Topology Question

Is there some sort of metric for the complexity of a boundary like this one for PA Distric 7, among those invalidated in PA (in a decision the Supreme Court today refused to review)?

I keep wondering whether there is an objective standard we can set, rather just the sort of "I know it when I see it" one that everyone seems to use.

One way I can imagine testing is to do it Monte Carlo style by drawing a series of lines from one random point in the shape to another random point in the shape and calculating which percentage of the lines cross the boundary at least once.  That metric for a circle or a rectangle would be zero, but would be very high for this shape.



  1. davesmith001:

    This is not an official measure, but I would start by looking at the ratio of the perimeter to the area.

  2. Indy:

    There are several ways to analyze the character of the boundary of an area, but the most common ones that comes up in discussions of Gerrymandering are various ways to measure the boundary's "compactness" and "convexity". See this article for a good intro.

  3. Chris:

    "One way I can imagine testing is to do it Monte Carlo style by drawing a
    series of lines from one random point in the shape to another random
    point in the shape and calculating which percentage of the lines cross
    the boundary at least once."
    Such a line has to cross the boundary an even number of times, or the two points are not in the shape.

  4. Solomon Foster:

    I like the idea, but how do you calculate EITHER of those values in these cases? Calculating the area looks about as hard as Warren's Monte Carlo proposal.

    I guess you could look at that graphic and count the red pixels for the perimeter and the white for the area?

  5. Solomon Foster:

    Now I'm wondering how these districts are officially defined... property parcels that are in the district? Survey lines around the outside?

  6. Chris:

    Courts have never been on the side of random "fairness" in drawing election boundaries. At times they have rejected fairness in order to guarantee minorities get reperesentatives. Mathematical fairness would be easy to achieve. If a minority were 40% of the population, fair boundaries might give no minority representatives, if they were only 40% of the voters in each district (and voters vote by majority/minority status). But if you concentrate minorities, they might elect 40% of the respresentatives. When you have multiple ways to count minorities (race, gender, age, etc.) the problem becomes unsolvable.

  7. Sam P:

    Your idea suggested triangulation (or tesselation) decomposing the area into a set of non-overlapping triangles and counting up the number of triangles, though there isn't a unique triangulation (unless the area originally is a triangle).

  8. roystgnr:

    Topologically that district is still homeomorphic to a circle.

    Geometrically, you have the usual trouble with standards: too many, not too few.

    Reock, Convex Hull, Polsby-Popper, Boyce-Clark, Length/Width, X-Symmetry, Significant Corners, Chambers-Miller Bizarreness...

  9. roystgnr:

    Oh, and I forgot the most important point: considering geometry alone isn't nearly enough to define a "fair" redistricting!

    Imagine the state of Red, whose one major city of Blue contains a third of its population, and which will have 3 representatives. You could divide Blue evenly between all three districts, which makes each district as representative of the whole state population as possible (and puts Red politicians in all 3 seats), or you could make Blue a single district and divide the rest of the state evenly (which gives us 2 Red and 1 Blue politician), and either division can be done with a very compact geometry. If there's a strong urban/rural political divide (such as, for example, almost everywhere in history), then the results will greatly depend on which division is chosen.

  10. mharris717:

    There are several objective measures of district shapes in this article:

  11. Mr. Generic:

    I'm not sure what's going on there, because the 16th, 6th, and 7th are all Lean Republican and all similar in demographics.

  12. Ken in NH:

    How about we fix Congress instead? The cap of 435 is arbitrary and helps representatives concentrate their power. It ensures that Congress remains a full time occupation filled with professional politicians instead of citizen-statesmen hoped for by the founders. It allows individual representatives to ignore the concerns of hundreds of thousands of their constituents and still get elected every two years. And it encourages the type of games you see with gerrymandering.

  13. Mark Roberts:

    The Polsby-Popper Test has been used in gerrymandering cases in the past. It's a mathematical method for calculating compactness. 0 for complete lack of compactness and 1 for maximum compactness.

  14. Matthew Slyfield:

    "Geometrically, you have the usual trouble with standards: too many, not too few."

    Easy to fix, pick one.

  15. Matthew Slyfield:

    "Such a line has to cross the boundary an even number of times, or the two points are not both in the shape."

    That's not strictly true when you allow more complex shapes than basic polygons. Consider a "bow tie" shape consisting of two equilateral triangles touching at one apex. If you draw a line from the center of one far face, through the connecting apex to the far face, it will intersect the boundary exactly 3 times.

  16. NL7:

    I've always thought the best method was to just say that other divisions, like counties, townships, cities, etc., cannot be split between districts except on the margin. So an electoral district has to fill up every township and every county before its boundaries can extend into a new township or a new county. For big cities, this would probably have to include wards. And maybe we could even do it with reference to zip codes.

