Math Digests December 2024

The hidden science swirling in “The Starry Night”

The Washington Post

The Van Gogh Gallery’s description of “The Starry Night” begins by calling your attention to Van Gogh’s brush strokes: “For the sky they swirl, each dab of color rolling with the clouds around the stars and moon.” In the last couple of decades, some physicists have wondered whether these swirls correspond to the physical phenomenon of turbulence. In this interactive piece for the Washington Post, Carolyn Y. Johnson tells the story of the attempts to mathematically compare “The Starry Night” to turbulence. To describe turbulence, Johnson turns to a poem by Lewis Fry Richardson: “Big whirls have little whirls that feed on their velocity, and little whirls have lesser whirls and so on to viscosity.”

Classroom Activities: scale, turbulence, fractals

  • (Mid level) Watch this video by Veritasium, up to timestamp 7:39. Reflect:
    • Which characteristics make flow turbulent, rather than laminar? Describe these characteristics in your own words.
    • Overall, what is the difference between turbulent and laminar flow?
    • Come up with three examples where you can see turbulent flow.
  • (Mid level) Read the article. Johnson writes that “Turbulence links together the motion of fluids at different scales of a system.”
    • What do you think this means?
    • How might this apply to the examples she cites of bumpy plane rides and ocean eddies?
  • (High level) The article gives one example of mathematics showing up in fine art through nature. Find a painting that includes a tree, and read this blog post by forest scientist Gabriel Hemery.
    • Count the “Orders” of the tree, and see if Leonardo da Vinci’s rule holds.
    • In her article, Johnson brings up the limitations of mathematically analyzing turbulence through a still painting. In this exercise, you are studying a two-dimensional picture of a tree, while a real tree is three-dimensional. How does this affect how you should interpret your results?
    • Can you think of any other limitations of your study?

—Leila Sloman

Animations of coiled hair for Black film characters improve with new algorithms

The Washington Post, November 30, 2024

It takes a lot of careful physics to bring animated hair to life. Thanks to computer scientists who wanted to see more Black hair textures represented, new algorithms now can simulate the physics of curly and coiled hair. The Washington Post’s Lizette Ortega talked with scholars and artists about how these new computer models recreate phenomena of coiled hair, such as “switchbacks”—when a curl twists into a new direction. “Hair within African diasporic communities has been an important cultural signifier,” said Shelleen Greene, an associate professor of cinema and media studies at UCLA. “It is a way of showing one’s culture, of representing oneself in space.”

Classroom Activities:  curve phenomena, geometry, calculus

  • (All levels) The article describes three “phenomena” of coiled hair.
    • List them and describe them in your own words
    • Sketch a picture based on the descriptions
  • (High level) Watch this Khan Academy video about the math of curvature, and answer the following questions.
    • Describe in your own words the meaning of a “radius of curvature.”
    • If a car has a turning radius of 30 feet, what is the curvature of that turn? (Use correct units.)
    • Which has higher curvature: tightly coiled hair or loose curls?

—Max Levy

Mathematicians have figured out the best sofa shape for moving around

New Scientist, December 9, 2024

In this article, Alex Wilkins covers a new paper that finishes off the so-called “moving sofa problem”—if you want to move a sofa that is as large as possible around a corner, how should the sofa be shaped? (In this problem, the hallway and the sofa are both two-dimensional.) In 1992, a mathematician named Joseph Gerver found a potential solution. In November 2024, Jineon Baek proved Gerver’s solution was right.

Classroom Activities: geometry, calculus

  • (All levels) Read Dan Romik’s blog post about the moving sofa problem.
    • Using Legos, toothpicks, or whatever else you have handy, build a hallway with a 90-degree corner and width 6 inches. Now, cut out the first three shapes using construction paper, rulers, and protractors:
      • Square
      • Semicircle
      • Two quarter-circles connected by a rectangle with a semicircular cutout
    •  By trial and error, figure out what measurements these shapes should have to maximize area, while still fitting around the corner.
  • (All levels) Try some alternative shapes. What happens if you have a rectangular sofa? Is it better for it to be long and thin, or closer to square? A circular sofa? Come up with some shapes on your own. Which measurements for these shapes work best?
  • (High level, Calculus) Consider Hammersley’s shape—the two quarter-circles, connected by a rectangle with a semicircular cutout.
    • Explain why the two quarter-circles should have radius 1.
    • Calculate the area when the semicircular cutout has radius r.
    • Show that r must be between 0 and 1.
    • Prove that the area is maximized when r is $2/\pi$.
  • (Mid level) In real life, sofas and hallways are three-dimensional. How does this affect the moving sofa problem? As a class, brainstorm a list of differences and similarities between the two-dimensional problem and the real-life problem.

—Leila Sloman

Would Donald Trump’s tariffs hurt US consumers?

BBC, November 26, 2024

President-elect Donald Trump plans to enact a tax on imports from countries including Canada, Mexico, and China. Although he claims this tax (called a tariff) will not affect consumer prices, “that was almost universally regarded by economists as misleading,” according to Ben Chu for the BBC. One think tank estimates the Trump tariffs will increase American household expenses by \$2,500 to \$3,900. This article describes the math behind tariffs and forecasts their impact on the domestic economy.

Classroom Activities: tariffs, economic math, supply and demand

  • (Mid level) An American company imports \$1 million of parts from Mexico. The parts are used to build 2,000 refrigerators, which are then sold for \$1,000 each. Assume a 25% tariff on Mexican imports. Answer the following based on your reading.
    • How much will the Mexican exporter company pay in tariffs?
    • How much will the American company pay in tariffs?
    • Assuming all other costs are equal, by how much does the cost increase per refrigerator?
    • If before tariffs the American company’s profit margin was 20% and the price stays the same, how much profit did they earn per refrigerator before and after the tariffs? How must the price change to keep profits constant?
  • (Mid level) Watch this short explainer of supply and demand.
    • Discuss how increased costs from tariffs might affect supply and/or demand.
    • Using information from the article, and this Wolfram graph tool simulating supply and demand curves, explain how the following possible consequences of tariffs could shift the price or supply/demand curve. How do these shifts translate to real-world consequences?
      • Decreasing the amount of imported goods
      • Tariff costs passed on to consumers
    • What are some other consequences of tariffs, and how would they shift the supply/demand curve?

—Max Levy

Teacher uses art of origami in geometry lessons (video)

NBC Des Moines, December 16, 2024

Origami, the practice of folding paper into intricate shapes, is more than an art to Iowa teacher Leann Ludwig. Ludwig uses origami to demonstrate geometry’s importance to her middle school students. “There are a lot of connections to engineering,” Ludwig told NBC Des Moines. Ludwig’s students are excited to learn math in a new way, and several have taken up origami as a hobby.

Classroom Activities: origami, geometry

—Max Levy

More of this month’s math headlines: