem.
- Angular_defect wikiPageWikiLink Angle.
- Angular_defect wikiPageWikiLink Category:Hyperbolic_geometry.
- Angular_defect wikiPageWikiLink Category:Polyhedra.
- Angular_defect wikiPageWikiLink Cube.
- Angular_defect wikiPageWikiLink Dihedral_angle.
- Angular_defect wikiPageWikiLink Dodecahedron.
- Angular_defect wikiPageWikiLink Elongated_square_pyramid.
- Angular_defect wikiPageWikiLink Euler_characteristic.
- Angular_defect wikiPageWikiLink Face_(geometry).
- Angular_defect wikiPageWikiLink File:Polydera_with_positive_defects_concave.svg.
- Angular_defect wikiPageWikiLink File:Polydera_with_positive_defects_convex.svg.
- Angular_defect wikiPageWikiLink Gaussian_curvature.
- Angular_defect wikiPageWikiLink Gauss–Bonnet_theorem.
- Angular_defect wikiPageWikiLink Geometry.
- Angular_defect wikiPageWikiLink Great_stellated_dodecahedron.
- Angular_defect wikiPageWikiLink Homeomorphism.
- Angular_defect wikiPageWikiLink Hyperbolic_triangle.
- Angular_defect wikiPageWikiLink Icosahedron.
- Angular_defect wikiPageWikiLink Internal_and_external_angle.
- Angular_defect wikiPageWikiLink Maxima_and_minima.
- Angular_defect wikiPageWikiLink Octahedron.
- Angular_defect wikiPageWikiLink Pentagon.
- Angular_defect wikiPageWikiLink Platonic_solid.
- Angular_defect wikiPageWikiLink Polyhedron.
- Angular_defect wikiPageWikiLink Saddle_point.
- Angular_defect wikiPageWikiLink Small_stellated_dodecahedron.
- Angular_defect wikiPageWikiLink Spherical_trigonometry.
- Angular_defect wikiPageWikiLink Square_pyramid.
- Angular_defect wikiPageWikiLink Tetrahedron.
- Angular_defect wikiPageWikiLink Toroidal_polyhedron.
- Angular_defect wikiPageWikiLink Turn_(geometry).
- Angular_defect wikiPageWikiLinkText "Angle deficit".
- Angular_defect wikiPageWikiLinkText "Angular defect".
- Angular_defect wikiPageWikiLinkText "Angular defect#Descartes.27 theorem".
- Angular_defect wikiPageWikiLinkText "Descartes' theorem on total angular defect".
- Angular_defect wikiPageWikiLinkText "Descartes' theorem".
- Angular_defect wikiPageWikiLinkText "angular defect".
- Angular_defect wikiPageWikiLinkText "defect".
- Angular_defect title "Angular defect".
- Angular_defect urlname "AngularDefect".
- Angular_defect wikiPageUsesTemplate Template:!.
- Angular_defect wikiPageUsesTemplate Template:Mathworld.
- Angular_defect wikiPageUsesTemplate Template:Other_uses.
- Angular_defect wikiPageUsesTemplate Template:Wiktionary.
- Angular_defect subject Category:Hyperbolic_geometry.
- Angular_defect subject Category:Polyhedra.
- Angular_defect type Polytope.
- Angular_defect type Redirect.
- Angular_defect type Surface.
- Angular_defect comment "In geometry, the (angular) defect (or deficit or deficiency) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the plane would. The opposite notion is the excess.Classically the defect arises in two ways: the defect of a vertex of a polyhedron; the defect of a hyperbolic triangle;and the excess also arises in two ways: the excess of a toroidal polyhedron.".
- Angular_defect label "Angular defect".
- Angular_defect sameAs Q2533936.
- Angular_defect sameAs Angula_difekto.
- Angular_defect sameAs Difetto_(geometria).
- Angular_defect sameAs m.031t0y.
- Angular_defect sameAs Дефект_(геометрия).
- Angular_defect sameAs Kotni_primanjkljaj.
- Angular_defect sameAs Дефект_(геометрія).
- Angular_defect sameAs Q2533936.
- Angular_defect wasDerivedFrom Angular_defect?oldid=707789253.
- Angular_defect isPrimaryTopicOf Angular_defect.