Math Digests June 2022

Enormous ‘rogue waves’ can appear out of nowhere. Maths is revealing their secrets.

National Geographic, June 6, 2022

If a giant wave hits a ship, and nobody survives to tell the tale, did the wave even exist? What if someone does survive, but the wave is simply too big to believe? In 1826, a French naval officer and scientist survived a 100-foot wave during a storm in the middle of the ocean. “His story, backed by three witnesses, seemed so outlandish that it was dismissed as fantasy,” writes Ally Hirschlag. But similar accounts of so-called “rogue waves” appeared many times through the years, and scientists now acknowledge their existence. Still, rogue waves—waves more than twice as tall as those around them—remain a mystery to physicists and mathematicians. Unlike tsunamis, the giant swells appear out of nowhere, often within just 10 to 15 seconds. In this article, Hirschlag describes recent research aiming to understand how rogue waves form, how common they are, and how to predict them in real time.

Classroom activities: waves, linear and nonlinear effects

  • (Mid level) One way that rogue waves can form is through linear addition, where waves of different sizes and speed overlap and combine. You can visualize this by imagining snapshots of two “waves”: $f(x) = \sin x$ and $g(x) = \sin (2x)$. Graph $f(x)$ and $g(x)$. Now graph $h(x) = f(x) + g(x) = \sin x  + \sin (2x)$. Discuss what you notice about the shape of this graph compared to $f(x)$ and $g(x)$.
  • (Mid level) Linear addition cannot explain every rogue wave because there are many variables in the ocean that can affect waves, including currents, differences in temperature, and the geology of the ocean floor—these give rise to nonlinear interactions.
    • Graph $k(x) = \sin x \times \sin (2x)$ and compare it to $f(x)$, $g(x)$, and $h(x)$ above.
    • Below are animations of a sum and a product of two solitary traveling waves. (In both videos, one of the component waves has half the amplitude and 1.5 times the speed of the other.) Discuss the qualitative difference between the two composite waves.

      Mathematica animations of traveling waves. On the left, two separate waves merge into one taller wave. On the right, a tall wave emerges out of nowhere.
      Left: The sum of two waves. Right: The product of the same two waves.
  • (All levels) Watch this video by Physics Girl about the science of rogue waves.
    • The video gives baking as an example of a linear process and watering a cactus as an example of a nonlinear process. What other examples of linear and nonlinear processes can you come up with?

Related Mathematical Moments: Knowing Rogues.

—Max Levy

Bender’s Serial Number In Futurama Has A Hidden Meaning

/Film, June 8, 2022

Math makes a cameo in the animated television show Futurama via an ordinary-looking string of digits. In the show, a robot named Bender has the serial number $1729$, which has a special property: It is the smallest number that can be written as the sum of two positive cubes in two different ways. In other words, two different choices of whole numbers $a$ and $b$ are solutions to the equation $1729 = a^3 + b^3$. This fact was discovered by Srinivasa Ramanujan, the prolific mathematician who was the subject of the 2015 film The Man Who Knew Infinity. Futurama writer Ken Keeler, who holds a PhD in applied math, suggested using $1729$ in the show because of its mathematical significance. In an article for /Film, Witney Seibold discusses $1729$ in Futurama and some riffs on the sums of cubes property.

Classroom activities: number theory, exponents, nested radicals

  • (All levels) Verify that $1729$ is the sum of two positive cubes in two different ways by checking that $12^3 + 1^3 = 1729$ and $9^3+10^3=1729$.
  • (Algebra) Show that $50$ is a sum of two squares in two different ways.
    • (Algebra, advanced) Find another number that is a sum of two squares in two different ways.
  • (Pre-calculus, advanced) Read the first part of the article “Three Puzzles Inspired by Ramanujan” in Quanta Magazine, up until but not including the second question.
    • Try to solve the first puzzle in the article.
    • Pair up with a classmate and discuss your thoughts. If you got a solution or partial solution, try to justify the steps that you took. If you got stuck, talk about a point you are stuck on.
    • Go over the solution as a class. Discuss any parts of the proof that seemed especially challenging or surprising.

—Tamar Lichter Blanks

How Antarctic Krill Coordinate the Biggest Swarms in the World

Scientific American, July 2022 issue

A flock of birds, a swarm of locusts, a crowd of people—large groups of animals often seem to move as a cohesive unit rather than as unconnected individuals. This phenomenon is called collective motion, and there’s a lot of math to model it. Now, researchers studying Antarctic krill say they have figured out how these tiny crustaceans form and maintain underwater swarms. Krill seem to pay the most attention to the neighbors swimming above and below them instead of those to the left and right—a behavior not seen in fish and birds that exhibit collective motion. In this article, Andrew Chapman explains how the scientists conducted the study and what questions remain unanswered.

Classroom activities: emergence, geometry, vectors

time (s) $x_9$ (mm) $y_9$ (mm) $z_9$ (mm) $x_{10}$ (mm) $y_{10}$ (mm) $z_{10}$ (mm)
0.000 98 -34 520 117 -18 503
0.100 97 -38 518 116 -21 500
    • Using the three-dimensional distance formula, calculate the distance between the two krill at $t=0.000$ s and at $t=0.100$ s. Did the two krill get closer to or farther away from each other during this interval?
    • Calculate the speed of krill #9 during this interval. (Hint: first calculate its speed in the $x$, $y$, and $z$ directions, then combine these three numbers using a formula similar to the distance formula.)
    • (Pre-calculus, advanced) Write down the velocity vector for krill #9, then convert it to spherical coordinates. Describe in words the direction that the krill is traveling.

Related Mathematical Moments: Getting It Together.

—Scott Hershberger

How math and language can combine to map the globe and create strong passwords, using the power of 3 random words

The Conversation, June 16, 2022

Random sequences of letters and numbers make great passwords because they are hard to guess. Unfortunately, they are also hard to recall. Consider the sequence “UKCIQCKDEWIAOMF.” It might take you a minute to memorize. How about “QUICK.WICKED.FOAM,” three random words made from the same characters? Much easier to remember—and still useful as a unique identification. A web app called what3words divides the entire surface of Earth into 3-meter-by-3-meter squares and uses three random English words to represent every square. The order of the words matters, and they don’t appear in alphabetic order based on longitude or latitude. So while quick.wicked.foam takes you to the US president’s Oval Office, wicked.quick.foam sits near a bus stop in Louisville, Kentucky. In this article, professor Mary Lynn Reed explains how what3words works: “The secret behind this power is just a little bit of math.”

Classroom activities: combinations, permutations, surface area, encryption, algorithms

  • (Low level) Form groups of 4. Each student can search their home address (or another place) using what3words. Then, they can tell their group 1) the what3words address and 2) the actual street address. After everyone has a chance to share, each student can try to recall their groupmates’ addresses. Which form of address is easier to remember? Discuss the pros and cons of this form of address encoding.
  • (Mid level, Geometry) Could the same what3words system that relies on 40,000 words also build a map of the surface of Mars? What about the Sun? (Hint: Follow the math as described in the article.)
  • (Upper level) There are many algorithms, or step-by-step processes, you could use to create a system pairing words with geographic locations.
    • Given what you know about what3words, brainstorm how you would assign individual sequences of three words to the 57 trillion squares on Earth. What are some potential problems with assigning the words completely at random?
    • From this blog post, read the section titled “The Algorithm” about the steps in the what3words algorithm. Discuss the purpose of each of the steps shown in the flowchart.

—Max Levy

Some more of this month’s math headlines:

There will be no July Math Digests. We will return with the August Math Digests in early September. Until then, explore coverage of the recipients of the 2022 Fields Medals in Nature, Quanta Magazine, and The New York Times.