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). 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 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 vertexEdit
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.
ExamplesEdit
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:
| 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 theoremEdit
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 4Template: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 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 figuresEdit
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).
| File:Polydera with positive defects convex.svg | File:Polydera with positive defects concave.svg |
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.
ReferencesEdit
NotesEdit
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BibliographyEdit
- Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton (2008), Pages 220–225.