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Descartes's Lost Theorem


The Euler characteristic

If you ask any mathematician for Euler's theorem, you will undoubtedly get it in the form

S - A + H = 2,

with an emphasis on the ``2''. This is a slight anachronism, because Euler would certainly never have thought of his theorem that way. What has happened in the last one hundred years is that many other shapes have been studied besides convex polyhedra, and (under certain natural hypotheses) it has been discovered that the analogue of S - A + H depends only on the topology: it does not matter how the space is dissected into 0, 1, 2, 3, ... dimensional elements. As long as they are topologically cells, i.e., like points, edges, faces, etc., and fit together roughly like the elements of a polyhedron, the alternating sum

(# of 0-dim elements) - (# of 1-dim elements) + (# of 2-dim elements) - (# of 3-dim elements) + ...

depends only on the topology of the space you start with. In honor of Euler's discovery that this number is always 2 for a convex polyhedron, it is called the Euler characteristic of the space.

Let us compute the Euler characteristic for a surface which is not a convex polyhedron.

          /         /          /|
         /    _____/_____     / |
        /    /          /    /  /
       /    / |________/    /  /
      /    /  /       /    /  /  
     /    /  /       /    /  /
    /    /__________/    /  /
   /          /         /  /
  /__________/_________/  /
  |                    | /

A cellular dissection of the 
surface of a torus.

This picture represents a dissection of the surface of a torus into cells. (The hidden sides are partitioned like the visible ones.) There are 24 vertices (0-dimensional cells), 36 edges (1-dimensional cells) and 12 faces (2-dimensional cells). The alternating sum is 0, which is the Euler characteristic of this surface. Note that if the top and bottom had been left as annuli (topologically not cells), the numbers would have been 16, 24, 12. The theorem requires cells.

The point of the topological invariance of the Euler characteristic is that any dissection of that surface will give the same answer. Clearly the answer does not depend on the faces being planar or the edges being straight lines.

For further thought. Cut this toroidal surface into triangles, and recompute the Euler characteristic. Calculate the Euler characteristic for the surface of a 2-handled torus (you should get -2). Generalize. Calculate the Euler characteristic of the Klein bottle.


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