Defect (geometry)

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In geometry, the defect of a vertex of a polyhedron is the amount by which the sum of the angles of the faces at the vertex falls short of a full circle. If the sum of the angles exceeds a full circle, as occurs in some vertices of most (not all) non-convex polyhedra, then the defect is negative. If a polyhedron is convex, then the defects of all of its vertices are positive.

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 cube is a right angle.

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 is 108°; three of these meet at each vertex, so the defect is 360° − (108° + 108° + 108°) = 36°.

Descartes' theorem

Descartes' 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.

A generalization says the number of circles in the total defect equals the Euler characteristic of the polyhedron.

A potential error

Image:Polyhedra with positive defects.PNG It is tempting to think (and has even been stated in geometry textbooks) that every non-convex polyhedron has some vertices whose defect is negative. Here is a counterexample. Consider a cube where one face is replaced by a square pyramid: this 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 non-convex, but the defects remain the same and so are all positive.