Team 1034 Page 5 of 21 to apply. However, they do not consider any other possibly important considerations for districts, such as: geographic freaures of the state or how well they encompass citi 1.2 Developing Our Approach Since our goal is to create new methods that add to the diversity of models available to a committee, we should focus on creating district boundaries independently of current divisions. Not only has this approach not been explored to its fullest, but it is not obvious why counties are a good beginning point for a model: Counties are created in the same arbitrary way as districts, so they might also contain biases, since counties are typically not much smaller than districts. Many of the division dependent models end up relaxin their boundaries from county lines in order to maintain equal populations, which makes the initial assumption of using county divisions useless, and also allows for gerrymandering if this relaxation method is not well regulated Treating the state as continuous (i.e. without preexisting divisions) does not lead any specific type of approach. It gives us a lot of freedom, but at the same time we can impose more conditions. If the Forrest and Hale etal. methods are any indication, we should focus on keeping cities within districts and introduce geographical considerations. (Note that these conditions do not have to be considered if we were to treat the problem discretely because current divisions, like counties, are probably dependent on prominent geographical features. Goal: Create a method for redistricting a state by treating the state continu- ously. We require the final districts to contain equal populations and be contiguous. Additionally, the districts should be as simple as possible(see $2 for a definition of simple) and optimally take into account important geographical features of the state 2 Notation and definitions contiguous: A set R is contiguous if it is pathwise-connected compactness: We would like the definition of compactness to be intuitive. One way to look at compactness is the ratio of the area of a bounded region to the square of its perimeter. In other words C PR where Cr is the compactness of region R, AR is the area, PR is the perimeter and Q is the isoperimetric quotient. We do not explicitely use this equation, but we do keep this idea in mind when we evaluate our model. simple: Simple regions are compact and convex. Note that this describes a relative quality, so we can compare regions by their simpliciTeam 1034 Page 5 of 21 to apply. However, they do not consider any other possibly important considerations for districts, such as: geographic freaures of the state or how well they encompass cities. 1.2 Developing Our Approach Since our goal is to create new methods that add to the diversity of models available to a committee, we should focus on creating district boundaries independently of current divisions. Not only has this approach not been explored to its fullest, but it is not obvious why counties are a good beginning point for a model: Counties are created in the same arbitrary way as districts, so they might also contain biases, since counties are typically not much smaller than districts. Many of the division dependent models end up relaxing their boundaries from county lines in order to maintain equal populations, which makes the initial assumption of using county divisions useless, and also allows for gerrymandering if this relaxation method is not well regulated. Treating the state as continuous (i.e. without preexisting divisions) does not lead to any specific type of approach. It gives us a lot of freedom, but at the same time we can impose more conditions. If the Forrest and Hale et.al. methods are any indication, we should focus on keeping cities within districts and introduce geographical considerations. (Note that these conditions do not have to be considered if we were to treat the problem discretely because current divisions, like counties, are probably dependent on prominent geographical features.) Goal: Create a method for redistricting a state by treating the state continu￾ously. We require the final districts to contain equal populations and be contiguous. Additionally, the districts should be as simple as possible (see §2 for a definition of simple) and optimally take into account important geographical features of the state. 2 Notation and Definitions • contiguous: A set R is contiguous if it is pathwise-connected. • compactness: We would like the definition of compactness to be intuitive. One way to look at compactness is the ratio of the area of a bounded region to the square of its perimeter. In other words CR = AR p2 R = 1 4π Q where CR is the compactness of region R, AR is the area, pR is the perimeter and Q is the isoperimetric quotient. We do not explicitely use this equation, but we do keep this idea in mind when we evaluate our model. • simple: Simple regions are compact and convex. Note that this describes a relative quality, so we can compare regions by their simplicity