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- Gödel’s incompleteness theorem in
*The Guardian* - “Fun with Math” in
*The New Yorker* - Rock-paper-scissors and evolutionary game theory
- “What physics owes to math”

Alex Bellos’s Monday puzzle in *The Guardian* for January 10, 2022 was derived from explanations of Gödel’s incompleteness theorem due to the logician Raymond Smullyan. (Smullyan published *Forever Undecided: A Puzzle Guide to Gödel *in 1987). The setting for the puzzle, as Bellos presents it, is a hypothetical island he calls “If.” Natives of If are either “Alethians” or “Pseudians;” they are indistinguishable except that Alethians always tell the truth, while Pseudians always lie. This sounds like the traditional Liar Problem, but there is a wrinkle: on the island is a Ledger where every native is listed, along with his or her tribe. Anyone can consult this Ledger. You arrive on If and a person, Kurt, comes up to you stating: “You will never have concrete evidence that confirms I am an Alethian.” The puzzle: is Kurt an Alethian, a Pseudian or neither? Think about it before checking Bellos’s solution and before reading on.

Now comes the connection with Gödel’s incompleteness theorem, which states, as Bellos puts it, “that there are mathematical statements that are true but not formally provable.” Suppose you are the first non-native ever to visit If, so you know that everyone you meet is Alethian or Pseudian. Kurt pops up and says, just as before, “You will never have concrete evidence that confirms I am an Alethian.” But now just as he speaks the Ledger burns to ashes.

Where are we? Kurt cannot be a Pseudian, because with no Ledger that statement has to be true. So you know Kurt is an Alethian. But you can never have concrete evidence of that fact because if you did, his statement would be false, and it can’t be false since he is Alethian. Think about it. [Thanks to Jonathan Farley for bringing this item to my attention. -TP]

Dan Rockmore’s contribution to “Talk of the Town” in the January 17, 2022 issue of the *The New Yorker* was an item titled “Fun with Math.” He recounts attending “a recent evening of math dinner theater” organized by Cindy Lawrence, director of New York’s National Museum of Mathematics. The event featured Peter Winkler, a mathematics professor at Dartmouth and expert on math puzzles. Among the puzzles and phenomena that Winkler served up for discussion during dinner:

- “On average, how many cards does it take to get to a jack in a shuffled deck of fifty-two cards?”
- “What’s the best way to use two coin tosses to determine which of two coins, one fair and one ‘biased,’ is fair?”
*Simpson’s paradox*in statistics, best described by an example: for Berkeley’s graduate programs in 1973, overall “men were admitted at a higher rate than women, but, program by program, women were admitted at a higher rate.” (See MinutePhysics for a more detailed explanation.)

Apropos of Simpson’s paradox Marilyn Simons, a guest with a PhD in economics, remarked, “I think that, to a lot of us who even *think* we know statistics, the way we process statistics is not deeply informed.” Elsewhere Rockmore quotes her as saying that her husband Jim (identified as “a financier and a former mathematician”) doesn’t like puzzles: “He says that if he works that hard he wants to get a theorem out of it.”

“Non-Hermitian topology in rock-paper-scissors games” by the three Tsukuba physicists Tsuneya Yoshida, Tomonari Mizoguchi and Yasuhiro Hatsugai was published January 12, 2022 in *Scientific Reports*. This is a physics article, but it applies a nice piece of mathematics, *evolutionary game theory,* to the familiar rock-paper-scissors game.

The game consists of two players; at a signal each shows a clenched fist (“rock”), a flat hand (“paper”) or a vertical hand with the first two fingers displayed (“scissors”). The winner (rock smashes scissors, scissors cut paper, paper covers rock) gets one point, the loser loses one. If both players show the same symbol, each gets zero.

The article contains this image:

Here R, P and S have to stand for rock, paper and scissors, but how is this diagram related to the game? We need to make a detour into e*volutionary game theory*. This is a method for simulating the process of evolution in populations. Here the population is split among three subspecies; let’s call them Ravens, a fraction $s_1$ of the population, Penguins with $s_2$, and Swifts with $s_3$, where the fractions $s_1, s_2, s_3$ add up to 1. These correspond to the three “pure strategies” in the game: at every encounter, a Raven will play “rock,” a Penguin will play “paper” and a Swift, “scissors.” The *state vector* ${\bf s}=(s_1, s_2, s_3)$ encapsulates the current mix in the population.

Evolution occurs in time. Suppose the population is in state ${\bf s}$ at some moment. Where will it be just a litle later? For example, suppose ${\bf s}=(\frac{1}{2},\frac{1}{2},0)$. That means half the population are Ravens and half are Penguins. So a Raven will meet a Penguin with probability $\frac{1}{2}$, and can expect to lose half a point. Likewise a Penguin will meet a Raven with probability $\frac{1}{2}$ and so can expect to gain half a point. The Penguins have an advantage. If the object is to model evolution, then the Penguin’s advantage in that state should translate into their population increasing at the expense of the Ravens. That gives a clue to the meaning of the red arrow at the point $(\frac{1}{2},\frac{1}{2},0)$: the population mix at that point is shifting to the right. More Penguins, fewer Ravens.

To make this more precise, keeping the language of evolution, we measure the *fitness* of one of the groups at some state ${\bf s}=(s_1,s_2,s_3)$ of the population by the expected gain or loss in points at the next encounter. So the fitness of the Ravens at state ${\bf s}$ will be the probability of meeting a Penguin times $-1$ plus the probability of meeting a Swift times 1. We write this as

$F(\mbox{Ravens}|{\bf s})= -s_2 + s_3.$ Similarly $F(\mbox{Penguins}|{\bf s})= s_1 – s_3$ and $F(\mbox{Swifts}|{\bf s})= -s_1 + s_2.$

Finally we set up a dynamical system by stating that the proportion of the population in any group will increase or decrease exponentially with growth coefficient equal to the fitness of that group (which can be positive or negative) at that instant in time. Writing that statement as a differential equation gives the *replicator equation* for rock-paper-scissors:

$$\frac{ds_1}{dt}= s_1(-s_2 + s_3), ~~\frac{ds_2}{dt}= s_2(s_1 – s_3), ~~\frac{ds_3}{dt}= s_3(-s_1 + s_2).$$

In vector form, the equivalent equation is

$$\frac{d{\bf s}}{dt}= (s_1(-s_2 + s_3), s_2(s_1 – s_3), s_3(-s_1 + s_2)).$$

Now we can interpret the first image in this item, which shows the state space for rock-paper-scissors as an evolutionary game. The arrows represent the direction of evolution, with the magnitude encoded by color saturation. The central cross marks the equilibrium $(\frac{1}{3},\frac{1}{3},\frac{1}{3})$; the blue loop is the solution curve obtained by numerically integrating the replicator equation starting at the point ${\bf s} = \frac{1}{3}(1-\delta, 1+\delta/2, 1+\delta/2)$, with $\delta=0.1$.

For this game, all the solution curves are closed loops. In fact, they are level curves of the function $f(s_1,s_2,s_3)=s_1s_2s_3$. This follows from the equality $\displaystyle{\frac{d{\bf s}}{dt}\cdot \nabla f= 0}$, which is actually fun to check. Try it.

The evolutionary game described here is just the starting point for the article by Yoshida *et al.* They consider perturbations of this system that break its symmetry—surprisingly, the resulting phenomena have parallels in condensed-matter physics.

On January 12, 2022 *Le Monde* ran a guest article with the title “What physics owes to math” (text in French) written by two physicists, Jean Farago and Wiebke Drenckhan, from the Institut Charles-Sadron in Strasbourg. The authors begin with the famous quote from Galileo about the Book of Nature being written in the language of mathematics, and go on to observe: “The millenia elapsed between the birth of mathematics and its use in physics demonstrate that this contiguity between natural phenomena and the mathematical laws of our human rationality was far from being obvious.”

