Exhibit: Identifying good regions

An early attempt to fix a bad case. These pictures were done at the earliest stages of writing this article while a lot of experiments were done (spring 2009). The question tossed around was whether to include an other point to the boundary to have the sum of curvature equal to 12? In the end, the solution was to assume that the complement is a smooth region too (defined in the article).

Good cases. Instead of reducing the region, the region is enlarged. The computer inspects whether the curvature stays the same. Here are cases where the curvature stays the same.

Bad deformation cases. The curvature changes.
The animations illustrate some deformation experiments in the exploration phase to find out, which regions are good. Looking at such cases allowed to formulate the conditions to be satisfied so that the curvature 12 theorem holds. Until December 2009, I still thought that listing all the deformation cases would provide a proof of the theorem and I had planned to list the cases or refer to a computer assisted analysis. Still needed however was an argument that one can always reduce the size of the region to apply the "decent argument". This difficulty turned out to be a blessing since it allowed to write down a proof free of computer assistance.