Editing Angular defect

In [[geometry]], the '''angular defect''' or simply '''defect''' (also called '''deficit''' or '''deficiency''') is the failure of some [[angle]]s to add up to the expected amount of 360° or 180°, when such angles in the [[Euclidean plane]] would. The opposite notion is the [[angle excess|''excess'']].

Classically the defect arises in two contexts: in the Euclidean plane, angles about a point add up to 360°, while [[Internal and external angle|interior angles]] in a triangle add up to 180°. However, on a [[Polyhedron|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). Also 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 [[Gaussian curvature|curvature]] of the polyhedral surface [[Dirac delta function|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 (geometry)|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 angle]]s of the [[cell (geometry)|cells]] at a [[peak (mathematics)|peak]] falls short of a full circle.

==Examples==

The defect of any of the vertices of a regular [[dodecahedron]] (in which three regular [[pentagon]]s 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 solid]]s:
{| 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's theorem==
Descartes's theorem on the "total defect" of a polyhedron states that if the polyhedron is [[homeomorphism|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>[[René Descartes|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 [[Isometry|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 uniqueness theorem]], 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 in a unique way as the surface of a convex polyhedron.

==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
|[[Image:Polydera with positive defects convex.svg|180px]]
|[[Image:Polydera with positive defects concave.svg|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 which 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===
<references />

===Bibliography===
*[[David Richeson|Richeson, D.]]; ''[[Euler's Gem|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]]