Defect (geometry)

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Defect (geometry)

In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, the defect (or deficit) of a vertex of a polyhedron

Polyhedron

|}A polyhedron is often defined as a geometry object with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is quite unsatisfactory....
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 angle

Dihedral angle

In geometry, the angle between two Plane s is called their dihedral or torsion angle.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection....
s of the cells

Cell (geometry)

In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object....
at a peak falls short of a full circle.

(According to the Oxford English Dictionary

Oxford English Dictionary

The Oxford English Dictionary , published by the Oxford University Press , is a comprehensive dictionary of the English language. Two fully-bound print editions of the OED have been published under its current name, in 1928 and 1989; as of December 2008 the dictionary's current editors have completed a quarter of the third edition....
, one of the senses of the word "defect" is "The quantity or amount by which anything falls short; in Math.

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Encyclopedia

In geometry

Geometry

Polyhedron

Dihedral angle

Cell (geometry)

In geometry, a cell is a three-dimensional element that is part of a higher-dimensional object....
at a peak falls short of a full circle.

(According to the Oxford English Dictionary

Oxford English Dictionary

Examples

The defect of any of the vertices of a regular dodecahedron

Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
(in which three regular pentagon

Pentagon

In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The internal angles in a simple pentagon total 540?....
s meet at each vertex) is 36°, or p/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 solid

Platonic solid

In geometry, a Platonic solid is a convex set polyhedron that is regular polyhedron, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruence regular polygons, with the same number of faces meeting at each vertex....
s

Shape	Number of vertices	Polygons meeting at each vertex
tetrahedron Tetrahedron A tetrahedron is a polyhedron composed of four triangle faces, three of which meet at each vertex . A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids....	4	Three equilateral triangles
octahedron Octahedron An octahedron is a polyhedron with eight faces. A regular octahedron is a Platonic solid composed of eight equilateral triangles, four of which meet at each wikt:vertex....	6	Four equilateral triangles
cube Cube A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....	8	Three squares
icosahedron Icosahedron In geometry, an icosahedron isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....	12	Five equilateral triangles
dodecahedron Dodecahedron A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....	20	Three regular pentagons

Descartes' theorem

Descartes' theorem on the "total defect" of a polyhedron states that if the polyhedron is homeomorphic

Homeomorphism

In the mathematics field of topology, a homeomorphism or topological isomorphism is a continuous function between two topological spaces....
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 4p radians). The polyhedron need not be convex.

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

Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
of the polyhedron. This is a special case of the Gauss–Bonnet theorem

Gauss–Bonnet theorem

The Gauss?Bonnet theorem or Gauss?Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry to their topology ....
which relates the integral of the Gaussian curvature

Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, ?1 and ?2, of the given point....
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

Polyhedra with positive defects

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

Cube

A cube is a three-dimensional space solid object bounded by six square faces, facets or sides, with three meeting at each wikt:vertex. The cube can also be called a Regular polyhedron hexahedron and is one of the five Platonic solids....
where one face is replaced by a square pyramid

Square pyramid

In geometry, a square pyramid is a Pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry....
: this elongated square pyramid

Elongated square pyramid

In geometry, the elongated square pyramid is one of the Johnson solids . As the name suggests, it can be constructed by elongating a square pyramid by attaching a cube to its square base....
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.

Defect (geometry)

Examples

Descartes' theorem

A potential error

External links