*The Ohio Star* ran “University System Weighs Gutting Math Standards After Students Keep Failing Algebra” on December 13, 2022. They were relaying information from a report by Suzanne Perez on the site of the NPR affiliate station KCUR datelined Wichita the day before. (The item generated significant local coverage and even made it to Fox News).

The story is that Daniel Archer, the vice-president for academic affairs for the Kansas Board of Regents, recently recommended to the Board that they consider not requiring College Algebra for so many of their undergraduates. As Perez explains, one reason for the change is that one in three students fail the course the first time they take it. Here is some background:

- College Algebra in fact covers high-school-level material; it is identical to Algebra 2, except taught much faster.
- The University of Kansas organizes its general graduation requirements into 6 goals, each with 2 outcomes. Goal 1, outcome 2 is Quantitative Literacy. The courses satisfying this outcome are listed on the site. Almost all have MATH 101 (College Algebra) as a prerequisite.

Archer is proposing a switch to Math Pathways, a program developed at Teacher’s College, Columbia, with the credo “algebra is designed to prepare students for calculus rather than for the type of math many students need in their majors, jobs, and lives.” Perez tells us that the program has helped Georgia State improve their graduation rate by 5% over the last 7 years.

While the population of the United States has approximately doubled since 1950, college enrollment has grown almost by a factor of 10. We cannot expect today’s students to have either the training or the goals of those who came to college back then. A pedagogically and scientifically informed reworking of the mathematics curriculum for the current audience is just a matter of common sense.

*Mind Matters News*, an online outlet of the Walter Bradley Center for Natural and Artificial Intelligence at Discovery Institute, posted “Are extra dimensions of the universe real or imaginary?” on December 25, 2022. The existence of extra spatial dimensions beyond the standard three sounds like science fiction, but it is a fundamental ingredient of much current research in physics. People are usually willing to include time as a fourth dimension (even though it seems quite different from left-right, up-down and forward-back), but they naturally balk at more dimensions for space. For one thing, where are they? The *Mind Matters News* posting is a look back at Margaret Wertheim’s essay “Radical Dimensions” (Aeon, 2018), which investigates this problem. They quote several paragraphs from Wertheim (in one she shows us how the very notion of an ambient, Euclidean “space” only dates back to the Renaissance) and link to several explanatory videos, which are unfortunately of unequal scientific quality. (The clip they link to of a 3-dimensional projection of a rotating 4-dimensional cube has nothing wrong with it, but their text: “There is also the concept of a fourth spatial dimension, often pictured as a tesseract, a four-dimensional cube” is meaningless). Another link, to The 11 dimensions EXPLAINED, has that same projection floating over a grassy field as example of what a 4-dimensional cube would look like as it passed through our space. This is wrong. The video shows a series of *projections* from 4-space to 3-space; but if a tesseract (a 4-dimensional cube) happened to pass through our space a spectator would see a series of three-dimensional *sections*.

Another possibility, as drawn by Tom Banchoff, is shown in “Understanding the hidden dimensions of modern physics through the arts”. *Mind Matters News* does have some useful links: *Flatland* (the movie) and an interview with Brian Greene (“How to visualize the 10 dimensions of String Theory”). I also recommend his TED talk on the topic.

An article in *PNAS* (November 23, 2022; picked up in *Physics Today*) analyzes the dynamics of honeycomb construction by setting a computer to do the same job. A regular hexagonal grid is the most efficient way of partitioning the plane into equal areas. (Presumably known to bees tens of millions of years ago, this was only proved mathematically in 1999). But what happens when that grid does not exactly fit in the physical space it occupies? The *PNAS* authors (Golnar Gharooni Fard, Francisco López Jiménez, and Orit Peleg of University of Colorado, Boulder and Daisy Zhang of Princeton University) set up experiments in which bees were presented with incomplete combs made up of misaligned structures that could not be part of the same hexagonal grid.

