Math Digests January 2024

How to Convince Your Flat-Earth Friends the World Is Round

Wired, January 26, 2024.

According to the American Physical Society, the ancient Greek philosopher Pythagoras suggested Earth was a sphere merely because he liked the idea. But there’s much more compelling evidence out there. In this article for Wired, Rhett Allain explains how to use scientific observation and mathematics to deduce that Earth is round, in two different ways.

Classroom Activities: geometry, trigonometry

  • (All levels) Read the section “Water Isn’t Flat” in the article.
  • (Mid level) Lake Pontchartrain is more than 40 miles wide. For simplicity, assume that when Allain took his photo, he was 40 miles away from a building on the opposite shore.
    • The path between Allain and the building forms an arc on the Earth. Prove that the length of a chord between Allain’s eye and the building is almost exactly 40 miles. Give your intuition as to why this is true.
    • Examine Allain’s diagram showing a right triangle with side lengths $R$, $s$, and $R+h$. Draw your own version and add a building across the lake. Find a right triangle that will help you calculate how much of the building is blocked by the horizon.
    • Assuming the distance between Allain and the building can be approximated as 40 miles, calculate the height of the part of the building that is blocked by the horizon.
    • The Pacific Ocean is about 12,000 miles across at its widest point. Repeat the calculation for someone looking at a building across the Pacific Ocean. Careful: The approximation that the arc length and chord length are the same no longer holds! How will you fix this?

—Leila Sloman

Billions of cicadas are set to appear in a rare ‘double brood emergence,’ scientists say

CNN, January 26, 2024.

Every year, inch-long insects known as cicadas emerge from the ground and take to the skies in a buzzy throng that numbers in the millions. They spend the next several weeks eating, mating, and laying eggs. The next generation will do the same; some descendants emerge the following year, and others, from a species known as periodical cicadas, come up only every 13 or 17 years. Biologists believe that periodical cicadas have evolved these long cycles to safeguard their offspring from predators. This spring, two distinct cycles, or broods, will coincide in an eruption of billions of once-dormant cicadas. “It’s a rare emergence of insects some are referring to as cicadapocalypse,” writes Kate Golembiewski. Her CNN article describes the once-in-a-lifetime insect event, and how mathematics offers an explanation for the strangely choreographed cycles.

Classroom Activities: prime numbers, data analysis

  • (All levels) Watch this Numberphile video about cicada math.
    • Describe in your own words the mathematical reason why 13 and 17 may have benefited cicada evolution. (Note: “The jury’s still out” on whether predators are truly the motivation for the cicada’s life cycle length, writes Golembiewski.)
    • What would be the next largest cycle length that offers the same evolutionary benefit?
  • (Mid level) The US is home to 12 broods of 17-year cicadas, and three broods of 13-year cicadas. Look up the 15 broods and what years they emerge.
    • On a spreadsheet, create a table showing when each brood (columns) emerges in a span of years (rows) from 2024 to 2224. When does the next overlap of broods occur?
    • Based on the math, when will these same two broods overlap again?
    • How many overlapping cicada emergences occur within 200 years?

—Max Levy

‘Magical’ Error Correction Scheme Proved Inherently Inefficient

Quanta Magazine, January 9, 2024.

In the 1940s, computer scientists began encoding data in a way that can reverse whatever data corruption might occur. Without that type of code, known as error correction, the data communication and storage your phone and computer rely on would be useless. As later researchers have improved on this idea, they have imagined new processes like “locally correctable codes,” resilient corrections that only query a few points of the data. “It’s as if you could recover any page torn out of a book by just glancing at a few others,” writes Ben Brubaker. There’s a catch, though: The examples of locally correctable codes that exist are terribly inefficient because they make the encoded message exponentially longer. Researchers have long hoped for a new algorithm that would skirt around this so-called exponential cost. This hypothetical algorithm would be able to correct data errors with only three queries. Yet now, Brubaker writes, computer scientists have proven that such a code is impossible. In this article for Quanta Magazine, we learn about work from theoretical computer scientists, and how the mathematics of error correction appear in disparate fields.

Classroom Activities: error correction, exponentials

  • (All levels) Read this explainer about error-correcting code from Brilliant, and familiarize yourself with the NATO phonetic alphabet table.
    • In groups of two, demonstrate the idea. Each person should write their own sequence of 16 random letters. Out loud, one person should communicate their secret message once, only using standard letters. Next, the other person communicates their own message using the NATO alphabet. Compare the success of communicating in each strategy.
    • Explain in your own words why communicating in code with the phonetic alphabet is less prone to misinterpretation.
  • (Mid level) As the number of bits $N$ to be relayed in a message grows, the number of bits in the error-correcting code grows exponentially in $N$. This is exponential complexity. Other computer science problems are mathematically more efficient, exhibiting linear complexity, logarithmic complexity, and polynomial complexity.
    • Using your own online research, describe linear, logarithmic, polynomial, and exponential complexity, and suggest a real-world example for each.
    • Write an example mathematical expression for each type of complexity.
    • Rank each type of complexity in terms of efficiency.

—Max Levy

These Numbers Look Random but Aren’t, Mathematicians Prove

Scientific American, January 30, 2024.

It’s nearly impossible to generate a truly random sequence of numbers. Those who’d like to do so settle for “pseudorandom” sequences, which are not actually random but have many of the properties of randomness. In recent work, mathematicians Christopher Lutsko, Athanasios Sourmelidis, and Niclas Technau found a large family of pseudorandom sequences. Lutsko describes the result in this article for Scientific American.

Classroom Activities: probability, statistics

  • (All levels) Read the first two sections of the article (through “Detecting Randomness”).
    • Describe in your own words the randomness test that Lutsko explains in “Detecting Randomness.”
    • Suppose you had a sequence of 10 whole numbers between 1 and 10. Come up with a procedure that applies Lutsko’s test to the numbers.
    • Read the third section (“Other Tests”). What does the second type of test capture that the first does not? Is there any property of a random sequence that is not captured by the two types of tests described?
  • (All levels) Come up with three ways to generate a sequence of 10 pseudorandom numbers between 1 and 5. (Each number should come up with equal probability $1/5$.)
    • Generate the sequences and plot them on a number line. Do they look random?
    • Apply the procedure you came up with in the first exercise. What did you learn about your sequences?
  • (Mid level) In past digests, we’ve analyzed statistics of coin flips. The same ideas apply here. Calculate:
    • The probability that if you randomly generate 2 numbers between 1 and 5, you get the same number each time.
    • The probability that if you randomly generate 3 numbers between 1 and 5, you get the same number each time.
    • If your sequences were truly random, how many times should you see the same number twice in a row? Three times in a row? Compare to your pseudorandom sequences.

—Leila Sloman

Mathematician explains resonant frequencies and how not to spill coffee

CDM, January 4, 2024.

Have you ever spilled some of your drink out of a mug while walking just a few steps? You might think it’s your fault for not being more careful, but it’s not. It’s just physics. Physicists use the term “frequency” to describe how fast oscillations come and go. A slow-swinging pendulum has a low frequency, and squeaky sound waves oscillate rapidly (at high frequencies). The drink sloshing back and forth in your mug has a certain natural frequency, too. However, math student Sophie Abrahams explains, when the pace of your arms as you walk matches this frequency, a problem arises. The frequencies are said to resonate, which increases the swinging. This article in CDM covers Abrahams’ explanation and discusses other examples of resonant frequency, such as breaking glass with sound.

Classroom Activities: resonance, frequency

  • (All levels) Watch the first video, “Why do I always spill my coffee?”
    • Test Sophie Abrahams’ theory by walking with a full mug of water in three different ways. First, walk at a normal pace holding the mug by the handle. Then, walk at a slow pace holding the mug by the handle. Last, walk at a normal pace holding the mug in whatever way you expect will minimize spillage. Note your observations of the fluid movement and discuss in class.
  • (Mid level) Watch the other videos referenced in the article. Define oscillation, amplitude, period, frequency, and resonance in your own words as they relate to the following contexts:
    • Pushing a person on a swing
    • Walking with coffee in a mug
    • The Rubens tube
    • Singing high notes by glassware
  • (High level) We can use trigonometric functions, such as sine and cosine, to express model oscillations and resonance. For the following questions, assume that the function $\theta = \cos(t)$ describes the angle ($\theta$) of a swing from vertical at time ($t$).
    • What is the frequency of the swinging function?
    • Describe in your own words the motion of the swing. What do the variables $\theta$ and $t$ represent? Graph the function, and describe what it says about the swing’s motion.
    • Which of the following terms can you add to the above equation to most accurately represent being pushed on a swing with resonance? Explain why and sketch the resulting graph.
      • $\frac{1}{2\pi}$
      • $\cos(t)$
      • $t$
      • $\frac{t}{4}\cos(t)$

—Max Levy

Some more of this month’s math headlines: