You have probably not thought much about geometry since your tenth-grade math class where you learned how to calculate the area of circle or a triangle. Have you retained anything else? Or even that?

Well into the 19th century, you would not be considered to be an educated person unless you were familiar with Elements by Euclid, a Greek mathematician from the third century BCE. This was not so much because most people found the details of geometry useful in daily life, rather because of the style of logic Euclid laid out in proving his classic theorems of geometry. That method of formal proof, from defining postulates to step-by-step moving them into “Q.E.D.” proven theorems, is foundational to understanding formal logic. [1]

Geometry is basically the mathematics of Shape, and that is the title of the latest book by University of Wisconsin mathematics professor Jordan Ellenberg. The book’s subtitle is “The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else” (Penguin Press, 2021). If that sounds a bit hubristic, he is in good company, because geometry literally means “the measurement of Earth.” And you thought it was just something about 𝛑.

My own comfort with geometry is a bit rusty. On my first “real job” in the automobile industry over 50 years ago, I used analytic geometry to program some of the first computer-aided design applications in the business. My task was often to figure out how to rotate and scale two-dimensional and three-dimensional representations of car parts and assembly tools using very expensive early 1970s computer workstations that literally had only 64 thousand bytes of random-access memory to work with. By contrast, my iPhone has 128 billion bytes of memory (I specialize in dead computer languages and obsolete programming skills and am available for hire).

The author’s 1970s-era Control Data-Lundy CAD workstation with 64 kilobytes of computer RAM memory, running an automobile assembly simulation.

This display task is now done in many gradients of color, rather than on my “green screen,” in exquisite detail in every (now digital) cel of every new animation film release. But the math is the same, some matrix algebra I learned in a freshman analytic geometry class in engineering school.

I describe Shape as a kind of “rambling romp” through this neglected discipline. It avoids most of the difficult math by telling historical vignettes that, after seeming to “wander in the wilderness” a bit, take us to practical applications. And they are all tied together by this notion of “measuring shapes” of stuff that you may not have realized were about “shape” at all.

And as for Euclid, we learn that his geometry only works in a flat world (or one nearly so in small distances). You may have learned that the interior angles of a triangle add up to 180 degrees, say an equilateral triangle with three 60-degree corners. But if I plot a course from my current location to the North Pole, turn 90 degrees, go an equal distance south (every direction is south from the North Pole), turn 90 degrees again and return to my home, I would have created a “triangle” on the Earth’s curved surface with three 90-degree corners, totaling 270 degrees. The Earth itself has a “non-Euclidean” geometry.

Written early in the coronavirus pandemic, Ellenberg spends a lot of time developing a view of the pandemic itself as “shape.” To get there we travel through a mathematician’s history of “random walks” through a spatial “network” that mosquitoes exhibit as they spread through an area, a form of probabilistic randomness that is measurable to a large extent by geometers.  Besides giving us the math behind “six degrees of separation from Kevin Bacon” and Burton Malkiel’s classic book on financial portfolio theory, A Random Walk Down Wall Street, [2] the random walk helped us to get a handle on the ways and the speeds by which communicable diseases can spread. Which is, in turn, only one degree of separation from the exponential math behind the fecundity of the coronavirus that I wrote about a lot last year (and its baaaaack!),

Along the way you learn about some forgotten geometers like Russian mathematician Andrey Markov (1856–1922), who developed his probability approach to geometry called Markov chains. Markov first developed this concept to postulate the shape covered by the paths of those aforementioned mosquitoes. It is now used by quantitative marketers to forecast business success, such as restaurants, in drawing customers from the competition (hint: we consumers are a lot like flitting mosquitoes). I have previously used Markov chains in this blog to get a handle on how political parties change their collective positions on various issues (like mosquitoes again).

More recently, I found Markov chains useful as a way of quantifying how people move from the “unvaccinated Covid restaurant” to the “vaccinated restaurant” (or more importantly, why they don’t move). In the scope of Ellenberg’s book, political movements like this one have distinctive “shapes.” Much of the last part of Shape applies geometry to the divvying up a state into legislative and electoral districts.

Mathematicians first turned this “art” into sophisticated math-enabled gerrymandering, where essentially “representatives choose their voters” by reshaping electoral districts, and can thus entrench rule by the minority in states like Ellenburg’s Wisconsin. Despite Wisconsin Democrats consistently pulling voting pluralities in most elections, five of eight U.S. congressional districts are currently held by Republicans, as well as a supermajority 61 out of 99 seats in the State Assembly.

A group of mathematicians, including Ellenberg, submitted an “amicus brief” advocating a more logical and more fair (a fuzzy term, that) approach to redistricting to be applied in two recent Supreme Court cases tackling state gerrymandering, one favoring Republicans and one favoring Democrats. The Supreme Court’s oral arguments got stuck on what Ellenberg describes as a dead-end foray into proportional districting, where, say, the party with 40% of the votes should theoretically get about 40% of the legislative seats.

The author says that only Justice Elena Kagan was able to fully grasp the mathematicians’ brief, which uses both the aforementioned Markov chain and random walk concepts. These probability tools are run through thousands of computer iterations to map out a sampling of the “universe” of potential House of Representatives districts that would fit all existing redistricting criteria, such as being contiguous, respecting established political boundaries, containing a near-equal population, etc.

Source: The Mathematicians’ Brief in Rucho v. Common Cause

That universe of possible qualifying districts plots out in one Democratic-leaning district shown above as the blue shaded area, which approximates a probability distribution. This gives the district a “probability shape” as well as a physical one. Theoretically, the Republican legislature’s redistricting plans for 2012 and 2016, the orange and purple dots above, fit somewhere in the tail of this distribution, but in its tiniest extreme. A judge’s re-drawn plan comes out well within the “near-normal” part of the distribution.

The mathematicians propose this technique to demonstrate that a judge could quite easily determine whether a plan is reasonable or not. The 2012 and 2016 plans would clearly not be reasonable, but you still could craft a Republican-leaning district inside the blue “reasonable” space. It would be, however, much closer to a “swing” district.

Unfortunately, the math-deficient Supreme Court instead decided to “punt” in both of these cases and say that they would not hinder extreme gerrymandering for party dominance purposes. This will not end well; redistricting will remain a battle of the computer models and raw political weight-throwing.

The world, Ellenburg suggests, has many “shapes” that are a stretch for human minds to grasp. The complexity of modern social media networks, for example, have a conceptual multidimensional “space” that can be parsed and simplified into simpler numerical representations with the right geometry. Granted, it is not “as easy as 𝛑.” But perhaps we should have paid more attention in geometry class. A bit of old Euclid’s method of logic would also help.

Notes:

1. Q.E.D. is the abbreviation for the Latin quod erat demonstrandum, which means “what was to be demonstrated.” It is a transliteration of a phrase with which the early Greek mathematicians ended their logical proofs. Portuguese/Dutch philosopher Baruch Spinoza audaciously ended his classic 1677 book Ethics with “Q.E.D.”
2. Malkiel, Burton. A Random Walk Down Wall Street: the Time-Tested Strategy for Successful Investing. W.W. Norton & Company, 2020.

Related posts:

• Kids, can you say Fecundity?
• The math of changing your mind
• Choosing your “Covid restaurant”