Summer 2015 Project

This summer research will involve formulating and proving theorems about packings. The primary tools/techniques/activities involved in this are the following:

1. 1)Exploring packings to find candidate optimal arrangements. Some of this might be done using the program that I wrote during my sabbatical at Cornell University.

2. 2)Analyzing geometric situations, like finding the analytic coordinates of the centers of the circles in the packings above.

The goal of the research is to prove that we have all the optimal arrangements for a fixed small number of circles in a container or class of containers. Further we also need to prove that the discovered arrangement are, in fact, optimal. I expect that this summer we will focus on the container rather than increasing the number of circles. We will begin the summer by reviewing techniques used by previous summer research students (this will take at most a few days).

Since coming to GVSU I have been involved in conducting research with undergraduates. I have mentored 13 summer projects resulting in several publications and awards for my students. In 2010 I became the co-director of the Research Experiences for Undergraduates (REU) Site in mathematics at GVSU. This is the latest in a series of such grants for GVSU dating back to 2000. The current grant is a three year grant from the National Science Foundation (NSF) and ends in 2015 (pending funding). Below is a description of my project for summer 2015. This was also the subject of the summer research with undergraduates in 2007, 2008, 2010, 2011, 2012, 2013, and 2014.

What Will This Research Involve?

Desirable Experiences For Applicants

Overview and Introduction

Suppose you have seven cans of soda pop, what is the smallest box with a square cross section that can hold them standing upright in a single layer? By examining the cross section of the box, you can see that this is equivalent to determining the smallest square that contains seven equal and non-overlapping circles. The optimal solution is the arrangement below. You might think you can rearrange the circles in to a more dense arrangement so that there is no `free` circle (like the one in the upper left corner), but you cannot. This is a result due to J. Schaer and Graham from 1965.

What this project is NOT...

Notice how the circles are equal in diameter and non-overlapping and are contained in a square -- that is they form an equal circle packing in a hard-boundary square. Over the summer, we will study an equivalent problem to the soda pop can packing one. We will let the equal circles expand, while fixing the size of the contain. More formally we will seek to determine the largest possible common diameter of a fixed number of circles that form a packing in some kind of container. This is a problem in discrete geometry and, in the case of a hard boundary square, optimal arrangements have been proven for at least 20 circles and there are conjectured optimal arrangements for all n up to well over 100 circles. Packing of equal objects in other hard-boundary containers have also been studied.

This research project examines packings of equal equals in containers other than a hard-boundary square. The main focus of this project will be to consider spaces obtained from square (or more generally parallelograms) where the top and bottom and the left and right sides are identified. This space is called a square flat torus and is represented by the dark square in the picture below which contains seven circles. Can you count the seven circles?

This is conjectured to be one of three equal density arrangements of seven equal circles on a square flat torus. Included in this picture, in addition to the seven equal circles and the torus, is the packing graph associated to this packing. A packing graph is a graph where the vertices are the centers of the circles and the edges connect tangent circles. This graph comes naturally with an embedding on the torus and its properties are very important in the approach that we will use to study these packing.

In past summer research, when we were studying packings on a rhomboidal flat torus, we discovered the following one parameter family of packings. We proved that the last frame is the optimal packing of 6 equal circles in this container.

Applicants should like to analyze geometric situations -- working with lines, angles and geometric objects will be important to success of this research. The ideal applicant will geometrically inclined. The desirable experiences for applicants to this project are a positive attitude toward geometric ideas and a bit of geometric background. The background could be just that you liked geometric reasoning in the classes you have taken (like calculus or linear algebra) or it could be your enjoyment of any kind of geometry course you have taken.

Equal Circle Packing

How To Apply

Applied. Yes, I really DO NOT care about applications of this to packing anything real. I enjoy the mathematical side of this and not anything practical that may come out of this. (There could be applications of this -- I do not know and I’m not looking for them.... If you happen to find some please let me know.)