Circle Packings – Week 1

While I did a little bit of mentored research during high school, this program is my first time doing real, full-fledged research at the undergraduate level. Outside of my normal academic work, my learning experiences have otherwise been mostly limited to extracurriculars and internships. While these have been invaluable, this year I realized how crucial it was that I gain research experience as well, especially as I move forward with my plans to attend graduate school next year. So, when the opportunity to partake in research at the intersection of my two main fields of interest — math and computer science — was presented, I was of course thrilled.

This summer I will be working under the guidance of Professor Bowers on a project that focuses on circle packings, which are configurations of circles that have a specified pattern of tangencies. The project requires substantial knowledge of computational geometry, and so this first week mainly consisted of gaining a solid background in an area of mathematics that is relatively new to me. I had a couple of crash-course sessions with Professor Bowers, and also read through an assortment of texts on computational geometry and circle packings.

In particular, we have been focusing on Voronoi diagrams, which are partitions of a plane into convex polygons, where each polygon contains a generating point, or site, and where every point in a given polygon is closer to its generating point than to any other.

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Figure 1: A Voronoi diagram

The dual graph of the Voronoi diagram, the Delaunay Triangulation, is another important topic that comes into play in circle packings. The Delaunay Triangulation is a triangulation of the convex hull of the sites of the diagram in which no point in the set is inside of the circumcircle of any triangle of the triangulation.

Delaunay_circumcircles_vectorial.svg.png Figure 2: A Delaunay Triangulation

In addition, we covered some basics of stereographic projections, Möbius transformations, inversive geometry, and parabolic, elliptic, and hyperbolic flows.

Once I had a better grasp on some of these topics, I then was able to dive a bit deeper and began reading a couple of research papers dealing with Voronoi diagrams, Delanauy Triangulations, and Möbius transformations on circle packings.

Furthermore, prior to this program, I had only coded in Python and C++, and so in addition to learning much of the necessary math concepts, I also have been spending some time getting acquainted with Java and working with the programming language/IDE Processing, which is a software created for data visualization.

After taking a step back and reflecting on the week, I realize just how much I have learned in such a short period of time. I look forward to continuing to learn more about circle packings and am excited for what next week has in store!

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