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Encyclopedia > Defect (geometry)

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. Geometry (Greek Î³ÎµÏ‰Î¼ÎµÏ„ÏÎ¯Î±; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ... This article is about the geometric shape. ...

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. In Aerospace engineering, the dihedral is the angle that the two wings make with each other. ... A cell is a three-dimensional object that is part of a higher-dimensional object, such as a polychoron. ... In geometry, a peak is an (n-3)-dimensional element of a polytope. ...

The defect of any of the vertices of a cube is a right angle. Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... This article is about angles in geometry. ...

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°. A dodecahedron is literally a polyhedron with 12 faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ... In geometry, a pentagon is any five-sided polygon. ...

## 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. For other things named Descartes, see Descartes (disambiguation). ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ...

A generalization says the number of circles in the total defect equals the Euler characteristic of the polyhedron. In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ...

## 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 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. Image File history File links Polyhedra_with_positive_defects. ... A cube (or regular hexahedron) is a three-dimensional Platonic solid composed of six square faces, with three meeting at each vertex. ... In geometry, the square pyramid, a pyramid with a square base and equilateral sides, is one of the Johnson solids (J1). ... Results from FactBites:

 PlanetMath: defect (74 words) Note that, in both hyperbolic and spherical geometry, the area of a triangle is equal to its defect. Cross-references: area, spherical geometry, hyperbolic geometry, radians, sum, angle, triangle This is version 7 of defect, born on 2006-07-19, modified 2007-06-26.
 Defect Detection Simulation SOftware (1541 words) The defect geometry itself is represented by a grid surface in the form of an elevation matrix. The Defect geometry program is a simple application that uses standard MFC GUI components, as well as an OpenGL display class to implement all 3D displays. The defect is a point defect, and it is evedent that this configuration is well suited for the detection of defects of this nature.
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