What is the mathematics behind districting?

This Nov.8 in California there is going to be a special election in which one of the ballot initiative (Prop 77) concerns how district lines are drawn. This got me thinking about whether there is any mathematical way of saying one has drawn district lines reasonably? The proposition suggests having three judges look at the map. Supposedly, Ohio is going
to try some scheme where they try to draw district lines so each district is as competitive as possible. I guess part of the issue
is what is being optimized. To me it seems that the whole point of districts is that the representative is representing some geographic
region of people who tend to have frequent interactions with each other
because their homes are close to each other. This is valuable because these people would have similar regional concerns. My guess is that this would lead to districts which are reasonably convex except for where they hit things like mountains or rivers. To keep things simple let's assume there are no such geographic features and that our state is a giant square. We want to have say 100 representatives in the assembly. (In contrast, California has 80 assembly and 40 senators.) Let's also assume that each voter lives in one fixed house and has a gaussian likelihood of interacting with a neighbor at radius r from him. Further assume that each voter except for voting preference is identical. Finally, let's pretend there is a density function d(x,y) which gives for each point on the square the density of voters at that point. How should the districts be drawn to maximize the amount of interactions people have with other people in their district? I have some sorta half-flaky ideas but was curious to see what other people would think up. Also, I would be interested in improving my formulation of the problem.

Chris

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