Compactness Measures for Legislative Districts

May 4, 2020   /  

Student: Isaac Weiss
Majors: Mathematics, Political Science
Minor: Computer Science
Advisors: Dr. John Ramsay, Dr. Bas van Doorn

Gerrymandering has been around for centuries and the past few decades have seen dramatic changes in the way that it has shaped our democracy. In this project, we will explore how the Supreme Court has shaped how we can utilize tools to help us examine gerrymandering, and, more importantly, we focus on building a compactness measure that will allow us to test constitutionality. We find that a weighted average of nearly six-tenths times a convex hull score plus four-tenths of the Polsby-Popper score should be our measure. We also discover a cut off system for determining constitutionality. Any district scoring less than 0.45, we should say is unconstitutional. We, finally, examine why we may not want to consider compactness measures at all.

Isaac will be online to field comments on May 8:
Noon-2pm EDT (PST 9am-11am, Africa/Europe: early evening)

93 thoughts on “Compactness Measures for Legislative Districts”

  1. Isaac, what a great combination of Math and Political Science. Your question slide made me laugh – how long did you have to search for districts to do that – or did you have to make some of them up?

    1. Karen,

      Thank you so much! There’s actually an entire font dedicated to taking congressional districts that look like letters and having them represent as such. So, it didn’t take to long at all to get that all set up, and they are all 100% real!

  2. Loved working with you and Dr. Ramsay on this project, Isaac. A really nice example of what a double major IS can be!

    1. Dr. van Doorn,

      It was a pleasure having you along for the ride. Thank you so much for all of the help.

  3. Thanks, Isaac, for an impressive presentation of your timely research. In New Hampshire, where I live, partisan Gerrymandering has enabled legislators to draw the lines of voting districts in a way that gives their party a built-in electoral advantage. This process has shaped the political climate for over 30 years and almost entirely benefited Republicans. I, and many others, see this inherent advantage, whereby a dominant political party can perpetuate its influence, as unconstitutional. In 2019, NH’s current Republican Governor Chris Sununu, vetoed bipartisan legislation that would have created an independent redistricting commission to shape the state’s district maps.
    What insight has your research into the mathematical parameters underlying the shape of voting districts given you into the future of the democratic process in the US? Do you consider math as a cornerstone to a more fair-minded and fair-handed democratic process?

    1. Eve,

      Unless we are able to severely limit the ability of state legislatures to draw their own districts, gerrymandering will continue to be a problem, regardless of the political party in charge. Mathematics is one of several tools that we can use to limit partisan attacks on democracy, but ultimately it is not likely to be the most effective at deterrence. The most effective things we can do as citizens is to vote out politicians who stand in our way of good districting reforms, or to do what we can to help put independent redistrict commissions on the ballot as referendums, allowing us to bypass supermajority legislatures.

  4. Hi Isaac, excellent work on this project! My question is what was the inspiration behind this project?

    1. Hi Bilford,

      I’m a double major in Mathematics and Political Science, so I wanted to find an Independent Study that effectively combined the two fields, while still being interesting (i.e. no polling data analysis). I quickly became aquatinted with the work of the Metric Geometry and Gerrymandering Group, who uses mathematics to solve gerrymandering. I was enamored by their work and wanted to use some of my mathematical knowledge to help solve important questions.

  5. I’m always impressed by your enthusiasm for the things about which you’re passionate. This is absolutely fantastic, and I’m grateful to have been able to see you present on it in person in the past.

    1. It’s been a pleasure getting to work with you these last four years! Thanks for stopping by

  6. Very interesting, comprehensive, and engaging. Congrats – well done! Best of luck…

  7. Thoughtful and informative. Great integration of different fields. Viva liberal arts!

    1. Dr. Bonvallet,

      Thank you so much! And thank you for taking the time to listen to my presentation

  8. Exceptional project and exceptional presentation, Isaac! What a great combination of your majors. I have always been amazed by your ability to take complicated math models and explain them in ways that everyone can understand!

    1. Jenni,

      Thank you! Like I say in the presentation, what’s the point in having a good model if no one can understand it. Communication in Mathematics and other disciplines is of great importance.

  9. Critical topic, great presentation!
    Coming from one of the most crazy congressional districts in the country (former PA-6) I am keenly aware of the importance of getting this right. Your work is needed for democracy – best wishes to you.

  10. Critical topic – and what a great presentation!
    Coming from one of the most crazy congressional districts in the country (former PA-6) I am keenly aware of the importance of getting this right. Your work is needed for democracy – best wishes to you.

    1. Julia,

      Thank you so much for taking the time to look at my presentation. Yes, PA-6 is a particularly good example of gerrymandering. Luckily, you all have been able to get new, more competitive maps.

  11. Isaac,

    So happy you sent me your link and presentation. I’m blown away by your presentation style, knowledge, and amazing work. Even though I have very little knowledge of the measures you spoke about, the way you explained many factors help me understand the overall project. I absolutely love your last slide, it suits your personality and love of thinking out of the box! I was going to ask about your inspiration but have read your responses already posted. I’m so excited for you as you walk into your future with this love of learning and sharing what you learn about pure math. I can’t wait to see what you do next.

    1. Lisa,

      Thank you so much! The last slide is built from an actual font that you can download. As such, I had to include it in my IS, and it’s the official font of my title page.

  12. Isaac: This looks like a great project that balances your interests in Political Science and Mathematics very well. I was glad to learn more about your work during this very busy year! Best wishes, and do keep us posted on your accomplishments in the future.

    1. Dr. Lantis,

      Thank you so much! It was truly an honor to be able to get to work so closely with my two favorite departments on campus.

  13. Hello Isaac! Amazing work, as is to be expected! My question is, how often would you consider that the three reasons you list at the end of the presentation actually impact the district lines? Would you consider that reasons such as political interest groups and minority interests justify many non-compact districts, or rather that these reasons are more often the exception than the norm? Thank you, have a wonderful life!

    1. Brandon,

      I know that these things impact district lines quite often. Consider the Illinois-4 from the presentation. This is an example of keeping political and minority interest groups together. It is easily justifiable to make strange looking districts in order to do this. In fact, the Voting Rights Act requires us to do this!

      More often, however, this is an exception and not the norm. Districts in suburban areas can reshape themselves to “pack” groups together to make republican legislators safe from voting behaviors (check out OH-1).

      If you want to read more about racial gerrymandering and how to detect strange districts that deny minority groups representation, check out LULAC v. Perry, and the Gingles Requirements.

  14. Nice job, Isaac! Very accessible video! Told you applied math is the way to go.

    1. But you can’t forget the pure math that was involved in dispersion methods!

  15. Clear and comprehensive as always. You mention that Baker v Carr found that gerrymandering “is not a political question.” What does this mean for trying to solve gerrymandering through legislation? Does it leave gerrymandering entirely in the jurisdiction of the courts?

    1. Cormac,

      Great question! To be a political question means that the courts are unable to make any type of ruling in the matter. Baker v. Carr allowed people to, for the first ever, sue states over the shape of congressional districts. Thus, judicable questions have relief available to plaintiffs in the court.

      One thing I didn’t talk about, is that Baker v. Carr is actually no longer standing precedent. Rucho v. Common Cause (RCC) overturned Baker v. Carr’s most important ruling. RCC states that gerrymandering is too complicated an issue for the Supreme Court, and thus must be a political questions. Justice can only be served by state courts who are equipped to deal with state issues.

      This leaves us with a big problem! States can deliver different amounts of justice in different areas of the country. Thus, as we have seen in other important Supreme Court cases, we may have to deal with the fact that there are different levels of justice applied to different areas of the country.

      But, for now, we will deal with these problems as they come.

  16. Excellent work, Isaac! It was a pleasure getting to know you these last few years, and best wishes for the future.

    “Completely Serious” Question: Could you use non-Lebesgue measurable districts to make copies of states?

    1. Dr. Fox,

      Thank you so much! I really enjoyed getting to work with you in class and in other projects, as well.

      As you know, AOC (Axiom of Choice for non-mathematicians, not congresswoman AOC) allows this. You could take a non-Lebesgue measurable district to make copies of states. I think the better question would be asking what you gain from accidentally (or maybe purposefully) doing so. 50 states is a lot, having 100 would double the size of the senate! More importantly, the work required to make a non-Lebesgue measurable district isn’t worth it, at least in my opinion

      1. I mean, you’d also duplicate the people in the district presumably. So, you could meta-gerrymander by only duplicating the people in the districts that are favorable to your party.

  17. Congratulations on an engaging I. S. presentation. Your enthusiasm for and commitment to the project are obvious. I am so glad that there was an opportunity for you to present it (and me to see it). It was well worth the wait. You did a great job of taking people though the process step-by-step. You took a subject that is a mystery to me (math), combined it with a subject that highly interests me (political science), and produced an accessible, interesting, relevant project. Congrats, again!

    1. Connie,

      Thank you so much for coming to my presentation. It was a pleasure getting to know you these past few years and be in class with you.

  18. What did you use to determine the Y/N values in your Compact column on your “Flagging” slide? An actual compactness measure? or the output of humans?

    1. Frank,

      I used the output of humans to determine Y/N for flagging. I had to cut talking about model validation, as well. Given more time and money, we could probably get a more expansive list of Y/N for more districts, to get a better sense of where a better cut off for constitutionality would be. My model is meant to mimic the “eyeball test” while giving a mathematical reasons for why things look the way they do and why our eyes don’t like them.

  19. Hi Isaac!

    Nice work!

