Definitions

Defect (geometry)

In geometry, the defect (or deficit) 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.

(According to the Oxford English Dictionary, one of the senses of the word "defect" is "The quantity or amount by which anything falls short; in Math. a part by which a figure or quantity is wanting or deficient.")

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 is 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 $pi,$ $4pi,$
octahedron 6 Four equilateral triangles $\left\{2 piover 3\right\}$ $4pi,$
cube 8 Three squares $\left\{piover 2\right\}$ $4pi,$
icosahedron 12 Five equilateral triangles $\left\{piover 3\right\}$ $4pi,$
dodecahedron 20 Three regular pentagons $\left\{piover 5\right\}$ $4pi,$

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. 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 Gaussian curvature is zero and the Gaussian curvature at a vertex is equal to the defect there.

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. This total will have one complete circle for every vertex in the polyhedron. Care has to be taken to use the correct Euler characteristic for the polyhedron.

A potential 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. Consider 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 non-convex, but the defects remain the same and so are all positive.