In
geometryGeometry 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) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the plane would. The opposite notion is the
excess.
Classically the defect arises in two ways:
- the defect of a vertex of a polyhedron;
- the defect of a hyperbolic triangle
-Euclidean geometry:In the foundations of the hyperbolic functions sinh, cosh and tanh, a hyperbolic triangle is a right triangle in the first quadrant of the Cartesian plane,...
;
and the excess arises in one way:
- the excess of a spherical triangle.
In the plane, angles about a point add up to 360°, while interior angles in a triangle add up to 180° (equivalently,
exterior angles add up to 360°).
In
geometryGeometry 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) means the failure of some angles to add up to the expected amount of 360° or 180°, when such angles in the plane would. The opposite notion is the
excess.
Classically the defect arises in two ways:
- the defect of a vertex of a polyhedron;
- the defect of a hyperbolic triangle
-Euclidean geometry:In the foundations of the hyperbolic functions sinh, cosh and tanh, a hyperbolic triangle is a right triangle in the first quadrant of the Cartesian plane,...
;
and the excess arises in one way:
- the excess of a spherical triangle.
In the plane, angles about a point add up to 360°, while interior angles in a triangle add up to 180° (equivalently,
exterior angles add up to 360°). However, on a convex polyhedron the angles at a vertex on average add up to less that 360°, on a spherical triangle the interior angles always add up to more than 180° (the exterior angles add up to
less that 360°), and the angles in a hyperbolic triangle always add up to less than 180° (the exterior angles add up to
more than 360°).
In modern terms, the defect at a vertex or over a triangle (with a minus) is precisely the curvature at that point or the total (integrated) over the triangle, as established by the
Gauss–Bonnet theoremThe Gauss–Bonnet theorem or Gauss–Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry to their topology...
.
Defect of a vertex
The defect of a vertex of a
polyhedronA polyhedron is often defined as a geometric solid with flat faces and straight edges .This definition of a polyhedron is not very precise, and to a modern mathematician is...
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 angleIn geometry, the angle between two planes 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
cellsIn geometry, a cell is a three-dimensional element that is part of a higher-dimensional object.- In polytopes :A cell is a three-dimensional polyhedron element that is part of the boundary of a higher-dimensional polytope, such as a polychoron or honeycomb .For example, a cubic honeycomb is made...
at a peak falls short of a full circle.
(According to the
Oxford English DictionaryThe Oxford English Dictionary , published by the Oxford University Press , is a comprehensive dictionary of the English language...
, 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
dodecahedronIn geometry, a dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid. It is composed of 12 regular pentagonal faces, with three meeting at each vertex, and is represented by the Schläfli symbol {5,3}. It has 20 vertices and 30 edges...
(in which three regular
pentagonIn 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°.- Regular pentagons :...
s 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 solidIn geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
s
| Shape |
Number of vertices |
Polygons meeting at each vertex |
Defect at each vertex |
Total defect |
tetrahedronIn geometry, a tetrahedron is a polyhedron composed of four triangular 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 |
|
|
octahedronIn geometry, 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 vertex.It is a 3-dimensional cross polytope.-Dimensions:...
|
6 |
Four equilateral triangles |
|
|
cubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
|
8 |
Three squares |
|
|
icosahedronIn geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of five Platonic solids....
|
12 |
Five equilateral triangles |
|
|
dodecahedronIn geometry, a dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid. It is composed of 12 regular pentagonal faces, with three meeting at each vertex, and is represented by the Schläfli symbol {5,3}. It has 20 vertices and 30 edges...
|
20 |
Three regular pentagons |
|
|
Descartes' theorem
Descartes' theorem on the "total defect" of a polyhedron states that if the polyhedron is
homeomorphicIn the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between two topological spaces that has a continuous inverse function...
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 characteristicIn 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 theoremThe 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 curvatureIn differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the...
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
cubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
where one face is replaced by a
square pyramidIn 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 C
4v symmetry.- Johnson solid :...
: this
elongated square pyramidIn 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. Like any elongated pyramid, it is self-dual.The 92 Johnson solids were named and described by Norman Johnson...
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.