# 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?

—Max Levy