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 <math>2\pi</math> times the Euler characteristic <math>\chi = 2</math>, so for a convex polyhedron the sum of the defects is <math>4\pi</math>, while a toroidal polyhedron has <math>\chi = 0</math> and total defect zero.
== Defect of a vertex == For a polyhedron, the defect at a vertex equals 2π minus the sum of all the angles at the vertex (all the faces at the vertex are included). If a polyhedron is convex, then the defect of each vertex is always positive. If the sum of the angles exceeds a full turn, as occurs in some vertices of many non-convex polyhedra, then the defect is negative.
The concept of defect extends to higher dimensions as the amount by which the sum of the dihedral angles of the cells at a peak falls short of a full circle.
==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: {| class="wikitable" !Shape !Number of vertices !Polygons meeting at each vertex !Defect at each vertex !Total defect |- |tetrahedron||4||Three equilateral triangles||<math>\pi \ \ (180^\circ )</math>||<math>4\pi \ \ (720^\circ )</math> |- |octahedron||6||Four equilateral triangles||<math>{2 \pi\over 3} \ (120^\circ )</math>||<math>4\pi \ \ (720^\circ )</math> |- |cube||8||Three squares||<math>{\pi\over 2}\ \ (90^\circ )</math>||<math>4\pi \ \ (720^\circ )</math> |- |icosahedron||12||Five equilateral triangles||<math>{\pi\over 3}\ \ (60^\circ )</math>||<math>4\pi \ \ (720^\circ )</math> |- |dodecahedron||20||Three regular pentagons||<math>{\pi\over 5}\ \ (36^\circ )</math>||<math>4\pi \ \ (720^\circ )</math> |}
==Descartes' theorem on total angular defect == {{About|angular defect|the radii of every four tangent circles satisfy a quadratic equation|Descartes' theorem|section=yes}} Descartes's theorem on the "total defect" of a polyhedron states that if the polyhedron is homeomorphic to a sphere (i.e. topologically equivalent to a sphere, so that it may be deformed into a sphere by stretching without tearing), the "total defect", i.e. the sum of the defects of all of the vertices, is two full circles (or 720° or 4{{pi}} radians). The polyhedron need not be convex.<ref>Descartes, René, ''Progymnasmata de solidorum elementis'', in ''Oeuvres de Descartes'', vol. X, pp. 265–276</ref>
A generalization says the number of circles in the total defect equals the Euler characteristic of the polyhedron. This is a special case of the Gauss–Bonnet theorem, which relates the integral of the Gaussian curvature to the Euler characteristic. Here, the Gaussian curvature is concentrated at the vertices: on the faces and edges the curvature is zero (the surface is locally isometric to a Euclidean plane) and the integral of curvature at a vertex is equal to the defect there (by definition).
This can be used to calculate the number ''V'' of vertices of a polyhedron by totaling the angles of all the faces, and adding the total defect (which is <math>2\pi</math> times the Euler characteristic). This total will have one complete circle for every vertex in the polyhedron.
A converse to Descartes' theorem is given by Alexandrov's theorem on polyhedra, according to which a metric space that is locally Euclidean (hence zero curvature) except for a finite number of points of positive angular defect, adding to <math>4\pi</math>, can be realized uniquely as the surface of a convex polyhedron.{{r|connelly}}
==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).
{| class=wikitable |+Polyhedra with positive defects |180px |180px |}
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=== {{reflist|refs=
<ref name=connelly>{{cite journal | last = Connelly | first = Robert | author-link = Robert Connelly | title = Convex Polyhedra by A. D. Alexandrov | journal = SIAM Review | volume = 48 | issue = 1 | date = March 2006 | pages = 157–160 | jstor = 20453762 | doi = 10.1137/SIREAD000048000001000149000001 | url = https://www.math.cornell.edu/~connelly/alexandrov.pdf | archive-url = https://web.archive.org/web/20170830001838/http://www.math.cornell.edu/~connelly/alexandrov.pdf | archive-date = 2017-08-30 | url-status = dead }}</ref>
}}
===Bibliography=== *Richeson, D.; ''Euler's Gem: The Polyhedron Formula and the Birth of Topology'', Princeton (2008), Pages 220–225.
==External links== {{wiktionary|defect}} *{{Mathworld | urlname=AngularDefect | title=Angular defect }}
Category:Polyhedra Category:Hyperbolic geometry