Dan Asimov <dasimov@earthlink.net> wrote:

2) refers to the fact that any simply-connected bounded region has a well-defined "eccentricity" ? the eccentricity of the unique best-approximating filled ellipse. Requiring that the eccentricity lie below some maximum value makes the region "sufficiently circular".

I for one have no idea how to find the best-approximating filled ellipse. Anyhow, why not just look at the ratio of the square of the perimeter to the enclosed area? Or maybe the ratio of the square of the perimeter of the convex hull to the original area? But I think you missed my point. My new approach is explicitly borderless, which is something I don't think anyone has looked at before, at least not in this list or anywhere else I've looked. Brent Meeker <meekerdb@verizon.net> wrote:

I don't think bad boundaries is the problem. In many cases geometric proximity isn't a measure of common interests. Sure it is for some local issues like getting pot holes fixed and managing elementary schools. But for Congress critters? Engineers, teachers, investors, doctors,...all living in the same area may have quite different interests at the national level.

You seem to be advocating syndicalism, in which people are grouped by occupation rather than location. That opens even more cans of worms than geography. How large is each occupation? For instance is "math teacher" the same occupation as "geography teacher"? Should "math teacher" be further subdivided into which branch of math they teach? What about people with multiple occupations or no occupation? And why occupation anyway, rather than, say, race, gender, age, net wealth, sexual orientation, IQ score, preferred temperature, urban vs. rural, favorite TV show, or countless other ways to categorize people. Maybe even political affiliation. Personally, I don't think anyone should ever represent a competent adult without the latter's explicit individual consent, revocable at any time for any reason. Just like hiring a lawyer. But I was hoping to discuss math, not politics. Henry Baker <hbaker1@pipeline.com> wrote:

Suppose that one had a very long thin state which forced all of the districts into *line segments* -- i.e., each person lives "closer" (under the chosen metric) to a state border than to any other person.

.... If we have enough population for 3 Representatives, then we need to find 2 dividing lines, which provides for 2 sources of non-uniqueness.

You're not going to address my proposal at all? I think in the general case (e.g. the people are not uniformly or symmetrically distributed) there would always be a unique solution, even in your unlikely case of a one-dimensional state. Similarly if the state was in disconnected or oddly-shaped pieces. Just so long as the state border isn't so convoluted that it becomes meaningless to ask whether any person of non-zero size is inside or outside the state. Mike Speciner <ms@alum.mit.edu> wrote:

So measuring the "goodness" of a districting probably requires various kinds of map data in addition to population data. Of course, if we had a way of secure remote voting, some of these considerations could go away.

Good point. Thanks for recognizing that we don't. As the paper "On Trusting Trust" showed, no computer can be trusted unless you personally designed, built, and programmed it, from the logic gates on up. Of course there could be non-computerized remote voting. Thane Plambeck <tplambeck@gmail.com> wrote:

this is a good resource created (i believe) by moon duchin, a mathematician at tufts ....

https://sites.tufts.edu/gerrymandr/resources/

Thanks. I see that there's no mention of anything like my idea there. Michael Kleber <michael.kleber@gmail.com> wrote:

It's worth remembering that congressional districts *must* be a union of census tracts (since otherwise the decennial "population" is undefined).

Thanks. I had wrongly assumed that they kept track of the home location of each individual. Wikipedia says that census tracts are divided into census block groups, which in turn are divided into census blocks. Whichever is the smallest for which population numbers are available to the public could be used in my suggestion in place of individuals. Of course they'd be weighted in proportion to how many people they contained, and all the people would be assumed to be in the center of the tract, block, or whatever. That would have the advantage that when a person moved, nobody would have to calculate for which district the sum of the squares of the distances from that person's new location to each of the people in the district was smallest, but would have the disadvantage that there are explicit borders. I was hoping to keep it completely borderless, just to see if it was possible, and just to be different.