Academic journal article Australian Mathematics Teacher

Angle Defect and Descartes' Theorem

Academic journal article Australian Mathematics Teacher

Angle Defect and Descartes' Theorem

Article excerpt

We discuss here a delightfully simple theorem of Descartes, which will enable us to determine easily the number of vertices of almost every polyhedron.

Angle defect

We define the angle defect at a vertex of a polyhedron to be the amount by which the sum of the face angles at that vertex falls short of 2[pi]. For example, for the regular tetrahedron, the angle defect is (in degrees) is:

360 - 60 - 60 - 60 = 180[degrees] = [pi].

The total angle defect of the polyhedron is defined to be what one gets by adding up the angle defects at all the vertices of the polyhedron. We call the total defect T.

Here is a simple exploratory exercise, good for class use!

Consider some simple polyhedra, and determine the angle defects and the total angle defect T. Can you make a conjecture about T? Use Table 1.

[TABLE 1 OMITTED]

A reasonable conjecture would seem to be that the total angle defect T is always 720[degrees] or 4[pi]. How might we prove this?

Descartes' theorem

Rene Descartes lived from 1596 to 1650. His contributions to geometry are still remembered today in the terminology "Descartes' plane". Descartes discovered that there is a connection between the total defect, T, and the Euler number V - E + F, where V, E, F denote the number of vertices, edges and faces of the given polyhedron. We might point out that strictly the Euler number is a relationship between the number of edges, vertices and regions ("faces") of a planar graph--a finite figure in the plane having no intersecting sides or loops. However, the number is easily applied to most polyhedra, for by looking 'through a face' of the polyhedron we can obtain a planar graph. For example, in the case of the cube, we obtain the diagram given in Figure 1.

[FIGURE 1 OMITTED]

For the cube we have V = 8, E = 12 and F = 6. In the above planar representation, we have V = 8, E = 12, and the number of regions (F) is 6. The (large) square face we are looking through is ignored, and replaced by the exterior region of the figure, thus retaining the F value. This representation of the cube (and similarly of other polyhedra) is called a Schlegel diagram.

We can now state Descartes' Theorem.

Descartes Theorem

The total angle defect of a polyhedron and the Euler number of that polyhedron are related by

T = 2[pi] (V - E + F) (*)

This means that for any polyhedron having Euler number 2, T = 4[pi], or 720[degrees]. (There are, in fact, occasional non-convex polyhedra which do not have Euler number 2.)

Proof of the theorem

Looking at the right hand side of (*), we can think of associating the quantity 2[pi] with each vertex, edge and face of the given polyhedron. Let us relate this idea to a particular face of the polyhedron (see Figure 2).

[FIGURE 2 OMITTED]

This face contributes 1 towards the number F, so we associate 2[pi] with the face. …

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