by Jonathan Kujawa
On a Friday night two weeks ago the US Supreme Court quietly announced they wouldn't hear a challenge to a lower court's order that North Carolina should redraw it's congressional districts. There wasn't much point in hearing the case. With Scalia gone the Court was widely expected to vote 4-4 on the case and ties go to whomever won the previous round. The case revolved around North Carolina's Districts 1 and 12. The lower court ruled that they were gerrymandered to pack more black voters into these districts. While gerrymandering is now a worldwide sensation, the US invented it and are true masters of the art.
Even a glance at NC District 12 makes it obvious that some fishy business is at work. A bow-legged cowboy could walk the length of NC 12 without touching either side:
North Carolina is hardly the only offender. Illinois District 4 is sometimes called The Earmuffs thanks to its bizarre shape.
It's one thing to eyeball a district and guess that the ghost of Governor Elbridge Gerry was at work, but I'd rather quantify somehow that a district is overly gnarled. As I pondered this last week I noticed that, in contrast to a nice, compact geometric shape like a circle or square, the long, twisted shape of NC 12 and IL 4 has a lot of perimeter for the given area. A reasonable measure of the disfigurement of a shape is the ratio of perimeter to area. In fact, for reasons we'll discuss in a moment, we want the ratio of the square of the perimeter to the area. Let's call this the distortion of the shape [1]. In formula form distortion is given by:
The reason we use the square of perimeter is that we want distortion to be what a geometer would call “unitless”. We want the distortion of a shape to be the same regardless of which units we use when measuring. Meters, miles, furlongs, and light years should all be equally good when calculating distortion.
To put it another way, we want distortion to measure an intrinsic quality of our shape and not depend on the scale we use. It shouldn't matter if we draw a square big or small when we calculate its distortion. Indeed, a square with side length x has perimeter 4x and area x2. The ratio of these two would be 4/x and would get ever smaller as the square grows in size. It would also vary if were to switch from meters to centimeters to millimeters. On the other hand, the distortion of every single square in the world is 16.