    There's probably still opportunity for gaming by picking entire towns and counties based on partisan affiliation. And it would might have to have a greater tolerance for imbalance between district population, just to make the districts work geographically.

    But if the districts have to be assembled in reference to divisions that exist for real purposes other than just electoral districts, it will be harder for politicians to game. I think it's unlikely that local government boundaries will be redrawn every decade, just for partisan gain. And if we do it with reference to zip codes, that's harder to change (though it injects politics into zip codes, which is bad).

    This has the benefit of not requiring any math and being pretty conceptually easy to check. "It contains only a portion of this town. Does it have any other partial towns? No? Then that's okay."

  17. MSO:

    It is silly to argue that one representative can represent 600,000 people in any meaningful way; courts that jump in to fix things is absurd. Realistically, each state should elect its representatives at large with each voter voting for one representative. In this way, voters can coalesce around common interests instead of a meaningless geometric association.

  18. mlhouse:

    Just let the legislatures draw whatever they want. If you don't like the results, vote them out.

    The other aspect is that gerrymandering is not "free". You improve competition in one district only by losing competitiveness in another.

    I once worked for a state legislative campaign. The Democrats had just redrawn the districts. THey took one township and small town, the most GOP areas in the county, and attached them to the county to the north which was much more conservative than our county. They then took several townships from not one, but two other counties and attached them to our district to add more Democratic voters. My candidate lost by less than 100 votes and with the one township from our home county he would have won.

    But the Democrats decided they would rather win one district and lose one district by a bigger margin, than lose both. They calculated the results down to the 100 vote level.

    Elections have consequences.

  19. marque2:

    It is interesting all the things that are taken for granted, when Democrates are in control and only become a problem when GOP suddenly takes control and does the same thing. And of course the lawsuit is in a district controlled by GOP as well. Hmm very suspicious.

    Here is an example Luis Gutiérrez, the left wing Democrat who ran out of the SOTU when they started chanting USA - from the very liberal Democrat controlled Illinois has a district which looks at least as bad as the one you highlight. Why isn't the press, or activist groups going after Illinois district 4? (see link to map below)'s_4th_congressional_district#/media/File:Illinois_US_Congressional_District_4_(since_2013).tif

  20. options:

    Start with a completely uniform n x m grid and only allow changes if both parties agree.

    Get rid of districts and allow voters in any state to elect candidates based on proportional voting.

    Get rid of representational democracy and embrace party votes via overall popular vote representation.

  21. Akiva:

    The "metric for the complexity of a boundary" is called the fractal dimension.

  22. Mark:

    You could divide the circumference by the area. The more complex the boundary, the larger the ratio. The closer to a circle, the smaller the ratio.

  23. jdgalt:

    I would measure it by drawing the smallest convex shape that includes an entire district, and dividing its area (excluding any parts outside the state) by the area of the district. Call this the district's red ratio. Then if the average red ratio of all districts > 1.25, I would tend to assume gerrymandering.

  24. Matthew Slyfield:

    It's doable, The area of complex polygons can be calculated, but you need actual measurements for the boundary line segments not just a low res graphic. This is something cartographers do all the time.

  25. kidmugsy:

    How would you allow for any physical features that might seem to be naturally candidates for inclusion in a boundary - rivers, watersheds, bogs, forests, canyons, railway lines, major roads?

  26. Mercury:

    Hang in there.
    When the open borders crowd gets their way Gerrymandering will go away - there will be no need for it.

    By the same token, all official opposition to voter ID laws will melt away as well, in fact it will be effectively illegal not to have your phone on your person and turned on at all time.

  27. Brad Richards:

    As others have written, it would be fairly easy to impose requirements based on the ratio of the circumference to the area. But a far, far better solution would be to eliminate the districts entirely and use proportional voting: Each party gets a number of representatives proportional to its share of the vote.

    Example: In a state with 10 representatives, R's get 50% of the vote, the D's get 40%, and SomeOtherParty gets 10%. Then the parties get 5, 4 and 1 representatives respectively. Which immediately shows why the established parties would fight this tooth-and-nail: This is a system which allows smaller parties to get into government, grow, and ultimately displace existing parties. This is the (very successful) basis of multi-party politics in all of Europe. The fact that it completely eliminates gerrymandering is just a bonus.

  28. John Dewey:

    Although this may make sense, I cannot imagine it ever being implemented. You could argue that it would be the political parties fighting it. But conservative voters in a conservative state would fight it. Liberal voters in a liberal state would fight it.

