Web Definition: Defect
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 non-convex polyhedra, then the defect is negative. If a polyhedron is convex, then the defects of all of its vertices are positive. 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 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° − = 36°. Descartes theorem on the total defect of a polyhedron states the if the polyhedron is homeomorphic to a sphere, the total defect, ie the sum of the defects of all of the vertices, is two full circles. The polyhedron need not be convex. A generalization says the number of circles in the total defect equals the Euler characteristic of the polyhedron. It is tempting to think that every non-convex polyhedron has some vertices whose defect is negative.
Read More »Dictionary Entry: Want or absence of something necessary for completeness or perfection - deficiency - opposed to superfluity. To fail - to become deficient. To injure - to damage.
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