Team 1034 Page 4 of 21 1 Introduction Defining Congressional districts has long been a source of controversy in the United States Since the district-drawers are chosen by those currently in power, the boundaries are often created to influence future elections by grouping an unfavorable minority demographic with a favorable majority; this process is called Gerrymandering. It is common for districts to take on bizarre shapes, spanning slim sections of multiple cities and criss-crossing the countryside in a haphazard fashion. The only lawful restrictions on legislative boundaries stipulate that districts must contain equal populations, but the makeup of the districts left entirely to the district-drawers. In the United Kingdom and Canada, the districts are more compact and intuitive. Their success in mitigating Gerrymandering is attributed to having turned over the task of boundary-drawing to nonpartisan advisory panels. However, these independent com- missions can take 2-3 years to finalize a new division plan, calling their effectiveness into question. It seems clear that the U.S. should establish similar unbiased commissions, yet make some effort to increase the efficiency of these groups. Accordingly, our goal is to develop a small toolbox that aids in the redistricting process. Specifically, we will create a model that draws legislative boundaries using simple geometric constructions 1.1 Current models The majority of methods for creating districts fall into two categories: ones that depend on a current division arrangement(most commonly counties) and ones that do not depend on current divisions. Most fall into the former category. By using current divisions the problem is reduced to grouping these divisions in a desirable way using a multitude of mathematical procedures. Mehrotra et al. uses graph partitioning theory to cluster counties to total population variation of around 2% of the average district size [8. Hess and Weaver use an iterative process to define population centroids, use integer programming to group counties into equally populated districts, and then reiterate the process until the centroids reach a limit 5. Garfinkel and Nemhauser use iterative matrix operations to search for district combinations that are contiguous and compact 3. Kaiser begins with the current districts and systematically swaps populations with adjacent districts [4 All of these methods use counties as their divisions since they partition the state into a relatively small number of sections. This is necessary because most of the mathematical tools they use become slow and imprecise with many divisions. (This is the same as saying they become unusable in the limit when the state is divided into more continuous sections.)Thus using small divisions, like zip codes which on average are 5 times smaller The other category of methods is less common. Out of all our researched papers d documentation, there were only two methods that did not depend on current state divisions. Forrest's method continually divides a state into halves while maintaining pop- ulation equality until the required number of districts is satisfied [4, 5. Hale, Ransom and Ramsey create pie-shaped wedges about population centers. This creates homogeneous districts which all contain portions of a large city, suburbs, and less populated areas [4 hese approaches are noted for being the least biased since their only consideration is population equality and do not use preexisting divisions. Also, they are straightforwardTeam 1034 Page 4 of 21 1 Introduction Defining Congressional districts has long been a source of controversy in the United States. Since the district-drawers are chosen by those currently in power, the boundaries are often created to influence future elections by grouping an unfavorable minority demographic with a favorable majority; this process is called Gerrymandering. It is common for districts to take on bizarre shapes, spanning slim sections of multiple cities and criss-crossing the countryside in a haphazard fashion. The only lawful restrictions on legislative boundaries stipulate that districts must contain equal populations, but the makeup of the districts is left entirely to the district-drawers. In the United Kingdom and Canada, the districts are more compact and intuitive. Their success in mitigating Gerrymandering is attributed to having turned over the task of boundary-drawing to nonpartisan advisory panels. However, these independent com￾missions can take 2-3 years to finalize a new division plan, calling their effectiveness into question. It seems clear that the U.S. should establish similar unbiased commissions, yet make some effort to increase the efficiency of these groups. Accordingly, our goal is to develop a small toolbox that aids in the redistricting process. Specifically, we will create a model that draws legislative boundaries using simple geometric constructions. 1.1 Current Models The majority of methods for creating districts fall into two categories: ones that depend on a current division arrangement (most commonly counties) and ones that do not depend on current divisions. Most fall into the former category. By using current divisions, the problem is reduced to grouping these divisions in a desirable way using a multitude of mathematical procedures. Mehrotra et.al. uses graph partitioning theory to cluster counties to total population variation of around 2% of the average district size [8]. Hess and Weaver use an iterative process to define population centroids, use integer programming to group counties into equally populated districts, and then reiterate the process until the centroids reach a limit [5]. Garfinkel and Nemhauser use iterative matrix operations to search for district combinations that are contiguous and compact [3]. Kaiser begins with the current districts and systematically swaps populations with adjacent districts [4]. All of these methods use counties as their divisions since they partition the state into a relatively small number of sections. This is necessary because most of the mathematical tools they use become slow and imprecise with many divisions. (This is the same as saying they become unusable in the limit when the state is divided into more continuous sections.) Thus using small divisions, like zip codes which on average are 5 times smaller than a county in New York, becomes impractical. The other category of methods is less common. Out of all our researched papers and documentation, there were only two methods that did not depend on current state divisions. Forrest’s method continually divides a state into halves while maintaining pop￾ulation equality until the required number of districts is satisfied [4, 5]. Hale, Ransom and Ramsey create pie-shaped wedges about population centers. This creates homogeneous districts which all contain portions of a large city, suburbs, and less populated areas [4]. These approaches are noted for being the least biased since their only consideration is population equality and do not use preexisting divisions. Also, they are straightforward