Euler�s formula. Euler characteristic. Triangulation of a surface. Compatible triangle orientations. Orientable surfaces.

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Euler’s formula. Euler characteristic. Triangulation of a surface. Compatible triangle orientations. Orientable surfaces.

Theorem 1. Euler’s formula for polyhedra. For any simple polyhedron

1) V - E + F = 2

where

V is the number of vertices

E is the number of edges

F is the number of faces

Proof

Def. Polyhedron. A solid bounded by plane faces. The intersections of the faces are the edges and the points where three or more edges meet are the vertices.

Def. Simple polyhedron. A polyhedron which is topologically equivalent to a sphere, a polyhedron with no holes in it.

One can confirm, if one wishes, Euler’s formula for the polyhedra shown in Fig. 1 by counting faces, edges and vertices. A polyhedron that is not simple is shown in Fig. 2. It has a hole running through the center of it. Euler’s formula applies only to simple polyhedra.

Euler’s formula in topology. Let us now consider what happens with an arbitrary topological transformation of a simple polyhedron into some other shape. For example, we could consider the polyhedron to be made of a thin sheet of rubber and stretch it into some shape such as a sphere, pear, or an elephant. When we do this the edges of the polyhedron are transformed into curved lines and faces into curved regions. However, it is important to note that under the transformation the number of vertices, edges and faces remain the same, the values of V, E, and F remain the same, and Euler’s formula still holds. Thus Euler’s formula has a much broader application than just the polyhedrons of geometry. It is a fundamental formula in topology. The quantity V - E + F is called the Euler characteristic of the surface. The Euler characteristic of a surface that has been overlaid by a network (i.e. divided up into a network of regions by means of vertices and arcs) is an invariant under a topological transformation.

Theorem 2. Let a closed surface of genus p be divided into a number of regions by means of vertices and edges i.e. by marking a number of vertices on the surface and joining them by arcs. Then

V - E + F = 2 - 2p

where

V is the number of vertices

E is the number of edges

F is the number of regions

Proof

● A surface has Euler characteristic 2 if and only if it is topologically equivalent to a sphere; it has Euler characteristic 1 if and only if it is topologically equivalent to the projective plane or to a disk (a circle and its interior); it has Euler characteristic 0 if and only if it is topologically equivalent to a cylinder, torus, Mobius strip, or a Klein bottle.

Def. Euler characteristic of a curve. Let a curve be divided into segments by points (vertices) such that each segment (along with its end points) is topologically equivalent to a closed line segment (can be continuously deformed into a closed interval). Then the Euler characteristic of the curve is the number of vertices minus the number of segments.

Def. Triangulation of a surface. To triangulate a surface is to divide it up into a network of triangles by means of vertices and arcs. A triangulation of a surface is a particular division of the surface into triangles. Surfaces overlaid by networks containing non-triangular regions can be triangulated by running diagonals from chosen vertices. All triangulations of a particular surface have the same Euler characteristic.

Triangle orientation. Any triangle can be provided with an orientation. By an orientation we mean that a definite direction has been given for traversing its boundary. The orientation of a triangle is given by listing its vertices in the direction of traversal. For example, the indicated clockwise orientation of the triangle ABC of Fig. 3 could be given as (ABC), (BCA) or (CAB).

Compatible orientations. Consider two oriented

triangles with a common side such as triangles ABC and CBD of Fig. 4. The orientations of two such triangles are said to be compatible if they generate opposite directions on the common side of the triangle. The orientations of the two triangles of Fig. 4 are compatible. The orientations of the two triangles of Fig. 5 are not compatible. In the plane or any other two-sided surface compatible orientations mean the triangles are traversed in the same direction (clockwise or counterclockwise).

Orientable triangulation. A triangulation of a surface is called orientable if the orientations of all the triangles in it can be so chosen that any two triangles adjoining in a common side are compatibly oriented.

Theorem 3. Every triangulation of a two-sided surface is orientable; every triangulation of a one-sided surface is nonorientable.

Because of this theorem two-sided surfaces are referred to as orientable and one-sided surfaces as non-orientable.

Theorem 4. Two surfaces are topologically equivalent (i.e. homeomorphic) if and only if

1) their triangulations have the same Euler characteristic

and

2) both are orientable or not orientable

References

1. James & James. Mathematics Dictionary.

2. Mathematics, Its Content, Methods and Meaning. Vol. III

3. Richard Courant, Herbert Robbins. What is Mathematics?