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Descartes's Lost Theorem

 

The polyhedral Gauss-Bonnet Theorem

The Gauss-Bonnet Theorem states that for any smooth surface S the integral of the Gaussian curvature is equal to 2 pi times the Euler characteristic of S. (This theorem is an elementary consequence of Gauss's local integral formula.)

For example, the sphere of radius R has constant Gaussian curvature 1/R2 and area 4 pi R2; the integral of the Gaussian curvature is 4 ,which is 2 pi times the Euler characteristic 2.

This theorem has a polyhedral version: For any polyhedral surface, the sum of the polyhedral curvatures is equal to 2 pi times the Euler characteristic.

For a convex polyhedron with Euler characteristic 2, this is just another statement of Descartes's Lost Theorem, but it can be applied more generally, for example, to a toroidal surface:

	   _____________________
          /         /          /|
         /    _____/_____     / |
        /    /          /    /  /
       /    / |________/    /  /
      /    /  /       /    /  /  
     /    /  /       /    /  /
    /    /__________/    /  /
   /          /         /  /
  /__________/_________/  /
  |                    | /
  |____________________|/
  

This polyhedral torus has 8 vertices 
where the curvature is 
pi/2,
8 vertices where the curvature
is 0, and 8 vertices where the 
curvature is -pi/2, 
Total = 0.

At each of the vertices around the outside there are three planar right angles, giving a polyhedral curvature of pi/2. At each of the vertices in the center of the top or bottom, the sum of the face angles is 2 pi, giving a polyhedral curvature of 0. At each of the eight vertices surrounding the hole, the sum of the face angles is 5 pi/2, giving polyhedral curvature -pi/2. The total curvature is 0.

It is not clear what appeal such a calculation would have had for Descartes. Even though tesselated tori were well known as mazzochi in Italian art of the fifteenth century, the classical subjects of interest to geometers remained convex polygons and convex polyhedra. Moreover, negative curvature would not have seemed a natural concept, since at the beginning of his career (he was twenty-four in 1620) he was reluctant to consider negative numbers at all. Finally, for Descartes the distinction between a vertex and the measure of the (planar or solid) angle at that vertex was not explicit; the lack of this distinction probably kept him from the combinatorial version of his theorem that Euler derived. Nevertheless, his Lost Theorem, now recovered, remains as indelible evidence of the geometrical power of this intellectual giant.

 

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