I'm asking this on behalf of Zach Weiner (actually it's my own initiative in order to promote this site). Original text is here, and is as follows:
Hey-- This is Zach from SMBC, and I have a math question you may find of interest. I only mention who I am because it relates to the idea.
I had an idea for a comic about gerrymandering. As you may know, gerrymandering is a significant social problem, in that it stifles voters' opinions. So, my idea was this: Why not make a rule that perimeter/area always has to be under a certain value. I figured this would limit how salamander-like the districts could be made. Then, I tried to figure out the math of this on the assumption that it was a simple calculus min/max problem. It seems not to be...
The biggest problem I'm running into is how to formalize the idea of a shape being weird. My intuition tells me that the lower perimeter/area is, the less weird the shape. I.e. a wacky snakey shape designed to get several populations will have a higher perimeter/area than a more reasonable district shape, which should be vaguely rectangular or circular. But, I don't know how to mathematize that. If that could be proved, you could probably figure out a reasonable ratio.