Exhibit: curvature 60
Here is an example for which the total curvature K(p) = 2 S_{1}(p)  S_{2}(p) is different from 60. The simplest example is a cylinder with 7 fold symmetry. The curvature is 56 here. All circles of radius 1 and 2 are circular. The graphs are relatively smooth. Still curvature 60 does not hold. Curvature seems to fluctuate too much. There are 14 points with curvature 2, 14 points with curvature 1 (a point with 2 spheres is illustrated here) and 2 points on the symmetry line with curvature 7. The 7 cylinder is the smallest example [here is the smallest]. Nostalgia: Blackboard pictures of late spring 2009 in School Street Arlington: [1], [2]. And relaxed work in Wallis of Summer 2009 [1]

Here is an example for which the total curvature K(x) = 2 S_{1}  S_{2} is equal to 60. Like for the icosahedron, where each point has curvature 5 and 12*5=60, the curvature 60 theorem holds for all fullerene or two dimensional graphs for which the degree is 5 or 6. Since most graphs I had build with geomags had curvature 60, I had hoped in Spring 2009 that a curvature 60 would hold. 2 years later, I still do not know exact conditions which make a curvature 60 theorem work. A two dimensional SMOOTH graph has curvature 60. What is condition SMOOTH My paper on flat regions shows that it might be subtle. We certainly need that spheres S_{1}(p) and S_{2}(p) are cyclic one dimensional graphs at each point. Its easy to see that this implies that each point has degree larger than 4. I know that XXX= "the combinatorial curvatures K_{1}(p) = 6S_{1}(p)" are all nonnegative" works but this is rather weak because such graphs are either flat tori with zero curvature or then the Euler characteristic is 2. (By the combinatorial Gauss Bonnet, the sum of the combinatorial curvatures is 6 times the Euler characteristic so that for Euler characteristic is negative, there are points with negative curvature).

this page posted, Mai 18, 2011