Math Digests November 2023

The Man Who Invented Fifteen Hundred Necktie Knots

The New Yorker, November 10, 2023.

When people say they know how to tie a necktie, they generally think of the one knot—maybe two or three—in their repertoire. Boris Mocka thinks of over fifteen hundred. New Yorker contributor Matthew Hutson recently introduced Mocka—a doorman in Hutson’s building—to experts in topology, a branch of mathematics. The mathematicians were fascinated by Mocka’s passion for creating new variations of loops and swoops. In this article for The New Yorker, Hutson writes about Mocka’s life, his designs, and about how the art of a necktie intersects with an important research area in mathematics called knot theory. “The Gardenia looks like a flower; the Wicker and the Mockatonic look like origami. The Riddler looks like a question mark, and the Exousia requires more than one tie,” writes Hutson. “Many more math techniques would be needed to describe Mocka’s art.”

Classroom Activities: topology, knot theory

  • (Mid level) Watch this Veritasium video about knot theory, and answer the following questions:
    • Why are all knots studied by mathematicians “closed loops?”
    • What makes two knots mathematically different?
    • What is the knot equivalence problem, and why is it so hard to solve?
    • What is a prime knot, and how is it different from a composite knot?
  • (Mid level) Read the New Yorker article. Based on your reading and the Veritasium video, what is one reason why it’s hard to define Mocka’s ties with knot theory?
  • (Mid level) Research 3 different knots that you can tie with neckties, or rope. Create them, and then identify which of these knots are cinquefoil, trefoil, or other categories you have learned in the lesson. Think about what the category for each means about their mathematical relationship. For example: the “four-in-hand” and “Windsor” knots are both trefoils, therefore they are mathematically identical.

—Max Levy

How maths can help you win at everything

The Guardian, November 5, 2023.

Mathematician Marcus du Sautoy has a message: “In love and economics, business and games, if you know your maths, you’ll end up the winner.” In an article for The Guardian, du Sautoy shares mathematical strategies for winning in a wide variety of situations, including some that don’t sound mathematical at first. Euler’s number $e$, for example, shows up in a strategy for finding the right number of people to date before settling on “the one.” Tools from calculus can help an entrepreneur figure out how many units to make of a new product. And statistically, a trivia buff playing “Who Wants to Be a Millionaire?” is better off asking the audience for help than phoning a friend. These and other real-world scenarios have rules and logical patterns that du Sautoy says people can tap into, mathematically, to succeed.

Classroom Activities: economics, game theory, probability

  • (All levels) Read the section of the article titled “How to win at economics.”
    • If you were Player A in this scenario, how much money would you offer? If you were Player B, would you accept the split suggested in the article? Do you agree that the golden ratio makes the split feel fairer?
    • For another example of Nash equilibrium, watch the Khan Academy video on the prisoners’ dilemma.
      • The video illustrates that from Al’s perspective, it’s better to confess, whether or not Bill does so. The same is true for Bill. But when they both confess, they get more jail time than when they both deny. Discuss: How can it be that when Al and Bill make their best individual decisions, it doesn’t lead to the best group outcome? Does this surprise you? Can you think of any other situations where something like this might occur?
  •  (All levels) Read the section of the article titled “How to win at games.” The article says that if you roll two dice, the most common total you will get is a $7$ because there are six ways to make $7$ out of two dice: $1+6$, $2+5$, $3+4$, $4+3$, $5+2$, and $6+1$.
    • How many ways are there to get a total of $2$ when rolling two dice? How many ways are there to get a total of $3$?
    • When you roll two dice, how many different outcomes are possible in total? What strategies did you use to try to figure this out?
    • (High level) If you roll two dice, what is the probability of rolling a total of $7$?

—Tamar Lichter Blanks

Diagnostic tests for rare conditions present a mathematical conundrum

Stat News, November 30, 2023.

Currently, the Food and Drug Administration allows labs that produce and then use their own diagnostic tests to proceed free from the usual regulations. This fall, the FDA proposed changing this policy, citing the tests’ frequent false positives. But the FDA, and the public, may find that they keep getting false positives even when diagnostic testing is subject to the strictest quality standards. In this article for Stat, mathematician Manil Suri and epidemiologist Daniel Morgan explain that false positives are par for the course when it comes to medical testing, especially when the condition being tested for is rare. “Practically all tests, not just [lab-developed tests], carry the risk of false positives, which can render the results effectively useless when the condition is rare enough,” they write.

Classroom Activities: statistics, probability

  • (Mid level) Learn about sensitivity and specificity of diagnostic tests with this short interactive lesson from SchoolYourself.
  • (Mid level) Navigate to Morgan’s online calculator, which shows the probability that a patient has a disease in a variety of situations.
    • Under “What disease are you testing for?” select “Flu (Influenza).” Under “What is the pre-test probability?” select “Non-Influenza Season.” Under “Symptoms,” select “No clinical symptoms.”
    • Before selecting any other options, scroll down to “What Test will you order?”, and read the numbers for sensitivity and specificity. Based on the “Chance of Flu” number given on the left-hand side, try to figure out how likely it is that a patient who has no symptoms actually has the flu if they test positive on each test
      • during flu season
      • not during flu season.
    • Now, work through all three tests. For each test, select “Result > If positive.” Look at the “Chance of Flu > After test” number. Does the result surprise you?
    • Repeat the exercise during flu season. How much do the numbers change? Why do you think they changed?
  • (Mid level) Now, create your own 2 x 2 grid as in the SchoolYourself lesson, with the columns “Actually has the disease,” “Does not have the disease,” and the rows “Tests positive,” and “Tests negative.”
    • According to Morgan’s data, during non-flu season, 0.5% of people have the flu, and during flu season, about 4% of people have the flu. Suppose you test 1,000 random people. Using the sensitivities and specificities listed on Morgan’s calculator, fill in the squares of your grid for each test, during flu season and not during flu season.

—Leila Sloman

This mathematician had another career: professional football player

NPR Short Wave, November 27, 2023.

When John Urschel started his Ph.D. in mathematics at the Massachusetts Institute of Technology, he already had a high-profile job playing football for the Baltimore Ravens. In this episode of Short Wave, Regina Barber interviews Urschel about how he balanced those two demanding careers. and about his current research in linear algebra.

Classroom Activities: linear algebra

  • (Mid level, Algebra II) Barber describes linear algebra by example: As the type of math used to solve systems of equations such as $x + y = 3$, $x – y = –1$.
    • Read Section 5.2 from this online textbook from LibreTexts to learn how to solve a system like this using a method called “substitution.” Do exercises 5.2.1, 5.2.2, 5.2.3, and 5.2.19.
    • Now read Section 5.3 to learn a method called “elimination”. Do exercises 5.3.4, 5.3.7, 5.3.10, and 5.3.22.
    • Solve Barber’s example using both substitution and elimination. Show your work.
  • (High level) In the episode, Urschel talks about “higher-dimensional versions of lines.” Before viewing the next activity, discuss the following questions in small groups, then answer individually.
    • What does he mean by “higher-dimensional”?
    • How would you define a “higher-dimensional version of a line”? Reminder: You can think about a line as a type of curve, or you can think about it in terms of the equation that defines it. You may find one of these perspectives easier than the other!
  • (Mid level, Algebra II) Generally, a “higher-dimensional line” refers to adding more variables into a linear equation. For example, $x + y + z = 1$ has three variables, instead of just two, and defines a plane in three-dimensional space.
    • Plot this plane in the GeoGebra 3D calculator. Do you think this deserves to be called a higher-dimensional line? Why or why not?
    • Play with the coefficients of the plane, and observe how the image in GeoGebra changes. How does the idea of a line’s “slope” generalize to higher dimensions?
    • Solve the following system of equations using both substitution and elimination.
      $$x + y + z = 1, \quad x + y – z = 1, \quad x + 2y + 3z = 3.$$

—Leila Sloman

Cancer has many faces − 5 counterintuitive ways scientists are approaching cancer research to improve treatment and prevention

The Conversation, November 1, 2023.

Cancer is not one disease. Cancer differs in the body parts it resides in, the genetic mutations that propel it, how it interacts with the body, and who it infects. This means that oncologists must think about cancer in many different ways. One of those ways is math. Of the five unconventional ways of thinking about cancer described by Vivian Lam in The Conversation, some are biological, like evolution and inflammation. But in the mathematical lens, one oncologist tells Lam that the mathematical randomness occurring in cancer biology, called epigenetic entropy, may be key to learning how to prevent cancer. “Epigenetic entropy shows that you can’t fully understand cancer without mathematics,” he says.

Classroom Activities: stochasticity

  • (All levels) The mathematics of cancer mentioned in the article deals with stochasticity, a word for randomness. Create a table with 100 rows and two columns. Each row represents the outcome of a coin flip—either “heads” or “tails.” In the first column, write what you expect 100 successive coin flip results to look like. Imagine that your goal is to convince someone that your list is really a list of 100 coin flips. In the next column, do 100 coin flips and record the outcome on each row.
    • Compare the two columns. How are they similar? How are they different? Did anything in the real coin flip surprise you?
    • Now, watch this video about stochasticity, which discusses the results of an experiment very much like the one you just performed on yourself. Discuss what you’ve learned.
    • Revisit the October digest on coin flipping for alternative activities and videos on coin flip outcomes and manual attempts to mimic them.
  • (All levels) Play this educational game about epigenetics, which will teach you the basic science of what is being randomized in this area of cancer.
  • (Mid level) The article describes how stochasticity appears in other processes like the stock market and epidemiology. Take a few minutes to research how stochasticity plays a role in both fields. What parameter is random in each field? How can we measure the randomness? Based on your reading, describe how the mathematics of epigenetics is related to the math of epidemiology and finance.
    • In what ways do the examples seem different?
    • Could researchers of one field (e.g. stochasticity in epidemiology) learn from the work and discoveries of researchers in another (e.g. stochasticity in cancer epigenetics)? Discuss why or why not as a group.

—Max Levy

Some more of this month’s math headlines