Week 6

I presented the counter example I found last week to Professor Luo, and he suggested that I try to find a counter example where the inversive distances are less than 1. I tried to do this by generating random triangulations of a equilateral triangle, and then assigning random inversive distances to the edges. However, I was unable to find a counter example where the inversive distances are less than 1. I will continue to explore this next week.

I was also exploring the theorem on Rigidity of Circle Packings, which states that if two circle packings have the same combinatorial structure and the same inversive distances, then they are equivalent up to a scaling for euclidean geometry and are completely equal in hyperbolic geometry. The theorem is proven for the case where the circle packings have inversive distances are greater than or equal to 0, but I am trying to see if the theorem holds for the case where the inversive distances greater than −1 (corresponding to the case where at least 1 circle intersect at angles greater than 90 degrees and less than 180 degrees ).