Whew! Interesting day around here yesterday, no? There’s more controversial topics out there: abortion, health care, gay marriage, Iraq, and a few others. But not many. It’s good for sparking discussion, but I also know that some large (probably majority!) fraction of you would prefer to hear about physics. Which is good because generally I prefer writing physics, and I know that I can get irritated when my otherwise favorite nonpolitical blogs go off on political crusades for causes I dislike. So as always I’ll try to continue to keep the partisan politics to relatively rare and easily skipped posts.

How about some of the *mathematics* of politics instead? I promise it’s not partisan!

As you know, the various states are divided into congressional districts which each have their own representative in congress. These districts are drawn in very arcane and shady ways, with the majority trying to carve out districts in a way to give them the most votes, the minority trying to do the opposite, and politicians of any party pushing those considerations to the side to give *their particular district* the greatest possible incumbency advantage for themselves. Gerrymandering is nothing new, but it’s especially bad these days.

There have been a number of mathematical proposals to fix this and take out any possibility of district manipulation. One of my favorite is the splitline algorithm, which works in the following way:

1. Start with the boundary outline of the state.

2. Let N=A+B where A and B are as nearly equal whole numbers as possible.

(For example, 7=4+3. More precisely, A = ⌈N/2⌉, B=⌊N/2⌋.)3. Among all possible dividing lines that split the state into two parts with population ratio A:B, choose the shortest. (Notes: since the Earth is round, when we say “line” we more precisely mean “great circle.” If there is an exact length-tie for “shortest” then break that tie by using the line closest to North-South orientation, and if it’s still a tie, then use the Westernmost of the tied dividing lines. “Length” means distance between the two furthest-apart points on the line, that both lie within the district being split.)

4.We now have two hemi-states, each to contain a specified number (namely A and B) of districts. Handle them recursively via the same splitting procedure.

My own state is a stellar example of gerrymandering:

Using the splitline algorithm, the districts would be much more fairly and sensibly placed:

The algorithm only takes into account the geometry of the state and the density of the population, and produces a unique result. It does so in a transparent way, and prevents any tweaking by elected officials. Certainly it would immediately result in true contests for many formerly entrenched officeholders. I doubt we’ll ever see anything like this implemented, but it’s an interesting application of mathematics to the process of fair governance even if it remains purely theoretical.