In geometry, the angular defect is the shortfall when some angles fail to sum to 360° or 180°, unlike in the Euclidean plane. For example, on a convex polyhedron, face angles meeting at a vertex add to less than 360°, creating a defect, while some vertices of a nonconvex polyhedron exhibit an excess by exceeding this sum. Similarly, angles of a hyperbolic triangle add up to less than 180°, indicating a defect, versus a spherical triangle with an excess. This defect corresponds to discrete curvature concentrated at vertices, linked by the Gauss–Bonnet theorem to a polyhedron’s Euler characteristic.
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:
| Shape | Number of vertices | Polygons meeting at each vertex | Defect at each vertex | Total defect |
|---|---|---|---|---|
| tetrahedron | 4 | Three equilateral triangles | π ( 180 ∘ ) {\displaystyle \pi \ \ (180^{\circ })} | 4 π ( 720 ∘ ) {\displaystyle 4\pi \ \ (720^{\circ })} |
| octahedron | 6 | Four equilateral triangles | 2 π 3 ( 120 ∘ ) {\displaystyle {2\pi \over 3}\ (120^{\circ })} | 4 π ( 720 ∘ ) {\displaystyle 4\pi \ \ (720^{\circ })} |
| cube | 8 | Three squares | π 2 ( 90 ∘ ) {\displaystyle {\pi \over 2}\ \ (90^{\circ })} | 4 π ( 720 ∘ ) {\displaystyle 4\pi \ \ (720^{\circ })} |
| icosahedron | 12 | Five equilateral triangles | π 3 ( 60 ∘ ) {\displaystyle {\pi \over 3}\ \ (60^{\circ })} | 4 π ( 720 ∘ ) {\displaystyle 4\pi \ \ (720^{\circ })} |
| dodecahedron | 20 | Three regular pentagons | π 5 ( 36 ∘ ) {\displaystyle {\pi \over 5}\ \ (36^{\circ })} | 4 π ( 720 ∘ ) {\displaystyle 4\pi \ \ (720^{\circ })} |
Descartes's theorem
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 4π radians). The polyhedron need not be convex.1
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 2 π {\displaystyle 2\pi } 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 4 π {\displaystyle 4\pi } , 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).
Polyhedra with positive defectsA 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.
Notes
Bibliography
- Richeson, D.; Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton (2008), Pages 220–225.
External links
Look up defect in Wiktionary, the free dictionary.References
Descartes, René, Progymnasmata de solidorum elementis, in Oeuvres de Descartes, vol. X, pp. 265–276 /wiki/Ren%C3%A9_Descartes ↩