Math Digests June 2024

Opinion | Using Math to Analyze the Supreme Court Reveals an Intriguing Pattern

Politico, June 2, 2024.

Much is made of the conservative-liberal divide on the Supreme Court. But in this article, legal commentator Sarah Isgur and economist Dean Jens argue — using math! — that the conservative group (currently comprising six justices) should be viewed as two distinct groups. Isgur and Jens argue that the main difference between these groups is institutionalism and show some of the numbers that they used in their analysis.

Classroom Activities: linear algebra, statistics

  • (Mid level) Read the article, then navigate to the graphic titled “Three Groupings on the Supreme Court Show Through in the 2023 Term.”
    • Describe, in your own words, the meaning of the number in top left corner of the graphic.
    • Examine the row representing Justice Sotomayor. Calculate the mean and variance of the data in this row. Calculate the mean and variance of the data in Sotomayor’s row within each of the clusters proposed by Isgur and Jens (this will give a different mean and variance for each of the three clusters.)
    • Randomly select two other justices besides Sotomayor and repeat the exercise on the data from their rows. Do your results support the clustering proposed by Isgur and Jens? Why or why not? As a class, vote on whether you agree with the 3-3-3 clustering.
  • (Mid level, Linear algebra) Isgur and Jens used a technique called principal component analysis. Principal component analysis involves finding eigenvalues and eigenvectors of a matrix of data.
    • Consider a hypothetical court that has three justices. Let $\mathbf{j}_i$ be the vector of how Justice $i$ voted on five separate cases, with each vote being either “yes” or “no.” Suppose
      $$\mathbf{j}_1 = \begin{bmatrix} \text{yes} \\ \text{yes} \\ \text{no}  \\ \text{no}  \\ \text{yes}  \end{bmatrix},\; \mathbf{j}_2 = \begin{bmatrix} \text{yes} \\ \text{no}  \\ \text{no}  \\ \text{no}  \\ \text{yes}  \end{bmatrix},\; \mathbf{j}_3 = \begin{bmatrix} \text{no} \\ \text{no}  \\ \text{yes}  \\ \text{yes}  \\ \text{no}  \end{bmatrix} $$
      Find the matrix whose $(i,j)$-th entry is the percentage of the time that Justice $i$ and Justice $j$ agree with one another. (The analog to the plot shown in the article.) Find the eigenvalues and eigenvectors of $M$.

—Leila Sloman 

Pokémon and Geometric Distributions

Numberphile, June 13, 2024.

Suppose that if you roll the number $N$ on a six-sided die, you win $N$ dollars. How much money would you expect to earn in a single roll? This scenario is an example of “expected value” in statistics. In this Numberphile video, we learn that expected value equals the sum of each outcome multiplied by its probability of occurring. This statistical method is useful in games such as Pokémon, where we can calculate the expected number of encounters it takes to “catch ‘em all” using geometric series.

Classroom Activities: expected value, geometric distributions

  • (All levels) Based on the rules outlined above, would you expect to win or lose money with the dice-rolling game if playing the game cost $30 for 9 rolls?
    • How much money would you win or lose? Show your work.
  • (Mid level) Watch the video. Explain in your own words:
    • Why does the mathematician in the video use an “infinite sum” when calculating expected value?
    • What is a geometric distribution?
  • (Mid level) How many encounters would it take to catch 5 different Pokémon? Assume that each Pokémon has equal likelihood of appearing. Show each step of your work.
  • (High level) Write out each step of the generalized example from the video of expected encounters for any number n of Pokémon. Explain why the mathematician compares the harmonic sequence to $1/x$.
    • Why does integrating $1/x$ give a useful approximation for expected value?
    • How many encounters do you expect it would take to catch 500 Pokémon?

—Max Levy

How we can understand our universe through math

Astronomy, June 1, 2024.

In the early 20th century, Albert Einstein presented a new theory of gravity that applied not just to our planet or our solar system, but to the universe. Einstein’s theory included mathematical terms for the familiar attractive forces of gravity, as well as for repulsive forces that prevented his mathematical universe from imploding due to gravity. This repulsion was represented by a cosmological constant, lambda. Scientists didn’t know what lambda represented in the physical world until many years later. “For physicists and mathematicians who work with these equations today, lambda represents dark energy,” wrote Steve Nadis and Shing-Tung Yau for Astronomy. In this article, Nadis and Yau describe how Einstein’s theory was connected to dark energy, as well as how mathematics predicted other important physics discoveries like black holes.

Classroom Activities: mathematical physics

—Max Levy

Japan’s birth rate falls to a record low as the number of marriages also drops

AP News, June 5, 2024.

Japan’s population is shrinking. As the population of Japan ages, young people are starting fewer and smaller families. Women in Japan average about 1.2 babies each in their lifetime, based on recent data — about 25% lower than the U.S. birth rate and about 45% below the global average. The new data makes 2023 the eighth consecutive year that Japan’s birth rate reached a new low. In this article for AP News, Mari Yamaguchi writes about the implications of low birth-rates on future economic predictions.

Classroom Activities: data analysis, compounding

  • (All levels) Based only on the numbers above, calculate the U.S. and global birth rates.
  • (Mid level) Read this MathWorld resource to learn more about compounding. Answer the following questions:
    • If Japan’s birth rate continues to fall 5.6% annually, when will the birth rate fall below 1?
    • By what percent would Japan’s birth rate need to increase annually to catch up to the global average within:
      • 10 years?
      • 5 years?

—Max Levy

University of Glasgow to celebrate 200th anniversary of Lord Kelvin’s birth

STV News, June 24, 2024.

June 26 was Lord Kelvin’s 200th birthday, and in celebration, the university where he spent most of his career launched a two-week exhibit titled “Lord Kelvin: Beyond Absolute Zero.” Two paintings made as part of the bicentennial highlight a project of Kelvin’s that was decidedly mathematical. In 1887, Kelvin proposed a shape which tiles all of three-dimensional space while keeping its surface area as small as possible. His project was misguided — he wanted to model an “ether” through which light could travel, when in fact no such ether exists — but he made long-lasting progress on the underlying math problem. It took 106 years to find a shape that both fills 3D space and has a smaller surface area than Kelvin’s.

Classroom Activities: tessellation, geometry

  • (All levels) A shape is called space-filling if copies of it can be packed together with no gaps or spaces in between the copies. For example, cubes are space-filling: You can stack them on top of and next to one another with no gaps. Are the following shapes space-filling?
    • Rectangular prisms

      Rectangular prism, drawn by Leila Sloman in TikZ.
    • Spheres
    • Equilateral tetrahedrons (shapes where each of 4 sides is an equilateral triangle).
      Equilateral tetrahedron, drawn by Leila Sloman in TikZ.

      (Note for teachers: This example may be hard to visualize without physical tetrahedrons for students to play with.)

  • (All levels) Kelvin thought his shape had the smallest surface area possible for a space-filling shape. Here is a list of space-filling polyhedra. Calculate the surface area of:
    • A regular hexagonal prism (all edge lengths are the same) of volume 1.
    • A cube of volume 1.
    • Using formulas (9) and (10) here, calculate the surface area of a truncated octahedron whose volume is 1. Kelvin’s shape was a curved version of this shape.

—Leila Sloman 

Some more of this month’s math headlines