Math Digests May 2023

50,000 Worms Tangled Up in a Ball Unravel in an Explosive Burst when a Predator Appears

Scientific American, April 27, 2023.

Groups of tiny aquatic worms known as California blackworms can cluster together to form big, entangled blobs. The worm balls are knotty and complex, and it takes several minutes for the worms to build the shape, yet if danger appears they can untangle in just milliseconds. In an article for Scientific American, Jack Tamisiea reports on a new paper in Science that analyzes the worms’ method for tangling and rapid untangling. The paper’s authors filmed the worms and used ultrasound imaging to see inside the knots. They then used topology and other mathematical methods to model the worm balls and the creatures’ skillful separation.

Classroom Activities: topology, knot theory

  • (All levels) Watch the video in this article from Georgia Tech about the worm knots, and notice the “helical wave pattern” shown at time 0:38. One of the authors of the California blackworm study told Scientific American: “If it spends longer winding in one direction before switching to winding in the other direction, you get the tangling behavior. … If the worm rapidly switches between winding clockwise and anticlockwise, you get untangling behavior.”
    • Discuss: Based on your own experiences with tangled materials, does this seem surprising to you? Why or why not?
  • (All levels) Give each student two lengths of rope, string, yarn, or other knottable item, and a secret knot randomly assigned from this list. Ask them to tie the ropes into their assigned knot, then trade knots with a partner. Now, each partner will try to untangle the knot they’ve been given.
    • Was it easier to make the tangle or to untangle it? What factors do you think might be involved in the difficulty of tangling and untangling? Does this affect the way you think about the worms tangling and untangling?
  • (High level) Read the first three sections of the introduction to knot theory from the Oglethorpe University Knot Theory website. Then try the trefoil knot activity. You may want to refer back to Section 3 of the introduction to help you answer questions 1-3.

—Tamar Lichter Blanks

Why Those Super Low College Admissions Rates Can Be Misleading

The New York Times, April 22, 2023.

Every winter, high school students wait anxiously to hear whether their college applications will bring good news or bad news. They try to glean some sense of how hard it is to get into a certain university by using metrics like the school’s acceptance rate, which quantifies what fraction of the total applications get accepted. This year, some colleges have reported record-low acceptance rates. Does this mean that getting into college is harder than ever? Not necessarily, writes Jessica Grose in an Op-Ed for the New York Times. This article uses college admissions statistics to describe how data literacy is important in everyday life.

Classroom Activities: statistical literacy, data analysis

  • (All levels) Answer the following based on the article:
    • List the reasons why the admissions data may be misleading.
    • List the reasons why it may be harder than ever to get into college.
    • Then, for each reason listed, come up with at least one source of data (e.g. “acceptance rates from the top 100 public universities”) that would confirm or quantify it. Which data sources are the most trustworthy? Form a hypothesis about whether college admissions have gotten harder.
  • (Mid level) In small groups, consider the importance of most or all of the reasons listed.
    • How would you weigh the importance of one reason compared to another?
    • Which reasons are the most objective?
    • Which are the most subjective?
    • Describe what forms of evidence can support subjective versus objective reasons.
    • (Hard) Now that you’ve studied all of your data, think about how you would prove or disprove your hypothesis. Discuss what challenges you would encounter when testing this hypothesis.
  • (Mid level) The article notes that the fraction of qualified applicants has gone down since the pandemic. Fill in each blank on the table below, and write a one-sentence analysis of each case based on its hypothetical data. If the college accepts 10,000 qualified students each year, what are the chances of a qualified applicant in each case?
Total Applicants (2022) Qualified Applicants (2022) % Qualified Total Applicants (2023) Qualified Applicants (2023) % Qualified
60,000 21,450 _________ 70,000 23,000 _________
60,000 25,632 _________ 60,500 25,633 _________
60,000 14,555 _________ 65,000 10,555 _________

—Max Levy

Why symmetry is so fundamental to our understanding of the universe

New Scientist, May 10, 2023.

Mathematicians and physicists often rely on symmetry to simplify their theories and calculations. Symmetry might explain mathematicians’ proclivity for exploring simple shapes like spheres, or it might be much more fundamental — and abstract — than that. The Standard Model, the starting point for the behavior of all known particles in the universe, is based on symmetries, writes Michael Brooks in this article. Symmetry may or may not remain a backbone of future theories. But mathematician Marcus du Sautoy’s words reveal that symmetry contains wonders in its own right: “The understanding of symmetry I have now is so much deeper and stranger, and it gives me access to symmetries that are so much more exotic than anything you can see with your eyes,” he told New Scientist.

Classroom Activities: geometry, symmetry

  • (All levels) Read this Math Salamanders lesson about regular and irregular shapes. Look at the “Regular and Irregular Shape Sheet”. What ways can you move the shapes so that they look the same after you moved them? That might be rotating them by 90 degrees, or flipping them upside down. These are the symmetries of the shapes.
    • What happens if you combine two of the symmetries of one shape — for instance, rotate it, and then flip it upside down. Is that a symmetry?
    • Can you draw an irregular polygon that has symmetry? Can you draw one that has rotational symmetry (i.e. you rotate it by a given amount, and it still looks the same)?
  • (High level) The symmetries of a shape, or other object, form an important mathematical structure called a group. Read pages 96-106 of these lecture notes by Emanuel Lazar for an introduction to groups and symmetry. What insight does the previous exercise give you into Example 18?
  • (High level) Symmetry does not only pertain to shapes. Sometimes, objects in a set are interchangeable (this happens in quantum mechanics). The symmetries then form a permutation group: The set of all possible ways to rearrange elements in the set. For instance, if you have three identical items in a row, you might swap items 1 and 3.
    • Figure out what permutations are possible on the set {1, 2, 3}. Now suppose these numbers represent the 3 sides of an equilateral triangle. What permutations correspond to symmetries? Does the symmetry group of the triangle capture all the permutations of the sides?

—Leila Sloman

Rhythm and Proofs: Finding music in mathematics

UC San Diego Today, May 1, 2023.

What does math sound like? In your classes, it might sound like the clacking of a keyboard, the scribbling of a pen, or the squeak of an eraser. But to some people, math can sound like music. That’s because music is math. The spacing between frets on a guitar. The relationships between musical keys. And the feeling is mutual: “I find (my) mathematics very musical. There is harmony (and sometimes dissonance) in the way different ideas and structures come together in a mathematical proof,” said Todd Kemp, Professor of Mathematics at the University of California, San Diego, in an interview with UC San Diego Today. In this article, Michelle Franklin interviews three mathematicians about how they see math’s relationship to music.

Note: The article attributes the quote “There is geometry in the humming of the strings, there is music in the spacing of the spheres,” to Pythagoras, but not enough evidence exists to prove that conclusively.

Classroom Activities: music, representation system

  • (All levels) Research famous mathematicians from history who were also musicians, and answer the following questions:
    • Which person’s accomplishments do you find most impressive, and why?
    • Which person’s accomplishments were most surprising, and why?
  • (All levels) Watch this video on math and music.
    • The video describes a magic number: the ratio between sequential steps within an octave. That ratio is approximately 1.0595. What would it be if there were 15 steps between octaves? 10? (More info here.)
    • The function $y=\sin(x)$ is a wave. Find two functions that are harmonics of $y = \sin(x)$.
  • (Mid level) One mathematician told the magazine “Both fields have their own representation system. Math has symbols and notation, and music has staffs and clefs.” Answer the following:
    • What can music communicate that math cannot?
    • What can math communicate that music cannot?
    • What can both representation systems communicate?

—Max Levy

How Bernoullis made their mark

Mint, May 18, 2023.

Lots of mathematical concepts are named “Bernoulli”: The sequence of rational Bernoulli numbers; Bernoulli distributions, which describes a random outcome that can be equal to either 0 or 1. I was vaguely aware that Bernoulli pops up in physics, too. But it never occurred to me that these Bernoullis might all be different people. Writer Dilip D’Souza had the same misconception, but in this article for Mint, he reveals that there were many Bernoullis — all related to one another, with complicated familial relationships and wide-ranging scientific interests. Several of the Bernoullis had a mathematical bent, especially toward probability theory. In these activities, we’ll explore some of the probabilistic concepts the Bernoullis discovered.

Classroom Activities: probability, Bernoulli distribution, Law of Large Numbers

  • (Probability) Watch this video explaining what a Bernoulli distribution is.
  • (Mid level) Calculate the following Bernoulli distributions:
    • The probability that a student randomly selected from your class has brown hair.
    • The probability that a randomly selected United Nations member state begins with the letter A.
    • The probability that two coin tosses in a row will come up heads.
  • (Mid level) What are the mean and variance of a Bernoulli distribution with probability of success p? (Answers here.)
  • Jacob Bernoulli proved the Law of Large Numbers. In his online probability textbook, Hossein Pishro-Nik describes this law as follows: “If you repeat an experiment independently a large number of times and average the result, what you obtain should be close to the expected value.”
    • (All levels) As a class, brainstorm how students would test this law in class.
    • (High level) Prove this law holds for an experiment that follows a Bernoulli distribution.
    • Try this lesson from Statistics Teacher, which uses Jimmy Fallon’s “Egg Roulette” game to explore the Law of Large Numbers.

—Leila Sloman

Some more of this month’s math headlines: