="Neil Sloane" <njasloane@gmail.com> Is there a good measure of how irregular a region of the plane is?
This is an interesting pure math question, which I'll leave t' y'all.
the motivation is to find a measure of how irregular (or gerrymandered) a congressional district is.
This application raises a different and wider topic that begs for better definition. Hopefully my analysis of the underlying problem definition will be sufficiently mathy and fun. If not please ignore with my apologies. Beyond calculating the salamander-nesses of various candidate districtings, we also must establish what the optimization criteria are by which to rank them. To address that we need to ask: just what problem is "boundary delimitation" trying to solve? Traditionally the process appears to involve two distinct operations: 1. Assign each individual to a set, call this their "electoral district" 2. Assign each point in a region to a set, call this its "geo-district" The result of #1 is the legislatively significant one, since electoral districts are allocated representatives to vote and otherwise act on behalf of the entire set. In contrast, #2 seems to be more of a (seemingly antiquated?) hack by which to implement #1, namely by associating geo-districts to electoral districts. The consequences and costs of this hack should be re-evaluated. ======== Take "fairness", for instance: One optimization criteria appears to be that folks want the result of #1 to be "fair". This seems to be interpreted to mean that the distribution of individuals within electoral districts matches that of the whole population (within random statistical sampling variance). In order to implement a "fairness measure" for use as an optimization goal, current practice seems to be to choose some arbitrary set of individual features and measure their uniformity. A perennial source of contention is choosing which features (such as party affiliation, race, language etc) are deemed relevant, and which features are suppressed (such as religion, IQ, favorite sports or drug allergies). This is inevitably a highly political question. There's a bug with representing individuals by groups, because as more and more features are added to ensure uniform representation the subpopulations defined by any given feature vector get smaller and smaller until finally they lose statistical meaning, since every individual is by definition unique once enough dimensions are used to classify them. Reductio ad absurdum (or maybe not) one could argue that, say, a-priori insurance rate settings based on a panel of classifiers are unfairly discriminatory because there is zero historical basis to which any actuarial science can be applied. Hence, because geographical distribution of traits is naturally non-uniform (eg language groups tend to affiliate in neighborhoods because it makes everyday intercourse easier), with enough identity-political balkanization in feature space any geo-districting scheme can be charged as "unfair". -------- So let's ask: why not just RANDOMLY assign people to political districts? By construction this should result in the FAIREST possible uniformity of representation. Geo-districting seems as quaintly dated as gluing colored stamps on paper envelopes and leaving them in boxes outside for liveried postal elves to whisk away. You could use a lottery, or SSN's mod X, or whatever (Kurt Vonnegut amusingly suggested giving everyone an arbitrary middle name, such as "flower" or "whale", so no one would be lonely for want of affiliation). On Election Day everyone in your group, call it District 9 say, would vote, and whoever wins would be District 9's electoral representative. No special interest advocacy, no endless arguing over gerrymandering minutiae, no hard math puzzles about smoothing out geographical subdivisions would be needed. And that would be "fair", right? ======== The problem of course is that random assignment merely results in a "quantized" equivalent of pure democracy, with the same attendant "tyranny of the majority" and other well-known horrors of inequity. People WANT non-uniform concentrations of representation, so that they can form more effective advocacy groups around their pet issues of vital importance to them. Which groups get to exist and prevail over others on irreconcilable differences is the essence of politics. Historically there was more of a tendency for like-minded subgroups to geographically concentrate, either by "right" of prior conquest or from people emigrating away from locations they found less congenial. Thus geo-districting was adopted, and has persisted. It is unsurprising that any such hoary institution will be gamed and gamed again as long as there are interested parties constantly pushing and pulling its details into fractal pretzels by means of incremental legislation. However simply low-pass filtering gerrymandered boundaries doesn't address the complete design problem, which is finding the "dialectic synthesis" between the desire for general fairness versus the necessary inevitability of irreconcilable interests. 'nuff said, I'll leave that puzzle t' y'all too.