Farago and Drenckhan mention that one of the most antonishing examples of the “intimate” relationship between mathematics and physics comes from complex numbers. Starting in the 16th century, mathematicians found that calculating solutions to polynomial equations with whole-number coefficients required the use of an ‘imaginary’ number $i$ with $i^2=-1.$ “How could anything be more abstract than this fictitious number, given that ordinary numbers always have a positive square ($2^2=(-2)^2=4$)!” But fast-forward to 1929 and Schrödinger’s equation $i\hbar\partial_t\psi=H\psi$, which doesn’t work without it. We read that no one was more surprised by “this irruption of $i$ in the corpus of physical laws” than Schrödinger himself, and that he described his reaction, in a footnote, by quoting an unnamed Viennese physicist, “known for his ability to always find the *mot juste*, the cruder the *juste*r,” and who compared the appearance of $i$ in that equation to one’s involuntary (but welcome) emission of a burp. Our authors add: “This shows us that a contiguity can sometimes also exist between humor and physics.” [My translations. -TP]

“What physics owes to math” could have mentioned an article from *Nature* last month: “Quantum theory based on real numbers can be experimentally falsified,” written by an international team with corresponding author Miguel Navascués (IQOQI, Vienna). Physical experiments are expressed in terms of probabilities, which are real numbers. So why can’t there be a “real” quantum theory? The authors show that complex numbers are actually needed, by devising “a Bell-like experiment, the successful realization of which would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.”

Math in the Media has moved to a new platform! In order to continue receiving each month’s posts via email, **you must visit the new website to confirm your subscription:**

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*The Guardian, *January 24, 2022

Time and time again, mathematical ideas developed decades or even centuries ago find unexpected—and profitable—uses in industry. In an opinion piece for *The Guardian*, mathematician David Sumpter gives a quick tour of how mathematics has transformed the modern world and speculates about which branch of mathematics will lead to money-making applications next. Fractal geometry, chaos theory, and random walks are all possibilities, he writes. “You don’t need to be a mathematical genius yourself in order to put the subject to good use. You just need to have a feeling for what equations are, and what they can and can’t do.”

**Classroom activities: ***fractals, probability, matrices*

- (All levels) Introduce students to fractals with K-12 activities from the Fractal Foundation. Topics include fractal triangles, coastlines, exponents, and more.
- Ask students to find examples of fractals in their own home or neighborhood and share them with the class.

- (Middle school) Introduce students to random walks using coin flips with this lesson plan from the National Museum of Mathematics.
- (Linear Algebra) When teaching eigenvalues and eigenvectors, use Google’s PageRank algorithm as an example (see these Cornell notes). Discuss: what are some other situations where a similar approach might be useful? What are the limitations of PageRank?
- (Advanced) Have students complete the exercises at the bottom of the Cornell notes.

*—Scott Hershberger*

*Slate, *January 14, 2022

During this winter’s Omicron wave, you may have taken rapid COVID-19 tests. But interpreting the results is not always straightforward. In an article for *Slate*, mathematician Gary Cornell explains two statistical terms and how they relate to COVID tests. A high *specificity* means that a test gives very few false positives. A high *sensitivity* means that a test gives very few false negatives. Rapid COVID tests have a high specificity, but a not-so-high sensitivity, Cornell writes—so a positive test result means you are almost certainly infected, but a negative result cannot give total confidence that you are in the clear.

**Classroom activities: ***statistics, specificity and sensitivity*

- (High school) Introduce students to sensitivity and specificity with this lesson plan from Penn State. The lesson also discusses
*positive predictive value*and*negative predictive value*, which answer an important question: What is the chance that a person who tests positive is infected, or that a person who tests negative is not infected? - (High school) A test’s predictive value depends in large part on how common the condition is in the population. (For example, a recent New York Times investigation found that prenatal tests for rare disorders give far more false positives than true positives.) For each of the following hypothetical diseases, suppose you take a test with 99.9% specificity and 80% sensitivity. If you test positive, what is the probability that you actually have the disease? If you test negative, what is the probability that you do not have the disease?
*(Hint: create charts like those in the lesson plan.)*- Disease W: prevalence is 10 out of every 100,000 people
- Disease X: prevalence is 10 out of every 1,000 people
- Disease Y: prevalence is 10 out of every 100 people
- Disease Z: prevalence is 40 out of every 100 people

Discuss how these results could be relevant during the different stages of a pandemic.

*—Scott Hershberger*

*Quanta Magazine*, January 13, 2022

Despite all we know about math, the field is still full of mysteries. Some may seem hopelessly abstract, but others have to do with fundamental concepts we all recognize, like primes. “Prime numbers are the most fundamental — and most fundamentally mysterious — objects in mathematics,” writes Kevin Hartnett*.* A prime number (like 3, 5, 23, or 419) is only divisible by 1 and itself. The mystery of primes is that they seem to follow no discernible pattern. Yet a 160-year-old idea called the Riemann hypothesis suggests that there *is* a pattern to be found, and mathematicians are hard at work trying to crack it. There is even a million-dollar prize on the line. In this article, Hartnett describes a groundbreaking new step toward solving this stubborn mystery.

**Classroom Activities: ***prime numbers, finding patterns, sequences*

- (All levels) Learn more about the Riemann hypothesis with these videos from
*Quanta Magazine*and Numberphile. - (Mid level) Write out the prime numbers under 100, one below the other. There should be 25 of them. One column to the right, in the space between each consecutive prime, write the result of subtracting the smaller number from the larger number. Do you see a pattern? Discuss why or why not. Compare this to the pattern you get when subtracting consecutive Fibonacci numbers.
- (High level) Compute the values of the function $f(n) = n^2 + n + 41$ for $n = 1, 2, 3, \text{and } 4$. Referencing your earlier work, what appears to be happening with this function? Can you find a counterexample to your conjecture?
- (High level) Have each student come up with their own simple rule to create a sequence of numbers. The rule should involve just addition, subtraction, multiplication, or division and should involve either one, two, or three consecutive terms of the sequence. Ask students to swap sequences in partners and see if they can figure out each other’s patterns.

*—Max Levy*

*The Conversation*, January 4, 2022

With a few folds, a piece of paper can become a piece of art—and maybe more. In an article for *The Conversation*, mathematician Julia Collins writes about how origami can inspire mathematical discovery. Collins starts with a small, parallelogram-shaped bit of origami called the sonobe unit. With six of those units, you can build a cube. With more of them, you can create other mathematical shapes such as Platonic solids, Archimedean solids, and Johnson solids. You can also explore principles in the mathematical field of graph theory, like the Four-Color Theorem, by building a shape out of sonobe units of different colors. Origami may even be useful for technology like unfolding solar panels in space.

**Classroom activities:** *origami, geometry, technology*

- (Middle school / high school) Follow Collins’ instructions or the video linked in her article to build sonobe units out of a square pieces of paper.
- Students can work alone or in groups to build cubes or more complicated geometric shapes out of the sonobe units.
- Discuss: what does it mean for a shape to have symmetry? Why does the sonobe unit lend itself to building objects with symmetry?

- (Middle school / high school) Watch the video “See a NASA Physicist’s Incredible Origami,” which is linked to in the article. What are some examples of technology that might be inspired or improved by origami?

*—Tamar Lichter Blanks*

*CTV News, *January 13, 2022

Throughout the COVID-19 pandemic, mathematical models have helped policymakers estimate infection risk based on factors like vaccination status, indoor versus outdoor setting, and crowd density. But one of the most important factors for determining transmission risk is also potentially misleading: positivity rate, or the percentage of tests that come back positive. The hyper-transmissible Omicron variant is straining test supplies, so a larger slice of positive cases is going unreported—skewing positivity rates. Mathematicians modeling the spread of COVID-19 are struggling to keep up. “We’re still adapting to flying blind in terms of reported cases,” one mathematician told *CTV News* reporter Sarah Smellie. In this article, Smellie explains how mathematicians need to adapt their models to keep up with the constantly changing pandemic.

**Classroom Activities: ***exponential growth, logarithms, data analysis*

- (Algebra II) The doubling times (how long it takes for the number of infections to double) for Omicron are “some of the fastest we’ve seen in the pandemic”—between 1.5 and 3 days in some regions. Imagine a city of 10 million people where two people are sick. If nobody is vaccinated or takes any precautions to prevent the spread, how many days would it take for 10% of all inhabitants to catch the disease if:
- The cases doubled every 2 days
- The cases doubled every 3 days
- Discuss the implications for public health interventions.

