Week 5

I began the week starting to explore the generalizations of Discrete Schwartz Lemma, particularly in the context of circle packings with inversive distances. I was specifically experimenting with a triangle with a single vertex in the center which are connected to all three vertices of the triangle, where the boundary edges have a inversive distance of (ie the circles corresponding to the boundary vertices are each mutually tangent to each other). I tried to understand the function where you give the coordinate of the center vertex and the radius of the center circle, and it outputs the inversive distances of the edge connecting the center vertex to each of the three boundary vertices. I showed that the for all points in the open disk contained by the circle intersecting the 3 vertices of the triangle, and thus the function is a local diffeomorphism. I then tried to show that the function is a global diffeomorphism, but I was unable to do so. I also tried to see if the function was injective, but I was unable to show that either. I will continue to explore this next week.

Getting frustrated with this approach, I tried to see if I can approach the problem computationally. I wrote a program to find counter examples to the Discrete Schwartz Lemma in the case of inversive distances where the inversive distances greater than 1. I wrote a program to generate random triangulations of a equilateral triangle, and then assign random inversive distances to the edges. I then checked if the Discrete Schwartz Lemma holds for the triangulation where are the interior vertices and are the boundary vertices, and if it does not, I output the triangulation. I was able to find a counter example where the lemma does not hold, but I am still trying to understand why it does not hold.