Math Digests March 2023

Pi Is Hiding Everywhere

Wired, March 14, 2023.

March 14 is “Pi Day,” a celebration of the mathematical constant $\pi$ (which rounds off to 3.14). While $\pi$ famously appears in equations that have to do with circles, its applications extend beyond the ones you might see in a geometry class. In an article for Wired, physics professor and science communicator Rhett Allain takes a deep dive into some ways that $\pi$ shows up in physics. The article explains why $\pi$ appears in equations that model the physical world, including equations for magnetic fields, oscillations, and even the uncertainty principle.

Classroom Activities: geometry, physics, astronomy

  • (Geometry) Allain describes how $\pi$ appears in a formula for measuring the intensity of sunlight. Work through the following steps to see how this formula works.
    • Write down the formula for the surface area of a sphere in terms of its radius.
    • According to a NASA fact sheet, the radius of the sun is about $695,700$ kilometers. Use this number to calculate the approximate surface area of the sun.
    • Allain writes that the sun emits almost $4 \times 10^{26}$ joules of energy every second. Based on the surface area that you found in the previous step, how much energy is being sent out of each square kilometer of the sun every second?
    • (Advanced) Read the section of the article titled “Pi and Symmetry” up through the paragraph that ends “…equidistant from the center of the sphere.” How would you calculate the intensity of sunlight as it reaches Earth? What information do you need?
    • In your own words, what does $\pi$ have to do with the intensity of sunlight?
  • (Trigonometry, Physics) Read the explanation of simple harmonic motion in the OpenStax physics textbook, up until the Khan Academy video.
    • Give a real-world example of a deformation, and give a real-world example of an oscillation. You can use examples from the book, or get creative with your own examples.
    • Based on the text, what does $\pi$ have to do with simple harmonic motion?
    • Watch the Khan Academy video embedded in the textbook up until time 7:12, then answer the question under the video about displacement, amplitude, and period. Using a calculator or other software, try graphing the function $A\cos(\omega t)$ for different values of $A$ and $\omega$.
    • (Calculus) Finish watching the Khan Academy video, then watch the other two videos in Khan Academy’s lesson about understanding harmonic motion. Discuss: where does the $2\pi$ come from in the formula for the period in simple harmonic motion?

—Tamar Lichter Blanks

Mathematicians have discovered the hidden patterns that exist within ‘chaotic’ crowds

Fast Company, March 3, 2023.

Have you ever been to a crowded event, like Comic-Con or a local concert, and watched the crowd as everyone exited at once? Despite the chaos, people manage to navigate without knocking each other over. How can this be? Math has the answer. The mathematical scientist Tim Rogers and his team recently discovered hidden mathematical patterns in human movement. They conducted an experiment that asked volunteers to walk within a maze of entrance and exit gates, finding that pedestrians subconsciously fell into orderly “lanes.” “At a glance, a crowd of pedestrians attempting to pass through two gates might seem disorderly,” according to Rogers. “But when you look more closely, you see the hidden structure. Depending on the layout of the space, you may observe either the classic straight lanes, or more complex curved patterns, such as ellipses, parabolas, and hyperbolas.” In this Fast Company article, Connie Lin describes Rogers’ work and inspiration.

Classroom Activities: Brownian motion, chaos

  • (Mid level) Ask the class to walk through the classroom doorway as quickly as possible without running or shoving each other. (Optional: Use a stopwatch to get the fastest time.) Take a video from behind. (If the class is small, consider inviting another class to join the activity.) Then read the article and play the video.
    • What did you notice? (e.g. roughly how many people were around, and how many people could pass through an entrance/exit at once)
    • How many “lanes” did you notice? What shape were the lanes?
    • Did the lanes persist throughout the video?
    • Discuss anything else you may have observed.
  • (High level) The researchers drew inspiration from Albert Einstein’s theory of Brownian motion. Watch this video for some ideas of how to simulate Brownian motion in class, or have them try the following recipe:
    • Mix one tablespoon of cornstarch with one cup of water to make a thin, milky solution. Add a few drops of green food coloring to the solution and stir. Pour the mixture into a clear container. Shine a flashlight on the container to illuminate the solution. Observe the movement of the particles in the solution under a microscope. Discuss how you expect the particles to be distributed if you let the experiment run for an hour, and why. (Hint: Should Brownian motion move the particles randomly or in a particular direction?) Compare what you observe to the video of the class moving through the doorway.

—Max Levy

Computers make snowplowing more efficient. Why don’t more cities use them?

The Washington Post, March 3, 2023.

