Math Digests May 2022

You Hear the Musical Saw. These Mathematicians Heard Geometry.

The New York Times, May 1, 2022

Have you ever wondered how instruments make music? Pluck a guitar string, and watch it vibrate. Vibrations reverberate in the instrument’s wooden body to sustain a note, and the geometry of the body and string influences the sound. The same mathematical music exists in other objects that may not seem musical—even construction tools. In an article for The New York Times, Nicholas Bakalar writes about the math of musical saws. Musicians have played the saw for around 200 years. They must bend the tool into an S shape and draw a bow across a precise sweet spot to sustain a note. Recently, researchers analyzed the math behind that bend. One scientist told Bakalar that the saw “sings,” like a violin: “Musicians have of course known this experientially for a long time, and scientists are only now beginning to understand why the saw can sing.” (See also this Harvard article.)

Classroom activities music, frequency and wavelength, geometry, inflection points

  • (All levels) Watch this video for an introduction to the science of vibration and music.
  • (Mid level) Answer these questions based on the article and its video of a woman playing the musical saw.
    • Does a curvy S shape create a higher or lower pitch than a relatively straight S shape? Explain why, using the equation from the first video.
    • How do you expect the sound of a saw with a thinner blade to be different from that of a saw with a thicker blade?
  • (Calculus) The musical saw can only make music if it’s bent in an S shape. That bend creates a “sweet spot,” Bakalar writes, where the curvature switches from positive to negative. Recall this concept from your lessons on inflection points, where a function’s second derivative equals zero.
    • Find the inflection point(s) for the function $y=\sin{(2x/3)}$ on the interval $(-\pi/2, \pi/2)$.
    • Find the inflection point(s) for the function $y=\sin{x}$ on the same interval, which represents a saw in a curvier S shape.
    • These two functions have the same curvature (zero) at the inflection point. But how do their second derivatives compare near the inflection point?
    • Discuss why these differences in the curvature of the saw might change its sound. (Hint: think about tension.)

—Max Levy

A Mathematical Formula for the Right Time to Show Up at a Party

The Atlantic, May 13, 2022

You’ve been invited to a party! You’ve figured out what to wear and what to bring—but when should you arrive? Too early, and you might enter an awkwardly empty room. Too late, and you might miss out on the best conversations. Maybe math can help. Mathematician Daniel Biss devised a (somewhat silly) formula to calculate your personally optimal arrival time. It includes seven factors, such as how punctual your friends are and how excited you are about the party. This article by Joe Pinsker makes Biss’s formula interactive and goes on to explore the tension between “clock time” and “event time.”

Classroom activities: mathematical modeling, algebra

  • (All levels) Have students fill out the interactive calculator (keeping track of their answers to each question) and share their results. Do the results make sense? Which factors seem to have the biggest impact on the number that the calculator spits out?
  • (Algebra II, Pre-calculus) Let’s analyze Biss’s formula, referring to your arrival time (in minutes) as $A$.
    • What happens when the Peer Group Punctuality ($\phi$) is 0? What if $\phi=10$? If all the other parameters are held constant, what type of function is the arrival time $A(\phi)$?
    • What happens when the Peer Group Accuracy ($\varepsilon$) is 0? What if $\varepsilon=10$? If all the other parameters are held constant, what type of function is the arrival time $A(\varepsilon)$?
    • The fear of being early ($E$) and the fear of being late ($L$) appear twice in the formula. How do these two variables interact in each term?
    • What are some limitations of the formula? Can you think of any factors related to party arrival time that matter to you but don’t appear in the formula?
  • (Upper level) Inspired by Biss’s equation, come up with a mathematical formula to model another type of decision that you often face in your own life.

—Scott Hershberger

Inflation may be easing — but low-income people are still paying the steepest prices

NPR, May 11, 2022

This year, the entire US has dealt with rising economic inflation at levels not seen in decades. Basically, the price of stuff is going up shockingly fast. A loaf of bread. Gasoline. A pair of jeans. Stuff that’s essential for people to subsist and hold down a job that allows them to subsist on their own. Over time, the prices naturally inch up—many economists consider around 2% per year to be ideal. But compared to this time last year, consumer prices are up an average of 8.3%, according to the most recent data. In this article, Scott Horsley discusses a possible fix with roots in economic math—raising federal interest rates—and why some communities are disproportionately burdened. “When inflation is high, everyone pays the price, but research suggests that lower-income families suffer the most.”

Classroom activities: economics, inflation, supply and demand, data analysis

—Max Levy

Leonardo da Vinci’s rule for how trees branch was close, but wrong

Science News, April 27, 2022

On a winter walk in the woods, you might notice that the bare trees around you have intricate shapes. The branches of deciduous trees fork into smaller and smaller offshoots until they end in narrow twigs. To describe how branches divide, Leonardo da Vinci proposed a mathematical rule: When a tree limb splits into smaller branches, the area of a cross-section of the big branch is the same as the sum of the cross-sectional areas of the smaller branches. In an article for Science News, James R. Riordon reports on a new paper that tweaks Leonardo’s rule. Instead of cross-sectional area, the researchers model tree branching using surface area, which uses information about the length of a branch as well as its diameter. The paper also incorporates more advanced techniques such as numerical Fourier analysis, but the researchers still consider their rule to be “Leonardo-like.”

Classroom activities: geometry, area, circles

  • (Geometry) Imagine cutting straight through a tree branch and looking at the flat cross-section of the tree that you expose. That flat shape is approximately a circle. The formula for the area of a circle is $\pi r^2$, where $r$ is the radius of the circle.
    • Find the area of the cross-section of a branch with radius $5$ cm.
    • Suppose that a branch with radius $5$ cm splits into two smaller branches, one with radius $4$ cm and one with radius $3$ cm. Check that da Vinci’s rule holds in this scenario.
    • (Advanced) If a branch with radius $r_1$ splits into two smaller branches with radii $r_2$ and $r_3$, how are these three radii related (assuming da Vinci’s rule is true)? Write down an equation that describes the relationship between these three numbers.
  • (Geometry) A typical tree branch is shaped approximately like a cylinder. Its lateral surface area (which is used in the new paper) is $2 \pi r \ell$, where $r$ is the radius and $\ell$ is the length of the branch. (This surface area doesn’t include the ends where the branch connects to the rest of the tree.)
    • Find the lateral surface area of a branch with radius $5$ cm and length $24$ cm.
    • (Advanced) Imagine taking the bark off of a branch by making a cut down the length of the branch, then peeling off the bark in one piece. The unrolled piece of bark will have the shape of a flat rectangle.
      • Let $r$ be the radius of the branch and $\ell$ be the length of the branch. What are the length and width of this rectangle of bark?
      • Compare the area of the rectangle to the lateral surface area described above. What do you notice about the two expressions, and what do you think is the reason for it?

—Tamar Lichter Blanks

Some more of this month’s math headlines: