When Math Gets Impossibly Hard {article w/surprising connection to Gerrymandering ! } (Quanta) [View all]

David S. Richeson
Contributing Columnist
September 14, 2020

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People use the term “impossible” in a variety of ways. It can describe things that are merely improbable, like finding identical decks of shuffled cards. It can describe tasks that are practically impossible due to a lack of time, space or resources, such as copying all the books in the Library of Congress in longhand. Devices like perpetual-motion machines are physically impossible because their existence would contradict our understanding of physics.

Mathematical impossibility is different. We begin with unambiguous assumptions and use mathematical reasoning and logic to conclude that some outcome is impossible. No amount of luck, persistence, time or skill will make the task possible. The history of mathematics is rich in proofs of impossibility. Many are among the most celebrated results in mathematics. But it was not always so.
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Many states require that districts be “compact,” a term with no fixed mathematical definition. In 1991, Daniel Polsby and Robert Popper proposed 4πA/P² as a way to measure the compactness of a district with area A and perimeter P. Values range from 1, for a circular district, to close to zero, for misshapen districts with long perimeters.

Meanwhile, Nicholas Stephanopoulos and Eric McGhee introduced the “efficiency gap” in 2014 as a measure of the political fairness of a redistricting plan. Two gerrymandering strategies are to ensure that the opposition party stays below the 50% threshold in districts (called cracking), or near the 100% level (stacking). Either tactic forces the other party to waste votes on losing candidates or on winning candidates who don’t need the votes. The efficiency gap captures the relative numbers of wasted votes.
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more: https://www.quantamagazine.org/when-math-gets-impossibly-hard-20200914/


Not much more on Gerrymandering, but lots of interesting background.