After a snowstorm hits and leaves a thick blanket in its wake, cities are left with a difficult math problem. They need to ensure roads get plowed as quickly as possible using a limited number of snowplows. What’s more, routes need to be designed with many details in mind, such as prioritizing high-traffic streets and roads to hospitals. In this article, Kasha Patel explains how mathematics and computer software are helping some cities solve this problem more efficiently.

Classroom Activities: optimization, graph theory

  • (Mid level) Patel relates route optimization to the famed “Königsberg bridge problem”, solved by Leonhard Euler. In class, read this Encyclopedia Britannica article about the Königsberg bridge problem. Ask students to solve the following questions:
    • Suppose the town of Königsberg wanted to plow its seven bridges without having any plow cross a bridge more than once. How many plows would they need to use? Justify your answer, and describe the routes of each plow using the bridge numbers in this map.
    • Suppose it costs the city \$500 to buy a snowplow, and \$100 to drive a snowplow across one bridge. (Assume it is free to travel between bridges.) Using the route you came up with in the previous problem, find the most cost-effective solution for the city. How much would a snowplow need to cost for your answer to change?
    • Now suppose it takes a snowplow one hour to cross a bridge, and each hour that the bridges are covered in snow costs the city $150. What is the best solution? At what cost per hour would your answer change?
  • (Mid level) Patel notes that several cities have declined to use computer software in their snowplow routes. When Shrewsbury, Massachusetts tried it, she writes, “the number of routes didn’t change, still requiring 33 snowplows. The computer-generated pathways also didn’t prioritize main roads as well as the status quo.” Ask students: Does this surprise you? Imagine that bridge 3 in the Königsberg problem was especially crucial. How would you try to prioritize it in a mathematical model?

—Leila Sloman

High Schoolers Prove the Pythagorean Theorem Using Trigonometry

Popular Mechanics, March 31, 2023.

At the AMS Spring Southeastern Sectional Meeting last month, one talk was given by an unlikely pair. Two high school students from New Orleans presented a proof of the Pythagorean theorem. Though the Pythagorean theorem has been proven many times, the two students, Calcea Johnson and Ne’Kiya Jackson, came up with a new argument using trigonometry. Darren Orf covers their talk for Popular Mechanics. (Although Orf writes that “almost none of [the proofs of the Pythagorean theorem]—if not none at all—have independently proved it using trigonometry,” there are other trigonometric proofs out there. Orf links to one from cut-the-knot.org; the same site lists a few others here.)

Classroom Activities: geometry, trigonometry

  • (All levels) The Pythagorean theorem relates the three side lengths in a right triangle. Learn about it with this online lesson from Leeds Beckett University. Do the practice problems and activities at the bottom of the lesson.
    • Have students draw their own right triangles using rulers and graph paper, without measuring. Then ask them to measure the lengths of the sides and verify that they satisfy the Pythagorean theorem.
    • Now, ask students to draw several copies of their triangles, cut them out, and use them to verify Proofs 3 and 4 at this link.
  • (Trigonometry) In their proof, Jackson and Johnson used a formula called the Law of Sines. The Law of Sines says that the length of one side of a triangle divided by the sine of its opposing angle is the same for all three sides of the triangle. Learn the Law of Sines here, and do the practice questions at the bottom.

—Leila Sloman

The Most Boring Number in the World Is

Scientific American, March 3, 2023.

We often think that mathematicians prefer objectivity to subjectivity. Square numbers are not “special” or “exciting,” they’re products of an integer and itself. But it turns out that mathematicians do gravitate toward certain numbers more than others. For example, the numbers $e \approx 2.72$ and $\pi \approx 3.14$ stand out. In an article for Scientific American, Manon Bischoff discusses how mathematicians analyzing a database of more than 360,000 integer sequences revealed a clear split, as certain numbers appeared in the database more frequently than others. “To mathematicians’ great surprise, research in 2009 suggested that natural numbers (positive integers) divide into two sharply defined camps: exciting and boring values,” writes Bischoff.

Classroom Activities: boring numbers, integers

  • (Mid level) Watch this Numberphile video about why 63 and -7/4 are special. In the video, 63 appears in the list of “Mersenne numbers.” Research Mersenne numbers and write out the first 10 of these numbers. Are most Mersenne numbers prime? Find examples of where Mersenne numbers are used in mathematics.
  • (High level) Bischoff writes that although the mathematician Godfrey Harold Hardy thought 1,729 was a boring number, Srinivasa Ramanujan knew that it was in fact special, remarking “It is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways.” Have students do research on why the following formulas create “special” numbers.
    • $2^n – 1$
    • $2^n + 1$
    • $2^{2^n}+1$
    • $n!+1$

—Max Levy

Some more of this month’s math headlines: