Sandi Xhumari

Equal Circle Packing In A Torus

What is the optimal way to pack equal circles into a container? A packing of equal circles into a container is an arrangement of circles such that any circle does not overlap another or the boundary of the container. A packing is optimal if it covers more of the space inside the container than any other packing. For example, packing equal circles into squares is a well studied mathematical problem.   In 1963, Graham found the optimal packing of six equal circles in a square.
In this presentation, we will explore packing six equal circles into a container called a triangular flat torus. We will show you the optimal packing in this situation and how we proved it to be optimal. The proof uses tools from several different mathematical areas including graph theory. Using numerous pictures, we will introduce you to all the basic concepts (including the notion of a triangular flat torus, an optimal packing and the graph of a packing) and guide you through our proof.

Faculty Mentor: William Dickinson, Mathematics


Sandi presented at MathFest 2008, the annual summer meeting of the Mathematics Association of America July 31-August 3 in Madison, WI.

Sandi's mentor, Prof. Dickinson, was interviewed about Sandi's project at Mathfest 2008.  His interview was published in SCIENCE vol 321.

Page last modified July 14, 2009