- (High level) Collecting enough data points is an important part of having a reliable model. To see why, gather two different colors of marbles (or pieces of paper or other item)—one will represent negative cases and the other positive. Place 10 marbles in each of four identical bags or boxes according to the following ratios of
*positive:negative*: 1:9, 2:8, 3:7, 5:5. Now, scramble the bags and remove one marble from each bag. Write down a guess of which bag corresponds to which ratio. Repeat this until no marbles remain. How many rounds did it take until you were correct about all of the bags? Discuss how this relates to the challenge of determining COVID infection rates by sampling from different areas of the country.*Remote-friendly version: Use**Wheel of Names**with 4 different ratios of names “positive” or “negative” instead of marbles and bags.*

**Related Mathematical Moments: ****Resisting the Spread of Disease.**

*—Max Levy*

**After twice being denied tenure, this Naval Academy professor says she is seeking justice**

The Washington Post, January 31, 2022**The Texas Oil Heir Who Took On Math’s Impossible Dare**

The New York Times, January 31, 2022**Why mathematicians sometimes get Covid projections wrong**

The Guardian, January 26, 2022**How Infinite Series Reveal the Unity of Mathematics**

Quanta Magazine, January 24, 2022**Take an online journey through the history of math**

Science News, January 18, 2022**Love Wordle? Here’s How to Use Math to Dominate Your Friends at the Viral Word Game**

Popular Mechanics, January 14, 2022**Fairer Elections in Pa. Could Depend on 12 Mathematicians**

NBC Philadelphia, January 17, 2022**ArXiv.org Reaches a Milestone and a Reckoning**

Scientific American, January 10, 2022**Secret maps revelation, testimony of mathematicians deal blows to GOP defense as NC redistricting trial concludes**

NC Policy Watch, January 7, 2022**Minnesota mathematicians, data scientists use new technology to shape political districts**

*Minneapolis Star-Tribune,*January 1, 2022

*The Conversation, *December 1, 2021

Pop superstar Adele has a habit of titling her albums in a peculiar way. Each album title is an integer that represents how old the singer was when she began writing the songs. David Patrick of the Art of Problem Solving searched the On-Line Encyclopedia of Integer Sequences to see if Adele’s albums match any known integer sequence. Nine sequences turned up—some of them easy to describe and others less so. For *The Conversation, *Anthony Bonato explains the rules underpinning one of the possible “Adele sequences.”

**Classroom Activities: ***sequences, prime numbers*

- (Middle school / high school) Explore the OEIS by having students come up with the first four terms of an integer sequence. They can choose four integers based on something in their own life, or they can choose four integers less than 40 that could represent the ages at which a musician releases new albums.
- Have students search the OEIS to see if their sequence matches any known sequences in mathematics, then share their findings with the class.
- Discuss: what do the results suggest about the mathematical significance of the “Adele Sequence”?

- (Number Theory, Problem-solving) The sequence A072666 that Bonato analyzes depends on the sequence of prime numbers. Ask students to prove that there are infinitely many primes using the following hints (full solution here):
- Assume that there are finitely many primes which can be denoted $p_1,p_2, \dots, p_n$.
- Consider the number $(p_1 \times p_2 \times \dots \times p_n) + 1$.

—*Leila Sloman*

*Science News for Students*, December 16, 2021

Turbulence in water affects where fish eggs end up. Massive icebergs drift on ocean currents. Wildfire smoke from the U.S. West can make its way to Europe. In all of these cases, mathematics helps researchers predict what will happen, offering potential solutions to environmental threats. As Rachell Crowell explains, the computer models used to simulate how objects drift depend on the math of *differential equations*. These equations relate quantities that change as time passes and vary at different locations in the environment. Crowell interviews several researchers about what they have learned about the science of drifting and why it is important.

**Classroom activities: ***mathematical modeling, differential equations, heat equation*

- (Algebra I, Algebra II) The article mentions that satellites helped reveal drifting wildfire smoke. Satellites are also useful in studying the fire itself. Explore one real-life example with this assignment from NASA’s Jet Propulsion Laboratory.
- (Differential Equations) The article also mentions that researchers use the heat equation to predict icebergs’ melt rate. When introducing partial differential equations, show students this 3Blue1Brown video that visually explains the heat equation.

*—Scott Hershberger*

*The New York Times*, December 10, 2021

Math shapes the world around us—and it shapes the *shapes* around us too. A trio of researchers from France and the United Kingdom are known for investigating the math and physics of how seashells form. Their newest advance deals with ammonites, an extinct group of mollusks. A type of ammonites called Nipponites had weird shells that twisted, turned, coiled, and bulged in unexpected directions. “The first time you look at it, it’s just this tangled mess,” mathematician Derek Moulton told reporter Sabrina Imbler. “And then you start to look closely and say, oh, actually there is a regularity there.” In this article, Imbler shares how Moulton’s team came up with a mathematical model that explains the origins of these ammonites’ weird shapes.

**Classroom activities: ***growth rates, symmetry, golden ratio*

- (All levels) Nipponites appear so strange because they are very asymmetrical, whereas nature is normally full of symmetry. Ask students why symmetry tends to be beneficial in the living world. (As an example, think of symmetry in birds and fish as they move.)
- (All levels) Nipponites provide one example of an initial symmetry giving way to an asymmetric result. Have students explore another example using a thin string. Hold the string above a table so it hangs freely. Now lower it gently so that one end touches the table, and continue to lower it until it has folded over itself a few times. Inspect the bundle and take a picture. Repeat this a few times and share your observations. Does the bundle look the same every time or does it appear unique? Discuss how this compares to what happens inside an ammonite shell.
- (Algebra II, Pre-calculus) To learn more about how a simple mathematical rule can give shells and flowers fascinating shapes, watch this Numberphile video on the Golden Ratio.

*Related Mathematical Moments:* Going Into a Shell.

*—Max Levy*

*Newsweek,* November 30, 2021

When Lotfi Zadeh first came up with the idea for fuzzy logic, a way of enshrining in mathematics the uncertainty and imprecision of life, it was not received well by everyone. But almost sixty years later, Zadeh’s impact on the world is indisputable. On November 30, a Google Doodle celebrated this impact, marking the 57th anniversary of Zadeh’s seminal paper, “Fuzzy Sets.” For *Newsweek*, Soo Kim articulates what was so important about that work: “Considered an early approach to artificial intelligence, Zadeh’s fuzzy logic structure formed the basis of various modern everyday technologies including facial recognition, air conditioning, washing machines, car transmissions, weather forecasting, stock trading and rice cookers.”

**Classroom Activities: ***set theory, logic, fuzzy logic*

- (High school) Teach an introduction to set theory with this online lesson, assigning the “Try It Now” boxes as problems. For extra practice, assign the following problems:
- Let $S = \{ 1, 10, 19 \}$. Write down all the subsets of $S$.
- Suppose the universal set is all the integers. Let $A = \{ \text{even integers} \}$ and $B = \{ \text{multiples of 3} \}$.
- Find 3 numbers in $A^c \cap B$.
- Find 3 numbers in $A \cap B^c$.

- (Advanced) Prove that for general sets $A$ and $B$, $(A \cap B)^c = A^c \cup B^c$.

- (High school) In classical logic, an item either belongs to a set or it doesn’t—there is no in-between. By contrast, fuzzy logic allows items to be partly in sets. Have students read this
*Britannica Kids*explainer on fuzzy logic. Ask them to think of three situations in which fuzzy logic might be more useful than classical logic and justify their claims.- (Advanced) Have students read the first four pages of Zadeh’s original paper and then come up with an example of two membership functions $f_A(x)$ and $f_B(x)$ in the real numbers such that $A$ is a fuzzy subset of $B$.

*—Leila Sloman*

*Nature, *December 1, 2021

Mathematics might sometimes seem very dry. You follow fixed rules to find right answers and check your work carefully to avoid wrong answers. That process doesn’t seem creative, but when it comes to discovering new proofs and theorems, creativity is arguably the most important trait. In an article for *Nature*, Davide Castelvecchi explains how math researchers have recruited a new ally to track down creative solutions to math’s mysteries—artificial intelligence. Researchers used machine learning, a type of AI, to comb through huge datasets and find patterns related to the study of knots and symmetry. It takes creativity and intuition to find these hidden patterns. “As mathematical researchers, we live in a world that is rich with intuition and imaginations,” one researcher said. “Computers so far have served the dry side. The reason I love this work so much is that they are helping with the other side.”