    IMO, political parties pretty much represent the attitudes of their constituents on important issues. Yes, political parties do cater to donors in order to get funds. But the opposing positions of the parties on many important issues reflect not so much the priorities of major donors but rather the priorities of the divided electorate. And those polarized voters would be foolish to give up an advantage to enable more proportionate reresentation.

  29. Dan Wendlick:

    My take on this is to make districting non-geographic. Either assign voters to districts randomly, or less practically but more intriguingly, by subscription. If a party wants to try to win every district 51-49, it could encourage it's voters to spread, or if a third party wanted to magnify its impact, it could encourage its voters to pack into a single district. You'd probably need to provide a three level preference and then random assignment to keep the districts roughly equal in size, but on an every 10-year basis it could work logistically, at least no les complicated than the current redistricting processes. As to constitutionality, Article I and the 14th Amendment should not be violated by such a system, as the voters themselves would be making the districting decisions, and there is no requirement for geographic districting. (just reasonable equal in size and non-discriminatory in makeup). Federal statutes might need to be changed, but that's the art of the possible.

  30. OpportunityCost:

    Resistance from the parties themselves notwithstanding, I see two reasons folks in the US would object:
    1. They want to vote for a particular person, not just for the Nth person on their party's list.
    2. They want a representative to be tied to their geographic district so as to deliver pork to them, not the state as a whole.

  31. Bram:

    Yes - following existing borders would be a good guideline.

  32. Bram:

    The tradition Dem drawn map has one conceded district out in the boonies where they jam in as many Republicans as possible and they barely bother putting up candidates. Then draw the rest of the districts with an urban slice of reliable Democrats that catches a swath of suburbs that can't ever outvote the Dem inner core.

    The tradition GOP map is the opposite. Jam all the urban Dems into the conceded district and draw the rest with suburban into rural areas.

  33. Chris:

    Or let every person select which district they want to be in regardless of their residence. Why should physical proximity be the deciding factor? And who cares about "one man one vote" if the person chooses to be in a district with a lot of others? Publish a list of districts, and a running total of those who've signed up for it. Let each voter choose which to be in.

  34. Solomon Foster:

    I actually know the math, if given a well defined polygonal boundary with XY coordinates. (Indeed, more than one way of doing the math.) But it's pretty fugly calculations on something with that irregular a shape.

    Huh. Actually, that thought leads to much simpler ideas. Like, every line between two boundary points must be entirely on or within the boundary. (Think this may be equivalent to a convex polygon?)

  35. Solomon Foster:

    In fact, I remember an article in National Review several decades ago where the Republicans were complaining about gerrymandering and the Democrats were explainly it was absolutely necessary to ensure that African-Americans were represented fairly. (Like a situation which was 60% white 40% AA with two districts split that way would most likely elect two whites, while if you split the region up so one district was 80% AA 20% white and the other was 100% white you'd get very different results.)

  36. Solomon Foster:

    Right, but the even number can be zero, which in fact is the preferred situation in this case.

  37. Don Malpas:

    If you could draw district lines, without any interference, how would you do it?

  38. Dan Wendlick:

    You can get the area numerically using a graphics program, as you could get the perimeter by counting pixels. I'd also try calculating the standard deviation of the distance from each perimeter pixel to the centroid of the enclosed area. Since the borders are drawn to enclose voters, you really don't get into the infinitely-small cases of fractal dimension.

  39. hcunn:

    At present, the Republicans are *slightly* worse offenders. Democrats unilaterally disarmed in California when voters in 2010 chose to put redistricting in the hands of citizens' commissions. Left-leaning activists have gained some influence in these commissions, but the gerrymandering is less egregious than it used to be.

  40. hcunn:

    Let the two leading parties submit competing redistricting plans to the voters. The minority party is likely to come up with an attractive logical plan, since that is the only way they would have a chance to win.

  41. MSO:

    "The majority of the ignorant is invincible, however, the tyranny of their reign is mitigated by their lack of consistency"

    Your plan would eliminate the inconsistency that limits the tyranny of the ignorant.

  42. marque2:

    Ha - California has the most gerrymandered districts it ever had. Yes we voted for a commissan of laymen to design the districts but they had no idea what to do - being laymen and all - and we're a bit lazy - being laymen and all- so they basically took a proposal by a group with a neural sounding name which was funded by the Democrat party. Once it was in place - Dems we're able to.get a super majority of state delegates because the GOP was drawn out of the Democrat party plan. Note maybe in 2021 the GOP might get a bit clever and send their own proposal to this team.

  43. wreckinball:

    That's the winner.

  44. WillusM:

    This was first my thought as well. You may also want to normalize by the perimeter/area ratio of the relevant zip codes (or similar) as well; to avoid terrain skewing the result.