Today a friend sent me the image you see below. At first glance I thought it might have to do with a terrible chemical spill, or perhaps a new kind of Rorschach test for geographers. But the text he sent along with it, an opinion piece by Thomas Friedman, brought everything into instant focus. The subject was politics. Or, more precisely, how to make democracy work by outlawing these bizarre and diseased shapes.
Fig. 1. Fourth Congressional District, Illinois
The image depicts the Fourth Congressional District of Illinois, not the subject of a chemical or psychological investigation. Why the strange shape? Why not a square or something chunky? Why these amoeba-like projections of political protoplasm? Well, once you think “amoeba,” you find the answer: the political protoplasm is protecting itself by a process known as “redistricting.” The result is a “gerrymandered district.”
The process works this way. An election is held. If the protoplasm wins an election, it gets to add surrounding territory to itself where there is food, that is, territory with voters that like it. This is called “ingestion.” Moreover, when the protoplasm covers dangerous territory, where voters live who do not like it, it can withdraw. This is called “rejection.” It is as distasteful and hurtful a process as it sounds. After each election, the amoeba changes shape, adding territory here, withdawing from territory there. Each time it wins, it becomes stronger, and so more likely to win again.
If this reminds you of the Blob, the movie, it should. And if it scares you, it should. Driven by its instinct to survive, our shape-shifting protoplasmic political animal can assume quite odd forms, with tentacles, antennae, and all manner of misbegotten pseudopods: protuberances, protrusions, and projections.
Fig. 2. Amoeba, from Mr. Lazaroff's Biology Class
The consequences of gerrymandering are well-known: a political ecology low in species diversity, a dysfunctional democracy where the nutcases rule. Hmmm…, this sounds familiar!
You ask, “fine, but where, dear blogger, is the advertised geometry? This is biology!” Well, it is simple. Those protuberant projections of political protoplasm are the problem. They must be eliminated by means seen to be fair by all: left, right, center, up and down. This is where the mathematics comes in. Mathematics gives a way of rating Congressional districts, a way of computing a numerical score of fairness, of lack of willful bias based solely on shape. The score is a number between 0 and 100, just like the test scores we used to get in school. A score of 100 is perfect; a score of 70 is OK, but anything too low we deem unacceptable.
To apply geometry to politics, we form a citizens’ commission, or whatever the law designates, to choose the district boundaries. They may do as they wish, subject to one proviso. The choice must conform, just as ours do, to a Higher Law. The Higher Law says: the score cannot too small, say, smaller than 20.
Fig. 3. Amoeba, score = 22
The rule for computing the score not a secret — there are ultimately no secrets in politics. But we want to keep up the suspense and so will divulge it only at the end of this article. A reward for the diligent reader!
Pending this revelation, let’s look at examples. Consider first a circle — an excellent shape for a Congressional district. Large or small, it is rated 100. Consider next a square, any square, small, medium, large, or humongous. Also a good district shape. It has has a rating of 78, regardless of size. What about the Fourth Congressional District? Well, I am working on that one. But preliminary estimates tell me that its score is veeery small. I did, however, do the computation — an estimate, really — for the amoeba, leaving out the vacuoles and two of the overlapping pseudopods. According the the worksheet (the figure with the grid, click to enlarge), the score is about 22. Much lower than the circle or square! Barely a pass!! In general, the more protuberances, protrusions, and projections, the skinnier the shape is overall, the smaller is its score
By Our Law, the amoeba-district is Approved. As for the Fourth Congressional District, No Way!
I very much like Thomas Friedman’s opinion piece. Let the center rise. Let it hold — indeed, let it hold sway! Let the nutcases fall by the wayside, to the left and the right of the road, where they can prattle, babble, and gesticulate to their heart’s content. But let’s bring some reason to public life, for God’s sake!
To end this essay we reveal the secret of computing the score. Take a shape in the plane, any shape. Find its perimeter, P. Find its area, A. Divide A by P-squared. Multiply by four times Π = 4 times 3.1416 = 12.5664. Then multiply by 100. The result is the score. A simple, apolitical formula. Let’s use it! Write your congressperson!!
[For more mathematics, you can also take a look at these highly recommended articles by Steve Strogatz]
Other links: Amoeba in Camille’s Sketchbook
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