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The Euler-Poincare Characteristic of Conjoined Polyhedra

The Euler formula for regular polyhedra is a marvelous result in topology to the effect that the sum of the number of faces and the number of vertices less the number of edges is equal to two. That net sum can be defined as the Euler-Poincare characteristic of a geometric figure. Then Euler's formula says that the Euler-Poincare characteristic for a class of polyhedra (ones without holes) is equal to two. There are some polyhedra for which the Euler formula does not hold. For example, consider two cubes brought together along one edge. The Euler-Poincare characteristic for this conjoined figure is three rather than two. As another example consider two tetrahedra which are brought together at one vertex. The Euler-Poincare characteristic for this figure is also three rather than two.

These two examples illustrate a property of the Euler-Poincare characteristic Χ( ). If A and B are polyhedra that are joined to create a polyhedron A∪B then

Χ(A∪B) = Χ(A) + Χ(B) − Χ(A∩B)

In the case of the two cubes joined at an edge, A∩B is an edge with two vertices. The Euler-Poincare characteristic of this edge and two vertices is -1+2=+1. Each of the cubes has characteristic two so the combined characteristic is four. This less 1 gives a characteristic of 3.

The two tetrahedron joined at a vertex have a characteristic which is 2+2−1=3. The Euler-Poincare characteristic of a point is one.

Now consider two cubes joined at a face. The Euler-Poincare characteristic of a square with its four edges and four vertices is 4-4+1=1. Thus two cubes joined on a face with the joint internal face maintained have an Euler-Poincare characteristic of 2+2-1=3.

Thus if a polyhedron-like configuration can be represented as two polyhedra being joined together then its Euler-Poincare characteristic can be computed easily using the above formula.

(To be continued.)


For more on polyhedra and thire Euler-Poincare characteristic see Euler Poincare.


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