Defect (geometry)

From Academic Kids

In geometry, the defect of a vertex of a polyhedron is the amount by which the sum of the angles 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.

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 the 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.

An error

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. Remove one square face from a cube — perhaps it can help to think of it as the top of a box — and put a new vertex at the center of the cube, and put in four new triangular faces, each abutting two of the other new triangular faces and one of the squares formerly adjacent to the square that was removed. Then four of the vertices of this new polyhedron are so shaped that they can occur only in a non-convex polyhedron. The defects of all of the vertices of this new polyhedron are positive, as can be checked easily.

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