    Given what you’ve learned about combining two disciplines through your IS and just through double majoring, do you feel that it’s important for politicians in positions of power to come from a wide range of disciplines?

    1. Riley,

      YES! We need politicians who have a better understanding of different lives. The more representative congress is, the more that Congress can understand the ongoings of their constituents. Gerrymandering makes it harder for low-income individuals to be successful in being representatives because money determines who gets elected. More wealthy people can forgo a paycheck or two to run for Congress.

      More importantly, having more information means that they’ll be able to take information from a different discipline and solve a problem in another. The Liberal Arts educational model strikes again!

  20. Neat project Isaac and congratulations to you! It is a great combination of your majors.

    You raise some interesting considerations and historical context. It really makes one think about the different criteria for districting and how that has even evolved over time.

    1. Dr. Sobeck,

      Thank you so much for taking the time to come to my presentation!

      The history of districting is fascinating. The first instance of Congressional gerrymandering comes from Patrick Henry (the man behind “give me liberty, or give me death”). He was so opposed to James Madison and his federalist supporters, that he gerrymandered as many anti-federalists into James Madison’s district. An impressive feat considering Congress didn’t even exist yet!

  21. Congratulations Isaac! It was a pleasure working with you on LIRTC, hearing your admissions tours in the Libraries, and attending the MLK Jr. Day Social Justice Dialogue in which you were a panelist. I hope your work helps with redistricting in the future!

    1. Irene,

      Thank you so much! It was a pleasure getting to work with you as well.

  22. Issac,

    This was really interesting! I am always fascinated and impressed by how double majors (particularly those in a hard science like mathematics) execute multi-major IS projects. A couple of questions: (1) How did you create/solicit your sample size (51) to ensure it was representative and valid? and (2) Did you do this analysis for all congressional districts in the US or a selected group that are typically viewed as gerrymandered?

    Be well, safe, and healthy,
    – Alex Jue ’10

    1. Alex,

      To answer your first question, my sample size is not incredibly representative. I focused more on political representation than other types of representation. I still had a wide range of racial, socio-economic, and age representation. As I note in my presentation, political representation is important for political problems like this. If we are to convince the courts that my model is sufficient to determine a constitutional violation, then I must show that my model does not favor any political party.

      Thus, to gain my sample, I did a combination of random sampling, where I asked people walking around campus to take a slightly lengthy survey, and I also visited all of the political groups on campus to ensure better representation of Republicans and Democrats. However, I cannot still ensure that my sample is valid. Future work should be conducted on ensuring that a larger, more representative sample is created.

      To answer the second question, I conducted this work over both a random selection of districts, 435 is a lot to work with, and I also selected some districts which are typically referred to as “highly gerrymandered.” The Illinois-4 is a great example because it looks to be highly gerrymandered, but is actually a constitutionally required district.

  23. Great work Issac! This is super interesting and you explain it really well, Congrats on everything!

    1. Thank you so much! It was a pleasure getting to meet you, and I’m sure we’ll meet again.

  24. Thanks for sharing your research, Isaac. Such a perfect combination of your interests. (To add in another of your Wooster skills, could you give the presentation while walking backwards?)

    1. Might take a little bit of practice, but I bet I could do it! Thank you for coming to my presentation!

  25. Isaac. Great job as always. It has been a delight working with you over the last couple of years. Teaming up with Dr. van Doorn and you for IS has been a big part of this. Congratulations on a stellar four years at Wooster. As good as your dad’s! : )

    1. Dr. Ramsay,

      It was a pleasure to get to work with you, as well! Thank you so much for your advice and helping to ensure that crazy ideas can be placed onto paper in a clear and concise way.

  26. Nice job Issac. Now I understand why it was hard to get my head around your quick explanation of what you were doing. You presented it here in a nice clear, concise, and comprehensible way. Hope your future job prospects match up to your abilities!

    1. Jim,

      Thank you so much! I’m glad that you enjoyed my presentation. I enjoyed getting to know you these last few years, and I will miss our weekly discussions.

  27. Isaac, this is very cool work, and certainly timely, with the 2020 US census in progress. Congratulations on your I.S., and we wish you the best with what comes next!

    1. Dr. Pasteur,

      Thank you so much! It was a pleasure getting to work with the Mathematics department these past four years.

  28. Excellent presentation and such a great way to really intertwine a double major. I am so impressed. It will be interesting to see how this plays out after the 2020 Census.

    I have done simple studies of this idea with my high school seniors. Would it be OK with you to show this in the future as an example of where math can take you in college? (I’m pretty sure one of your parents could send it to me.)


    1. Karen,

      Thank you so much! You are more than welcome to share this with your students to talk about mathematics!

  29. Wonderful job, Isaac! This is definitely a stellar example of an IS that brings together two majors and that is timely and accessible…one we can describe to prospective students for many years to come 🙂

    All the best!

  30. Woah, super interesting! I’m happy you mention that these terms need to be more easily understood for the wider communities the research benefits, not just mathematicians. One question though.

    What is the simplified version of (4x^2y)/(8xy^2)? Please help my final is due in 10 minutes.

    1. Marcel,

      Communication of research is almost, if not more important than the research itself. We cannot convince people how important the research is if we cannot tell them about it in a way that they can understand and conceptualize, regardless of their education.

      To answer your single question, (4x^2y)/(8xy^2)=x/(2y). Good luck on your final!

  31. Isaac,

    What a fun, interesting, and wonderfully interdisciplinary project! I love how you demonstrate the strengths and the limitations of viewing this issue through only one of these fields’ lenses. Bravo!

    1. Dr. Krain,

      Thank you so much! I’ve always been a big proponent of the Liberal Arts educational model. I’m glad that Wooster gave me a chance to demonstrate its effectiveness.

  32. Great work Isaac, you are well-spoken and interesting to listen to as always!!

  33. It was really great, Isaac, to see where your project went after advising you in JR IS! Understanding and improving how we draw congressional districts in this country is a vital question for US politics – and I love how you bring your expertise to thinking about how to do this. Well done and congrats on completing a fantastic double major IS!

    1. Dr. Bos,

      Your advising my JR IS was helpful in ensuring the success of this research! Thank you so much for all of your help.

  34. Congratulations, Isaac! I’m so glad I was able to see the progression of this project. Please stay in touch with us! You will be missed here at Wooster. -J. Bowen

    1. Dr. Bowen,

      Thank you so much! It was an honor getting to work with you and get to know you these past couple of years. Good luck in your new position!

  35. Isaac- Does the Heine-Borel theorem apply in all topological spaces? What a beautiful merging of the two disciplines here! Looks like another fantastic job! Congratulations!! I have enjoyed working with you these past few years and we will miss you next year. Keep in touch!

    1. Dr. Pierce,

      No. The Heine-Borel theorem does not apply in all topological spaces, and as mathematicians, we should be careful to not imply that it does. My comment in the video was more to poke fun at the idea of mathematical compactness. However, even if we do live in a topological space where the Heine-Borel theorem holds, it would still be insufficient to be able to detect unconstitutional gerrymanders.

      Thank you so much for being a fantastic professor, I will certainly miss being at Wooster, and have enjoyed my time there.

  36. Isaac-
    This is a really cool project. It is actually fascinating that both Democrats and Republicans are in such close agreement on what is a compact district. I really liked how you approached building the model based on what people perceive as compact.

    Wish you the best of luck with your future endeavors.

    1. Dr. Ondercin,

      I was elated to see those results! It helps to demonstrate that both political parties have similar views on compactness. If we wish to receive relief from the courts with regards to compactness, we need to show that no model dramatically favors any one political party; we now have the evidence to show that this is the case.

  37. Great presentation, Isaac!
    It was great to work with you for LIRTC in the past couple of years. I wish you the best as you are beginning the next chapter of your life journey!

  38. I really appreciate that you are trying to tackle social issues with mathematical principals. Watching this research evolve over the years has been fascinating and I can’t wait to see what you do with it.

    1. Dr. Weiss,

      Thank you so much for your support over the years. This would not be possible without your help.

  39. Hi Isaac,

    This was a great combination of subjects! I particularly appreciated your insight that creating an approach to districting that is both robust and widely accepted will require bringing together an “eyeball” test and a mathematical measure. You did a wonderful job explaining this, as well. Here’s to good math and to equitable voting systems!!

    Wishing you all the best, Pres. Bolton

    1. President Bolton,

      Thank you so much for stopping by! I’m glad you enjoyed my presentation.

  40. Informative job Isaac,

    As a nurse, I do not often think about political and mathematical areas of study. Your research; however, reminds me of how much influence these areas have on my daily life after all. Love your enthusiasm!

  41. Awesome, awesome work Isaac! Sorry I’m just now getting the chance to check out your presentation. You did a brilliant job presenting – very interesting! Thank you for taking the time to share your IS with us. Very proud of you! Cheers!

    1. Mel,

      Thank you so much for stopping by. It was a pleasure getting to know you these past few years!

  42. Congratulations Isaac for presenting your interdisciplinary work in such an accessible manner! You will make for an excellent teacher.

    1. Dr. Visa,

      Thank you so much for stopping by! And thank you for your kind words.

  43. Great presentation! I miss you. Glad I got to see you during spring break. Best wishes

    1. Lori,

      Thank you for stopping by! I miss seeing you and everyone else from the wellness center. I’m glad that I got to see you all one last time during spring break, as well.

  44. Great job, Isaac! Thank you for shedding light on this important topic. Congratulations and I wish you well! Also, I love the enthusiasm:)

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