Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and EverythingElse

by Jordan Ellenberg

What Lincoln had taken from Euclid (or what, already existing in Lincoln, harmonized with what he found in Euclid) was integrity, the principle that one does not say a thing unless one has justified, fair and square, that one has the right to say it. Geometry is a form of honesty.

He was startled by how easy he found it all now, and ascribed this to the fact that nobody was making him do it: “education must be chiefly self-education, during or after school, or it will never approach completion at all.” On this point, no one who teaches math can really disagree. I wish my explanations at the blackboard were so magisterially clear and my path through the material so efficient and direct that students could walk out of their fifty-minute hour with me in a condition of complete mastery. Not so. Education, as Ross understood, is self-education. Our job as teachers is, yes, to explain—but our job is just as much a species of marketing. We have to sell students on the idea that it’s worth it to spend the time outside of class it takes to truly learn this stuff.

[T]he thoughts of pure mathematics are true, not approximate or doubtful; they may not be the most interesting or important of God’s thoughts, but they are the only ones that we know exactly.

Alexander Hamilton complained in Federalist #22: Every idea of proportion and every rule of fair representation conspire to condemn a principle, which gives to Rhode Island an equal weight in the scale of power with Massachusetts, or Connecticut, or New York; and to Delaware an equal voice in the national deliberations with Pennsylvania, or Virginia, or North Carolina. Its operation contradicts the fundamental maxim of republican government, which requires that the sense of the majority should prevail. . . . It may happen that this majority of States is a small minority of the people of America; and two thirds of the people of America could not long be persuaded, upon the credit of artificial distinctions and syllogistic subtleties, to submit their interests to the management and disposal of one third.*

History has dehypotheticalized Hamilton’s angry worry; the twenty-six smallest states, whose fifty-two representatives make up a majority of the Senate, speak for just 18% of the population.*

It’s not just the Senate. Each state, however small, gets at least three votes in the Electoral College, which ultimately decides the presidency. Wyoming’s 579,000 people—about as many as live in greater Chattanooga—share three electoral votes among them, which means each electoral vote represents about 193,000 Wyomingites. California has almost 40 million people, so each of its fifty-five electoral votes stands for more than 700,000 Californians.

But many of the justices persisted in treating proportional representation as the issue they were being called on to decide. Neil Gorsuch worried that, if he decided against North Carolina, “We’re going to have to, as part of our mandatory jurisdiction, in every single redistricting case, look at the evidence to see why there was a deviation from the norm of proportional representation. That’s—that’s—that’s the ask?”

Justice Gorsuch: From—how much breathing room, from what standard? And isn’t the—isn’t the answer that you just—I understand you don’t want to give it, but isn’t the real answer here breathing room from proportional representation up to maybe 7 percent? Ms. Riggs: No.

And Gorsuch, Samuel Alito, and to some extent Brett Kavanaugh, with help from Paul Clement, collaborate on building a fictional version of the case in which the plaintiffs are asking the court to impose some form of proportional representation on the states.