**Classroom activities** *machine learning, knots*

- (All levels) The machine-learning algorithm helped researchers solve a question about knots that eluded researchers for decades. Read more about this study’s “knot” result and/or watch this Numberphile video about knot theory.
- (Middle level) A central question that researchers ask is whether a set of knots that appear different are actually equivalent. With a shoelace, string, or yarn, try making some of the distinct knots found here. Discuss what properties make them different. (For example, how many times the thread crosses over itself.)
- (High level) Discuss
*why*machine learning can speed up innovation, based on this article and others. (Here’s a primer on machine learning.) One researcher told the writer: “Without this tool, the mathematician might waste weeks or months trying to prove a formula or theorem that would ultimately turn out to be false.” If artificial intelligence can help find patterns quickly, what other areas of science or society could benefit from these quick solutions? In what situations would machine learning be more dangerous or unethical?

*Related Mathematical Moments:* Being Knotty.

*—Max Levy*

**Our Favorite Things: Math and Community in the Classroom**

Short Wave, NPR, December 28, 2021**The Year in Math and Computer Science**

Quanta Magazine, December 23, 2021**How Mathematicians Cracked the Zodiac Killer’s Cipher**

Discover Magazine, December 18, 2021**Google Doodle honors French mathematician Émilie du Châtelet**

CNET, December 16, 2021**Mathematician Hurls Structure and Disorder Into Century-Old Problem**

Quanta Magazine, December 15, 2021**Quantum physics requires imaginary numbers to explain reality**

Science News, December 15, 2021**Shirley McBay, Pioneering Mathematician, Is Dead at 86**

The New York Times, December 14, 2021**Abstractions Are Good for Goodness’ Sake**

The Wall Street Journal, December 10, 2021**Using Math to Rethink Gender**

Short Wave, NPR, December 1, 2021

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- The exponential function in
*Slate* - New mathematical ideas from artificial intelligence
- Math history online

The online magazine *Slate* invited Gary Cornell to explain to all of us why the Omicron variant of Covid-19 was spreading so fast. (This was back on December 17, 2021). He contributed “The Math That Explains Why Omicron Is Suddenly Everywhere” with an illustration containing a graph like this one:

Cornell explains that the current pattern of Omicron cases doubling every two days is an example of *exponential growth,* and that we humans are not wired to process that phenomenon reliably: “when it comes to exponential growth, your gut feelings *are* going to be wrong and you need to stop and do some (elementary) math.” The characteristic of exponential growth is that the increase during the next time period is proportional to the value right now. To illustrate this phenomenon Cornell retells the story of the king and the chessboard, which ends up with the king owing his opponent one grain of rice on the first square, two on the second, four on the third, and so on, totaling “more than 18 quintillion grains of rice, which would roughly cover the planet and would be the world’s output of rice for about 1,000 years.” He reminds us that cases of the Omicron variant, which were doubling every two days, have the same growth potential. And even if the hospitalization rate is as low as 1%, after less than two weeks the number of hospitalizations would be greater, day by day, than the number of infections from two weeks before.

“For the first time, machine learning has spotted mathematical connections that humans had missed.” This is the beginning of Davide Castelvecchi’s news piece in the December 9, 2021 issue of *Nature*. The article Castelvecchi refers to, “Advancing mathematics by guiding human intuition with AI,” was published in the same journal a week before. The authors were 11 members of Alphabet Inc.’s Deep Mind laboratory in London working with three academic mathematicians from Oxford and Sydney. The team, led by Deep Mind’s Alex Davies and Pushmeet Kohli, states: “We propose a process of using machine learning to discover potential patterns and relations between mathematical objects, understanding them with attribution techniques and using these observations to guide intuition and propose conjectures.” They present two examples, one involving knots and one representation theory, of their process in action.

The authors set up a general framework for approaching the process, as follows. They take $z$ to be a variable element of some class of mathematical entities (knots, polyhedra, groups, …) and consider two “mathematical objects” $X(z)$ and $Y(z)$ associated with $z$. Are these two objects related, and how? They think of such a relationship as a function, and ask if there is an $f$ such that $f(X(z))\approx Y(z)$ for all $z$.

They give an example of what they have in mind. Let $z$ range over the set of 2-dimensional polyhedra; they set $X(z)$ to be the vector $(V(z), E(z), Vol(z), Surf(z))$, where $V$ and $E$ are the numbers of vertices and edges, $Vol$ and $Surf$ the volume and the area of the polyhedron; and $Y(z)$ to be the number $F(z)$ of faces of $z$. In this case Euler’s formula $V-E+F=2$ means that the function $f(X) = (V, E, Vol, Surf)\cdot (-1, 1, 0, 0) +2$, where $\cdot$ is the dot-product of vectors, does the trick, since in fact $-V+E+2 = F$. Notice that in this example the volume and surface area are not part of the calculation. Eliminating spurious information is part of the “attribution techniques” mentioned above, used in an iterative process of refining the form of the relation to be found.

The training process relies on a large set of specimen values of $z$, a large bank of possible functions as candidates for $f$, and an initial guess of which objects $X(z)$ and $Y(z)$ have the potential of being part of a true mathematical statement of the form $f(X(z))= Y(z)$. If the machine finds an $f$ that works more often than expected by chance, the human partners examine the form of $f$ to see how it can be improved. Besides discarding spurious variables as above, a technique the authors use is *gradient saliency:* imitating optimization in calculus by examining the derivative of outputs of $f$ with respect to the inputs. “This allows a mathematician to identify and prioritize aspects of the problem that are most likely to be relevant for the relationship,” they write. They emphasize the interactive aspect of the iterative process, where “the mathematician can guide the choice of conjectures to those that not just fit the data but also seem interesting, plausibly true and, ideally, suggestive of a proof strategy.”

Applied to knot theory, the process yielded a theorem the authors describe as “one of the first results that connect the algebraic and geometric invariants of knots.” They comment: “It is surprising that a simple yet profound connection such as this has been overlooked in an area that has been extensively studied,” echoing Euler’s remark about his $V-E+F=2$ discovery: “It seems extremely amazing that, while Stereometry along with Geometry has been studied for so many centuries, nevertheless some of its most basic elements have been unknown until now” (Demonstratio … , p. 141).

Castelvecchi tells us that Alex Davies, one of the project leaders from Deep Mind, “told reporters that the project has given him a ‘real appreciation’ for the nature of mathematical research. Learning maths at school is akin to playing scales on a piano, he added, whereas real mathematicians’ work is more like jazz improvisations.”

“Online exhibit adds up the history of mathematics” by Erin Blackmore ran in the *Washington Post* on November 28, 2021. Blackmore reports on a collaboration among the National Museum of Mathematics, Wolfram Research, and the Overdeck Family Foundation. The History of Mathematics Project has nine interactive exhibits (Counting, Arithmetic, Algebra, Geometry, etc.), each with around six items. Each item has extra graphics, interpretations, and an interactive component. For example, the Rhind Papyrus appears in the Arithmetic exhibit along with a Wolfram-powered app that visualizes how each of the fractions 2/(odd number between 3 and 101) can be expressed, Egyptian-style, as a sum of fractions with numerator 1. If you want, you can see the new denominators in the Egyptian hieratic script used on the Papyrus. “Whether you come to try your hand at some ancient math homework or to enjoy imagery from artifacts from around the world,” Blackmore writes, “you’ll come away with a greater appreciation of how math developed — and how much modern math owes to our brainy ancestors.”

*Quanta Magazine,* November 18, 2021

No hot dog is complete without a bun. So it’s maddening that you typically have to buy eight buns at a time, but you can’t buy exactly eight hot dogs. Getting one package of each will leave you with two extra hot dogs and no buns*—*useless leftovers. Patrick Honner explains how the Chinese Remainder Theorem helps you avoid this situation. If you buy the right number of packages, you’ll end up with exactly one bun for each hot dog. Even when there are a few extra hot dogs or buns hanging out in the fridge, the Chinese remainder theorem guarantees you’ll be able to make things work out. If that’s too much math for you, there’s also an easier solution: In my experience, veggie sausages come in packages of four.

**Classroom activities: ***number theory, modular arithmetic, cryptography*

- (High school) Have students read the article and complete the exercises at the end.
- (High school) Honner mentions how the Chinese remainder theorem comes into cryptography. Have students use the A1Z26 cipher to decrypt codes in class:
- Break students into pairs. Have each student think of a secret word and encode it using the A1Z26 cipher. Then, have them encode it even further by finding the numbers mod 4 and mod 9. Thus the word CODE, encoded as (3-15-4-5), becomes (3-3-0-1) mod 4 and (3-6-4-5) mod 9.
- Have students trade their codes mod 4 with their partner and try to break them. After 5 minutes, allow them to share the code mod 9. According to the Chinese remainder theorem, they should now have enough information to break the code.
- Repeat the activity with new secret words, but this time have students share the code mod 9 first. Have students write about which way was easier and explain the reasoning they used.

*—Leila Sloman*

*Spectrum News 1, *November 4, 2021

Every 10 years, all the states in the US go through the process of *redistricting:* drawing new lines for congressional and state legislative districts based on the latest census data. The current redistricting cycle has been extremely contentious, with partisan map-drawers in many states creating maps heavily biased in favor of their political party. North Carolina is a perfect example. When mathematician Jonathan Mattingly’s team compared the state’s proposed maps to tens of thousands of alternatives, “We found that the map that has been proposed for the North Carolina House really dramatically under-elects Democrats. We have similar analysis for the Senate.” The mathematicians’ methods, which have already featured prominently in court cases in recent years, provide rigorous quantitative evidence that the maps in North Carolina and elsewhere intentionally give one party an unfair edge. (A *Raleigh News & Observer *article also quotes Mattingly.)

**Classroom activities: ***gerrymandering, voting, geometry, Markov chain Monte Carlo*

- (Middle school) Teach students about gerrymandering with this lesson plan from Illustrative Mathematics. The lesson begins with simple examples of elections for a school mascot and school board, then gives students the opportunity to draw district maps that favor one candidate over another.
- (Middle school / high school) Play this gerrymandering game by the Julia Robinson Mathematics Festival with 28 levels that increase in difficulty. Have students work in groups, then share their solutions with the class.
- (High school) Teach students about gerrymandering with this lesson plan from Corwin. The worksheets include problems related to contiguity, the efficiency gap, and two mathematical measures of compactness.
- (Advanced) Introduce the concept of Markov chain Monte Carlo methods, which underlie the mathematical analysis of the North Carolina maps mentioned in the article.

*Related Mathematical Moments poster and interview: *Countermanding Gerrymandering.

*—Scott Hershberger*

*Science News*, November 10, 2021

At the center of an oyster is something beautiful: a pearl. At the center of a pearl is something less so: a misshapen lump of debris. For years, scientists have wanted to figure out how oysters grow perfectly round pearls over irregular bits of junk. Now, the mystery is solved, Rachel Crowell writes. The answer is a mathematical pattern found commonly in nature, called 1/f noise or *pink noise*. Oysters build tiny layers of mineral and protein, called nacre, around the debris or grains of sand, gradually smoothing out the odd shape. Each layer of nacre has a different thickness that depends mathematically on the one below it. This *inversely proportional* relationship also shows up in seismic activity, classical music, and heartbeats.

**Classroom activities: ***pink noise, inverse proportionality, waves*

- (All levels) Hear what pink noise sounds like on Wikipedia, and watch videos about pink noise in human biology and music production.
- (All levels) Discuss why this sound is called pink noise. (Hint: sound and light are both waves. What’s the difference between white light and pink light?)
- (Middle level) When teaching students to graph equations like $y=1/x$, use pink noise as an example application. (Note that most graphs of pink noise appear linear because the axes are on a logarithmic scale.)
- (High level) What color might you associate with a sound that has a high intensity at middle frequencies but zero for high and low frequencies? What about a sound that has a low intensity at low frequencies but a high intensity at high frequencies? (Answers in the “Colors of Noise” Wikipedia page)

*—Max Levy*

*The Conversation*, November 21, 2021

Is math real, or is it just a way of describing the world? According to Sam Baron, an Australian professor, math is as real as the paper, the pen, and the brain with which you do it. Baron recently wrote an article for *The Conversation* in which he argues that math is an essential part of nature, not just a way of describing it. He draws upon examples of math in nature to prove his point. Bees make six-sided cells to store honey since hexagons form the most efficient tiles. Periodic cicadas evolved to emerge every 13 and 17 years because prime numbers help them avoid the more regular feasting patterns of predators. “The world has two parts, mathematics and matter,” Baron writes. “Mathematics gives matter its form, and matter gives mathematics its substance.”

**Classroom activities:** *tiling, prime numbers, nature*

- (All levels) Hexagons can tile a flat surface (cover it entirely with no gaps) to infinity. According to the
*honeycomb conjecture*in math*,*the hexagon is the best shape for tiling a plane, in the sense that hexagons require the shortest total perimeter to cover a given area. Bees use this fact to build efficient honey storage.- Watch this It’s Okay to Be Smart video called “Why Nature Loves Hexagons.”
- What other regular polygons can tile the plane? Using sturdy paper like construction paper, cut out an equilateral triangle, a square, a pentagon, a hexagon, and an octagon. Use each shape to sketch a tiling of a plane (if you can!).
- (Upper level) Once you’ve finished your tilings, measure the total perimeter and area of each design. Can you see why bees benefit from using hexagons?

- (All levels) Suppose that a group of cicadas emerges every 13 years, and their two predators have lifecycles of 3 and 4 years.
- How often will the cicadas face the threat of the 3-year predator?
- What about the 4-year predator?
- How often will the cicadas face
*both*predators in the same year? - How would the answers change if the cicadas emerged every 10 years instead?

- (High school) Using Baron’s article and this TED-Ed video to start the conversation, discuss whether you think math is invented or discovered.

*—Max Levy*

*NPR,* November 2, 2021

Since 2011, middle school students have entered their science projects in the Broadcom MASTERS competition in hopes of a $25,000 prize. This year for the first time, a young mathematician won the contest. Nell Clark covered Akilan Sankaran’s accomplishment for *NPR*. Akilan, who is 14, came up with an algorithm to quickly find “anti-prime” numbers, numbers with many prime factors. To do this, he used a function whose output depended on an integer’s factorization properties. In the end, the program he wrote delivered mind-bogglingly large anti-prime numbers in a short time. This isn’t just a mathematical curiosity: “highly divisible numbers are useful in computing because they can be used to divide data among computer processors, Akilan explains.”

**Classroom activities: ***number theory, prime factorization*

- (Middle school) The mathematical definition of an anti-prime (or highly composite) number is one that has more divisors than any smaller whole number has. Ask students to find all seven anti-prime numbers from 1 to 40 based on this definition (without looking at the preceding link!). (Hint: 1, 2, and 4 are the first three)
- (Middle school) Assign this prime factorization practice from Khan Academy.
- (High school) For his project, Akilan came up with two functions
__,__$f_s(n)$ and $S_k(n)$ (slide 3), that he used to find anti-prime numbers. He also compared them to the divisor function $d(n)$. On slide 5, he shares some properties of these functions.- Have students prove the divisor function properties in the first and second rows of the table on slide 5: $d(p) = 2$ for any prime $p$, and $d(mn) = d(m)d(n)$ when $m$ and $n$ don’t share any prime factors.
- Have students give an example to show that if $m$ and $n$ share a prime factor, then the multiplicative property of $d$ fails (that is, $d(mn)\neq d(m)d(n)$).
- (Very hard) Have students try to prove the smooth function properties in the first and second rows: $f_s(p) = 1+ 1/p$ when $p$ is a prime, and $f_s(mn) = f_s(m)f_s(n)$ when $m$ and $n$ don’t share any prime factors.

*—Leila Sloman*

Some more of this month’s math headlines:

- Who Was Lotfi Zadeh? Google Doodle Honors the Azerbaijani American Scientist

Newsweek, November 30, 2021 - Curious Nature: Nature is a mathematician

Vail Daily, November 20, 2021 - The Mathematician Who Delights in Building Bridges

Quanta Magazine, November 17, 2021 - Cancers are in an evolutionary battle with treatments – evolutionary game theory could tip the advantage to medicine

The Conversation, November 16, 2021 - Richard Rusczyk’s Worldwide Math Camp

The New Yorker, November 12, 2021 - Mathematician with Kolkata roots solves a century-old maths problem

The Telegraph India, November 11, 2021 - Benjamin Banneker: A Life Told in Cicada Years

Ask Nature, November 9, 2021 (reprinted by Scientific American) - Mathematicians Find Structure in Biased Polynomials

Quanta Magazine, November 9, 2021 - California Tries to Close the Gap in Math, but Sets Off a Backlash

The New York Times, November 4, 2021 - Surprising Limits Discovered in Quest for Optimal Solutions

Quanta Magazine, November 1, 2021

The Lorenz attractor is probably the most famous chaotic dynamical system. It is the trajectory of the solution to a nonlinear system of three differential equations, discovered by Edward Lorenz (1963) in his attempts to model weather; it has come to epitomize the “butterfly effect.” (In this awesome animation you can watch the trajectory take shape). The trajectory of a three-dimensional chaotic dynamical system like the Lorenz system evolves in a region of 3-space that can be collapsed onto a 2-dimensional *branched manifold* (a smooth surface except along a 1-dimensional branching locus where it has Y-shaped cross-sections).

In an article in *Chaos* published October 12, 2021 (see the arXiv preprint), an international team (Gisela D. Charó, Mickaël D. Chekroun, Denisse Sciamarella, and Michael Ghil) examined random time-dependent perturbations of the Lorenz system. For each instant $t$, their program produces a 2-dimensional cell complex (“[a] set in phase space that robustly supports the point cloud associated with the system’s invariant measure”) that plays the role of the underlying branched manifold. They track changes to the topology of that complex by computing the rank of its *first homology group* (here’s where algebraic topology enters the picture). For 2-dimensional sets like these, that rank is just the number of “holes.” For example, the branched manifold of the original Lorenz attractor can be continuously squeezed down to a figure-eight. Topologically speaking, it has 2 holes, so the rank of its first homology group is two.

The authors consider applications to Lorenz’s original problem, weather: “A fairly straightforward application [of this methodology] to the climate sciences might clarify […] the role of intermittent vs. oscillatory low-frequency variability in the atmosphere. […] Such phenomena include the so-called blocking of the westerlies and intraseasonal oscillations with periodicities of 40–50 days. They remark: “The framework introduced in this article to characterize such changes in topological features appears to hold promise for the understanding of topological tipping points in general.” (An online news service put two and two together and came up with: “Algebraic Topology Could be Used to Predict if and When Earth’s Climate System will Tip.”)

In conclusion: “We have concentrated throughout much of this paper on problems related to the climate sciences […]. With all due modesty, it is not unlikely — considering the great generality of topological methods — to expect the results obtained herein to have some applicability to other areas of the physical, life and socioeconomic sciences.”

It has been known at least since Jean Piaget’s 1948 work with Bärbel Inhelder, *La représentation de l’espace chez l’enfant,* that children start with basic topological distinctions (circle $\neq$ annulus) before more detailed or quantitative ones (circle $\neq$ triangle). So it is surprising to learn, from an article in *Child Development* (September 27, 2021), that in their peripheral vision, children under 10 do not take advantage of topological cues the way older children and adults do. As the authors, a Shenzhen-based team led by Lin Chen and Yan Huang, put it: “This study demonstrates that the peripheral vision of children aged 6–8 years functions differently than for adults when discriminating geometric properties of objects, that is, topological property (TP) and non-TP shapes.” The team worked with a large group of subjects: 773 children aged 6-14 and 179 adults. In their main experiment, they used two types of figures (one resembling letters and the other made of arrows and triangles) to test “topological and nontopological discrimination in the central and peripheral visual fields.”

Stimuli were presented in pairs, one stimulus on each side of the central fixation point. Participants, keeping their focus on the central point, had to record whether the two stimuli were the same or not by pressing one of the two specified keys on the keyboard. Reaction times were measured for each trial. In total, there were four types of combinations, that is, two eccentricities (central and peripheral) and two discrimination types (TP and non-TP).

The variable that the team measured, the *normalized TP priority effect,* was “computed from normalized reaction time differences between non-TP and TP trials,” so it measured how much a topological distinction speeded up the discrimination between two shapes.

As announced, the results show that whereas adults and children over 10 process TP differences faster than non-TP differences, both in their central and peripheral visual fields, this effect is almost completely absent in the peripheral vision of children aged 6-8.

]]>French Emperor Napoleon had a theorem, U.S. President Garfield had a proof and now Russian Prime Minister Mikhail Mishustin has a geometry problem. The story was reported by Alex Bellos in his Monday puzzle column in *The Guardian *on September 20, 2021. “Earlier this month, Russia’s Prime Minister, Mikhail Mishustin, marked the first day of the school year by visiting a sixth form maths class at one of his country’s top science-oriented schools.” The school is the Kapitsa Physics and Technology Lyceum in Dolgoprudny, a town about 20 kilometers north of Moscow; sixth form corresponds roughly to Senior year in high school. During his visit he posed a geometry problem to the students.

Bellos shows us a photograph of Mishustin at the board with a diagram

of the problem in which he has drawn lines through the point and the

ends of the diameter and marked the inscribed angle thus formed as a right angle.

Bellos gives us a hint: it may help to remember that the three altitudes of a triangle (altitude: “the line from a corner that meets the opposing side at a right angle”) meet in a single point. Give yourself a few minutes to think about the problem and the hint before looking at Bellos’s presentation of the solution.

University of Wisconsin professor Jordan Ellenberg posted “The Math of the Amazing Sandpile” in the online magazine *Nautilus* on October 6, 2021. A mathematical sandpile is essentially a cellular automaton (like Conway’s“Game of Life”) where grains of sand populate the squares of a grid according to the following rule: No square can have more than three grains, and if a fourth grain is added, then the four disperse, one to each of the four adjacent squares. Ellenberg sketches out what happens if two adjacent squares on an otherwise empty grid both start with 4 grains:

Ellenberg first shows us the amazing complexity this simple rule can engender. Suppose we put a large number of grains on a single square in the middle of an infinite, empty grid. “You might imagine you’d end up with a big smooth pile of sand, with a big area near the center of dots maxed out with three grains of sand. You’d imagine wrong. Here’s what you get:”

This is one of several such pictures in Ellenberg’s article. He explains: “These images were generated by Wes Pegden, a math professor at Carnegie Mellon whose work with Lionel Levine and Charlie Smart […] of Cornell stands at the leading edge of sandpile studies. Pegden has interactive pictures of the billion-grain sandpiles on his website. There, you can zoom in and wander to your heart’s content.”

Ellenberg next tells us about the *dynamics* of sandpile behavior. A stable configuration turns out to have a density of about 2.125 grains/square. As he remarks, this critical threshold is “the dividing line between quiet and chaos.” The phenomenon of this *critical state* can be investigated on a finite grid, where a grain dispersed over the edge just disappears. Suppose we start with an empty grid, and add sand, grain by grain, at a square in the center. “For a while, the pattern of sand expands, looking a lot like Pegden’s pictures above […] . But once the sand gets to the rim, the story changes. The pile approaches an equilibrium, where sand drops off the edge at the same rate you add sand to the center, and the density holds steady at the critical value. Of course, there may be local fluctuations, denser and less dense patches that come and go as the system evolves; but on averge over the whole table, the number of grains per dot will hover around 2.125.”

Ellenberg links to a “hypnotic movie” of a sandpile at its critical state, which he credits to R. M. Dimeo at NIST. He remarks: “This looks like a living process to me. And that’s no coincidence. The notion of self-organized criticality is one popular way to think about how the rich structures of life might have emerged from simple systems that automatically seek the critical threshold.”

[You may enjoy working out some sandpile moves with pencil and paper, for example

or an avalanche (both on a potentially infinite grid)

or

on a 5$\times$5 table.]

]]>*High Country News*, September 29, 2021

It’s fat bear season. Every fall, bears in Alaska’s Katmai National Park and Preserve gorge themselves before hibernating. Hundreds of thousands of people tune in to livestreams and vote for their favorite fat bear. But *how fat *are they really? In the spring, scientists have to tranquilize the bears and use helicopters to get an accurate measurement. In the fall, when they reach peak plump, it’s often too hard to reach them. But, as Kylie Mohr writes in *High Country News*, there may be a clever workaround: lasers. Light detection and ranging devices, or “lidar,” can scan a fat bear and tell us its volume, which can be used to calculate its weight if its density is known. Katmai’s Joel Cusick came up with the idea and tried it on a bear named Otis. “I got a laser return from the butt of Otis,” Cusick told Mohr. “I thought, ‘Wow, this just might work.’”

**Classroom activities:** *density, pre-algebra, algebra *

- (All levels) Meet the bears on the Fat Bear Week website
__.__ - (Lower level) In the article, we learn that one past competition winner, Bear 747, weighed 1,416 pounds and had a volume of 22.6 cubic feet.
- What is the average density of this big bear?
*Hint: the units for density for this problem are pounds per cubic foot.* - How much would a bear with the same density weigh if it measured 20 cubic feet in volume?
- What would be the volume of a bear with the same density that weighs 1000 pounds?

- What is the average density of this big bear?
- (Middle Level) Bodies are mostly water. The density of water is 62.4 pounds per cubic foot. How does this compare to the density that you calculated for Bear 747? Discuss how different parts of the body, such as fat, might affect density.
- (Higher level) Now, let’s assume that we don’t know a bear’s overall density, but we do know its volume and the density of three components.
*Density of water: 62.4 pounds per cubic foot**Density of fat: 56.2 pounds per cubic foot**Density of fur: 75.0**pounds per cubic foot**Volume of bear: 21.1 cubic feet*

Assume that a bear’s volume is 60% water, 30% fat, and 10% fur. How much does the bear weigh?

*—Max Levy*

*Scientific American, *October 14, 2021

When do polynomial equations have solutions made up of integers or rational numbers? In this article for *Scientific American*, Rachel Crowell describes the current state of affairs on these questions. In 1970, mathematicians proved that there is no algorithm for showing whether or not certain systems of equations could be satisfied by integers. But the same question in systems with several variables is still unanswered. Although these problems have remained unsettled for a long time, Crowell writes, what’s really interesting are the methods that mathematicians develop to solve them, which often end up having unexpected connections to other areas of mathematics.

**Classroom activities: ***polynomials, geometry, Diophantine problems*

The following exercises are designed to give students a taste of Diophantine problems using a simple example: a unit circle centered at the origin. For a slightly more difficult version, try using $x^2+y^2=5$.

- (Algebra 2) Consider the polynomial equation $x^2 + y^2 = 1$. Can you find a solution $(x,y)$ where $x$ and $y$ are integers? How confident are you that you found
*all*the integer solutions? - (Algebra 2) Draw the curve $x^2 + y^2 = 1$ on a graph (either on paper or digitally). Does this make it easier to find all the integer solutions?
- (Number theory, pre-calculus) Watch this video by Michael Penn showing how to find rational solutions to $x^2 + y^2 = 1$. Have students use the technique to find 3 more rational solutions.
- (Introductory programming) Have students write a computer program that checks all combinations of $x,y,z$ each running from 1 to 25 to find integer solutions of $x^2 + y^2 = z^2$.

*—Leila Sloman*

*Nautilus,* October 6, 2021

The “abelian sandpile” is an abstract model of a pile of sand. Despite its simple laws, the abelian sandpile gives rise to beautiful and complicated behavior. With the help of images and videos, Jordan Ellenberg discusses this behavior and the resulting “self-organized criticality.” When the sandpile has limited room—like if it spills over the sides of a table—it will always end up at a critical density of 2.125 grains of sand per spot. At this density, writes Ellenberg, “There’s constant activity, but the activity is somehow organized and structured.” This may just seem like another curiosity of mathematics, but “self-organized criticality is one popular way to think about how the rich structures of life might have emerged from simple systems that automatically seek the critical threshold,” writes Ellenberg. Maybe abstract math isn’t so different from real life, after all.

**Classroom activities: ***mathematical modeling, dynamics*

- (Pre-calculus) Introduce the rules of the abelian sandpile in class. Assign the question Ellenberg poses in the article: “Check that two adjacent dots in the interior sandpile can never be empty at once.”
- (Pre-calculus) Ellenberg links to Wes Pegden’s interactive gallery of abelian sandpile simulations on an infinitely large grid. Pegden shows the abelian sandpile on different types of grids, with different numbers of grains of sand (“chips”). Have students create an abelian sandpile by hand using any small tokens as chips:
- Place 16 chips on the center point of a square lattice.
- Move 1 chip from that point to each of the neighboring lattice points.
- Repeat step 2 with any lattice point that has
**4**or more chips. Continue until every lattice point has fewer than**4**chips. - If desired, repeat with 32 chips or on a triangular or hexagonal lattice. (For a triangular lattice, the number in bold in step 3 should be
**6**. For a hexagonal lattice, it should be**3**.) - Compare the resulting patterns to the ones in Pegden’s gallery.

- (Pre-calculus) Ellenberg writes: “The abelian sandpile model doesn’t even try to capture the behavior of actual physical materials.” Ask students to identify one way in which real sandpiles differ from the abelian sandpile, and have them modify the rules of the abelian sandpile to try to account for this difference. Is their new model still abelian?

*Related Mathematical Moments poster and interview:* Piling On.

*—Leila Sloman*

*Ars Technica*, October 5, 2021

The latest Nobel Prize in Physics recognizes *chaos and complexity.* To physicists, chaos is when seemingly simple systems begin acting erratically. Similarly, complex systems have many moving parts that interact, yielding unpredictable consequences. This year’s Nobel recipients helped prove that these ideas explain and predict how our climate is changing. But this complexity affects more than just physics: “mathematics, biology, neuroscience, laser science, materials science, and machine learning, to name a few,” according to writers at *Ars Technica*. For example, geometric patterns emerge in flying flocks of birds. Climate patterns emerge when gases mix and heat under radiation. “All of these systems seem very different on the surface, but they share a common underlying mathematical framework.” Making predictions in complex systems is extremely difficult—the math can be too hard for the world’s most powerful computers to handle. But such wide-ranging applications, this math is crucial.

**Classroom activities: ***complexity, chaos, synchrony, butterfly effect*

- (All levels) Watch this YouTube video by Veritasium about the science of synchronization, which is related to chaos. As you will see in the video, some systems that begin in total disorder will always drift into a coherent pattern—synchronized behavior like thousands of fireflies flickering simultaneously.
- (All levels) Read more about the Nobel Prize in Physics awarded to Giorgio Parisi, Syukuro Manabe, and Klaus Hasselmann. In this article on
*The Conversation*, an atmospheric scientist writes about how mathematical modeling made all weather and climate forecasting possible. In this*Quanta Magazine*article, students can learn more about how complex physics and math have helped create climate models. - (Middle school) Use this spreadsheet-based example to explain the butterfly effect.
- (Middle school / high school) Use this lesson plan from Shodor to lead students through interactive web games related to chaos and probability.
- (Middle school / high school) Play the Chaos Game. This game shows how patterns can emerge from a chaotic or complex system with seemingly random rules. (You will need some dice and markers.) The same website has additional chaos/complexity games.

**Related Mathematical Moments poster and interview: ****Predicting Climate.**

*—Max Levy*

Some more of this month’s math headlines:

- An elusive equation describing bird eggs of all shapes has been found at last

*Science News,*October 29, 2021 (see last month’s Math Digests) - Where Transcendental Numbers Hide in Everyday Math

Quanta Magazine, October 27, 2021 - How Tadayuki Watanabe Disproved a Major Conjecture About Spheres

*Quanta Magazine*, October 26, 2021 - How To Calculate Exactly How Much Halloween Candy You Should Buy for Trick-or-Treaters

*Popular Mechanics*, October 25, 2021 - Mathematician Lily Serna wants you to think again if you reckon you’re not ‘a maths person’

*ABC Australia*, October 21, 2021 - 4 moves to make math visible with kids, using counters

*The Conversation,*October 21, 2021 - Data Drives the World. You Need to Understand It

Time, October 20, 2021 - How Wavelets Allow Researchers to Transform, and Understand, Data

Quanta Magazine, October 13, 2021

*The New York Times, *September 11, 2021

You probably don’t want to get up close to an erupting volcano to study the magma “bombs” that it shoots out. Luckily, the power of mathematics allowed researchers to discover why some volcanic bombs fall to the ground without exploding—despite the pressure of the steam inside them. In this article, Robin George Andrews explains how mathematicians built a model to simulate volcanic bombs’ in-flight pressures and temperatures. Differential equations, thermodynamics, and conservation laws all played key roles.

**Classroom activities: ***mathematical modeling, ideal gas law, linear equations*

- (Algebra) The volcanic bombs mathematical model, while quite complicated, relies on other well-known models. One of these is the
*ideal gas law*: Under many conditions, the behavior of gases can be approximated by the equation \[PV=nRT,\] where $P$ is the pressure, $V$ is the volume, $n$ is the number of moles (how much gas is present), $R$ is the ideal gas constant (a constant of proportionality), and $T$ is the temperature.- If a gas sample in a 1.0-liter container has a pressure of 2.0 atmospheres at room temperature, what will be the pressure if the same amount of gas is confined to a 0.5-liter container at room temperature?
- Suppose that one container has 12 moles of gas A, and another container has 4 moles of gas B (both at the same temperature). If the pressure of gas A is twice as large as the pressure of gas B, and the volume of gas A is 30 liters, what is the volume of gas B?
- The ideal gas law doesn’t take into consideration the size of molecules or the interactions between them. Given these simplifications, in what situations would the ideal gas law
**not**be a good approximation of reality? - Explore this intuitive simulator of the ideal gas law.

- (Algebra I) Use lava flows to practice using linear equations with this activity from Science Friday.

*—Scott Hershberger*

*The Guardian*, September 12, 2021

In this article for *The Guardian*, Michael Brooks explores the myriad ways that algebra—particularly linear algebra—keeps society ticking. Every day, algebra and related mathematical tools are used to solve logistics problems all over the world: deciding how to package goods, the routes delivery workers should take, airline schedules, and more. Brooks’ story makes it clear that this ancient subfield of math is still essential in the modern world.

**Classroom activities: ***algebra, geometry, travelling salesperson problem, factorials*

- (Algebra, geometry) In discussing the history of algebra, Brooks mentions a medieval text entitled “Problems to Sharpen the Young” that contains several era-appropriate word problems.
- After solving some of the problems, ask students to write their own modern-day versions of these word problems. Have pairs of students exchange their new problems and try to solve them.

- (Pre-calculus) Imagine a truck departing Albany, NY with packages to be delivered to all 48 of the US mainland’s state capitals before returning to Albany. Have students try to guess the shortest route. Afterwards, compare their result to the answer found by data scientist Randy Olson. How close were they?
- When teaching permutations and combinations, ask students to calculate the number of possible routes using a tool like WolframAlpha. If a supercomputer can test 200 quadrillion ($2\times 10^{17}$) routes per second, how long would it take to test all of the possibilities by brute force? Compare this to the age of the universe.

*Related Mathematical Moments posters and interviews:* Trimming Taxiing Time, Scheduling Sports.

*—Leila Sloman*

*The Guardian*, September 22, 2021

When you hear a song that’s just plain catchy, it’s borderline impossible not to share it. Music, it turns out, can spread faster than even the most contagious diseases. In an article for *The Guardian*, Linda Geddes writes about how a mathematical model used to predict the spread of disease also fits the viral spread of tunes. The researchers analyzed how songs grow in popularity through social dynamics. They calculated a factor from epidemiology called the basic reproduction number, $R_0$, which quantifies how contagious something is. The mathematical model, called an SIR (susceptible-infectious-recovered) model, even revealed clear differences between genres. Electronica happens to be the most contagious, with an $R_0$ of 3,430. (The $R_0$ for measles is 18 and for COVID-19 is around 6 or 7.) Of course, that doesn’t mean that nobody is immune to certain genres, one disease modeler told Geddes. “My nan, for example, is particularly resistant to the infection of trap and dubstep.”

**Classroom activities: ***exponential growth, modeling in Excel*

The rapid spread of a song or disease is described in its initial phase by exponential growth. In this exercise, we will explore the exponential equation $y=a\cdot 2^{bx}$ (where $a$ and $b$ are constants and $x$ is a variable).

- (All levels) Watch this 3Blue1Brown video about SIR models.
- (All levels) Make a table with three columns: $x, y_1=2x,$ and $y_2=a\cdot 2^{bx}$. In the $x$ column, write the integers 1 through 10 on separate rows. Now, assuming that $a$ and $b$ are both equal to 1, fill in the values for $y_1$ and $y_2$. Notice how quickly the exponential function grows compared to the linear function. Discuss the mathematical reason why this happens.
- (Middle School) In the above situation, which change will make the exponential model grow faster in the long run: increasing $a$ from 1 to 12, or increasing $b$ from 1 to 2? Why?
- (High School) Have the students create the model of $y=a\cdot 2^{bx}$ on a spreadsheet. (Here is a handy guide for making spreadsheets on Excel or Google Sheets.) Make one column for $x$ (with values from 1 to 10), then one column for $y$. Let’s assume that $a=2$ and $b=1$. For the $y$ column, use an Excel formula to let Excel calculate the values. Working together or using online resources (such as this one from Microsoft), plot your data. Repeat with different values of $a$ and $b$ and plot on the same graph to compare different exponential curves.

**Related Mathematical Moments poster and interviews: ****Resisting the Spread of Disease.**

*—Max Levy*

*Quanta Magazine,* September 23, 2021

Groups are abstract objects that math students usually don’t encounter until university. They encompass a wide range of sets: the integers, the complex numbers, invertible $n\times n$ matrices, continuous functions on the real numbers, and permutations of $n$ objects, just to name a few. So, what do they have to do with polynomials? Patrick Honner illustrates the connection with the roots of unity—the complex numbers that solve polynomial equations of the form $x^n-1=0$—and discusses how it is fleshed out in Galois theory. This branch of math utilizes group theory to show that it is impossible to solve most polynomial equations using algebraic operations.

**Classroom activities: ***complex numbers, roots of unity, group theory*

- (Pre-calculus) Assign the exercises at the end of the article.
- (High school) Teach students about groups using this online encyclopedia. Assign the following questions:
- Is the set of integers with the operation of addition a group? Why or why not?
- Is the set of integers with the operation of multiplication a group? Why or why not?
- Consider the polynomial $x^2-4$. What are its roots? Do they form a group with the operation of multiplication? Why or why not? What’s different about this set, compared to the roots of unity?
- Come up with your own example of a group (other than the ones already mentioned!).

*—Leila Sloman*

*ZME Science, *September 10, 2021

A bird egg is deceptively complex. To biologists, an egg both incubates life and represents a single giant cell. To engineers, eggshells are comically fragile, yet can withstand the weight of a hen. To mathematicians, an egg’s shape appears simple, yet almost indescribable. Researchers have long relied on known math functions for spheres, ellipsoids, and ovoids to estimate an egg’s geometry. But for many different egg shapes, these formulas just don’t quite fit. Recent research finally cracks the general formula for all egg types. This new “egg-quation” works by adding an extra math function onto the existing formula for 3D ovals. The addition captures the complicated *pyriform*—a shape seen in king penguin eggs, for instance—that is round on one end and pointed on the other. The finding will be useful to study evolution, design bio-inspired structures, and create better food packaging. (The research was also covered by Sci-News.com.)

**Classroom activities: ***symmetry, geometry, functions*

- (All levels) Birds lay all sorts of eggs. Ural owls lay almost spherical eggs, emus lay ellipsoid-shaped eggs, ospreys lay ovoid eggs, and king penguins lay pyriform eggs. Figure 7 in an earlier paper cited by the researchers shows the corresponding shapes in 2D: circles, ellipses, ovals, and pyriforms.
- Where are the axes of symmetry for each?
- Is one of these shapes
*more*symmetrical than the rest? - How do the other shapes compare in terms of their symmetry?
- (High school) For the two shapes that have the same type of symmetry (oval and pyriform), discuss why you think one is more complicated to describe mathematically than the other.

- (High school) The mathematical formula described in this new study depends on four variables: the egg’s length $L$, its maximum width, the location of the line of maximum width, and the width at a distance $L/4$ from the pointed end. Put students in pairs and have them do the following. Using graph paper (hidden from your partner), sketch out some shape that is reasonably simple, yet more complicated than a circle or regular polygon. Now, try to come up with words and numbers to describe your shape. Put it to the test by having your partner draw your shape using
*just your description*. Compare and discuss what conditions make this task easy or hard.

*—Max Levy*

Some more of this month’s math headlines:

**Infinity Category Theory Offers a Bird’s-Eye View of Mathematics**

*Scientific American*, October issue**Math is Personal**

*The Atlantic,*September 25, 2021**The Calculus of a Shower That’s Either Too Hot or Too Cold**

*The Wall Street Journal*, September 23, 2021**Mathematician Answers Chess Problem about Attacking Queens**

*Quanta Magazine*, September 21, 2021**How Ancient War Trickery Is Alive in Math Today**

*Quanta Magazine*, September 14, 2021**COVID-19 breakthrough data triggers common statistical mistake, researcher says**

*WCVB*, September 9, 2021**Thai-based restaurant in the US offers free Wi-Fi password in mathematical equation; social media users stumped**

*First Post*, September 7, 2021**Virginia wants to prevent gerrymandering. Can a mathematician help?**

*The Washington Post,*September 3, 2021

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