How would a computer solve the same problem? The team started by taking a clue from the bees: how many new cells they introduced to complete the comb. They began by distributing that number of points randomly into the space between the two partial combs, and then constructing a *Voronoi diagram* using those points and the centers of the cells along the problem edges.

This gives them a starting configuration, and now the computer goes to work to find out how to place the points so as to use the least amount of material (wax, when bees are doing the work). To make this into a more standard math problem, they use a special *potential function* $f(r)$, “a variation of the Lennard-Jones potential, known to produce hexagonal lattices in the absence of constraints”, depending on a distance variable $r$. The potential function is adjusted so as to have a minimum at $r=5.4$mm (“the distance between the center of the cells built by bees under no geometric frustration,” measured experimentally). The computer is tasked to find the minimum of the function $\sum f(d(p,q))$ where $d$ is distance and the sum is taken over all pairs $(p,q)$ of centers of adjacent Voronoi cells in the part of the grid being constructed. For a problem like the one considered above, there are some 30 new cells, so this is a function of some 60 variables. A minimum configuration is found using a simulated annealing process, an iterative method borrowed from theoretical physics.

*The Conversation*, December 30, 2022.

What kind of information is most helpful to predicting the outcome of a soccer game? There is lots of data you could try to incorporate in a prediction, but the best information comes from one of the humblest measurements: The number of goals historically scored by each team. Other kinds of data have been analyzed, like amount of time in possession of the ball, or number and quality of goal-scoring opportunities. But these don’t add much to predictions about who will win, writes Laurence Shaw in *The Conversation*. That’s because only scored goals have a clear and certain impact on the game’s outcome. This failure of flashy statistics reminds us of what really matters for mathematical prediction. “A model that only uses goals to predict future games may seem remarkably simple, but its effectiveness lies in understanding what makes for good statistical analysis: high quality data, and lots of it,” writes Shaw.

**Classroom Activities:** *statistics, modeling*

- (All levels) Shaw writes: “As far back as 1968, a statistical study was unable to find any link between shots, possession or passing moves and the outcomes of football matches.” Does this surprise you? Why or why not?
- (Mid level) In chess, the Elo rating (named after Arpad Elo) can be used to predict who will win a game. The Elo rating is even simpler than the model discussed in the article: It only depends on whether you win, lose, or draw games. Check out this table showing how the difference between your rating and your opponent’s rating corresponds to the chance that you’ll win a game.
- If your rating is 1800, and your opponent’s is 1600, what are the chances you’ll win?
- What if your rating is 1500 and your opponent’s is 1600?

*—Leila Sloman*

*Quanta Magazine*, November 28

Much like a computer algorithm or instruction booklet, the fewer steps your brain takes to complete a task, the faster the results. And whether you’re a soccer player chasing a ball or a rat who just spotted a snack, you benefit from quick dialogue between your senses and muscles. In this article for *Quanta Magazine,* Kevin Hartnett discusses a new study of how the brain calculates motor control. Just thinking “stop” or “go” is not enough. “If you just take stop signals and feed them into [motor control region], the animal will stop, but the mathematics tell us that the stop won’t be fast enough,” a neuroscientist tells Hartnett. The real trick to quicker reflexes is about calculus.

**Classroom Activities: ***calculus, rate of change*

- (All levels) Introduce students to calculus with this video: How to use calculus in real life.
- (Mid level) The article explains that movements depend on a “rate of change” between opposing inhibitory and excitatory signals. Imagine a simplified scenario where there is only one type of signal. In each pair below, choose which scenario has the largest rate of change.
- A signal of 100 units followed 1 second later by a signal of –99 units
**or**A signal of 100 units followed 1 second later by a signal of 10 units - A signal of 50 units followed 10 seconds later by a signal of 30 units
**or**A signal of 50 units followed 1 second later by a signal of 30 units

- A signal of 100 units followed 1 second later by a signal of –99 units

*—Max Levy*

*Discover*, December 16, 2022

A new paper has made progress on the “squaring the circle” problem, a question that dates back to the ancient Greeks. The modern form of the problem is to cut a circle into pieces and rearrange those pieces into a square, without any gaps or overlaps. (The ancient Greek version was to construct a square with the same area as a given circle using only a compass and straightedge. It is now known to be impossible.) This problem is difficult, but using advanced techniques, mathematicians have cut the circle into complex pieces that make it work. This year, András Máthé, Jonathan Noel, and Oleg Pikhurko found yet another way to square the circle, as Stephen Ornes writes for *Discover*. Still, there’s no easy solution: they use a mind-boggling number of pieces — around of them.

**Classroom Activities:** *algebra, geometry, irrational numbers, pi, golden ratio*

- (Algebra, Geometry) The ancient Greek problem of “squaring the circle” was impossible because is a kind of number called a transcendental number, as mathematician James Grime explains in a video for Numberphile.
- Watch the Numberphile video. Notice that the video discusses three kinds of numbers: constructible numbers, irrational numbers, and transcendental numbers.
- According to the video, are there any irrational numbers that are constructible? Are there any transcendental numbers that are constructible?

- (Algebra, Geometry) Read this Quanta Magazine article by Dave Richeson about some other impossible problems in math.
- (Geometry) Show that if a cube has side length $1$, then a second cube with twice its volume has side length $\sqrt[3]{2}$. Based on the article, why does this make “doubling the cube” impossible?
- (Algebra) Richeson writes about a problem from ancient Greece involving the golden ratio . It can only be solved if there is a number for which and are both integers.
- Prove that is a root of the polynomial .
- (Advanced) Is rational or irrational? What does this say about the ancient Greeks’ problem?

*—Tamar Lichter Blanks*

*Inside Climate News*, December 27, 2022.

With effects of climate change already apparent, scientists fear the catastrophic effects of “tipping points” — points at which environmental damage creates a self-perpetuating cycle. In a recent paper, scientists studied how to predict when these tipping points will occur, reports Charlie Miller in this article. If this research can help stop humanity from crossing a tipping point, huge amounts of environmental destruction could be prevented. But as scientist Michael Oppenheimer told *Inside Climate News*, “Don’t expect clear answers anytime soon … The awesome complexity of the problem remains, and in fact we could already have passed a tipping point without knowing it.”

**Classroom Activities: ***equilibria, climate modeling*

- (All levels) Miller writes: “The study’s authors use the analogy of a chair to illustrate tipping points and early warning signals. A chair can be tilted so it balances on two legs, and in this state could fall to either side. Balanced at this tipping point, it will react dramatically to the smallest push.” Identify whether the following systems have tipping points, and if so, what they are:
- A ball rolling around on a hilltop
- A ball rolling around inside a bowl
- A rocket launching into space

- (Mid Level) For more on math and climate prediction, read “Climate modelling made easy”, by Chris Budd for
*Plus Magazine*.- (Calculus) Miller mentions melting ice caps as a system subject to a tipping point. For more on this process, advanced students can read Marianne Freiberger’s article “Maths and climate change: the melting Arctic”, also from
*Plus Magazine.*

- (Calculus) Miller mentions melting ice caps as a system subject to a tipping point. For more on this process, advanced students can read Marianne Freiberger’s article “Maths and climate change: the melting Arctic”, also from

*—Leila Sloman*

*The Conversation*, December 15, 2022

If you’re a competitive person, you probably feel that the whole point of playing a game is to win — even at Christmas party games. In the game Bad Santa (also known as White Elephant or Yankee Swap), each person brings an anonymous gift, and gets a chance to open a gift from the pool of presents. The twist is that people can steal each other’s gifts. Some lucky players take advantage of this twist to snag their favorite item in the bunch, but others may end up with their least favorite. “It’s a good alternative to buying a gift for everyone, and a great way to ruin friendships,” writes Joel Gilmore, a mathematician from Griffith University who wrote about the game for *The Conversation.* If you want to win the best presents next year, it helps to understand favorable strategies. In this article, Gilmore describes running computer simulations of the game to find the most fair rules and the most successful strategies.

**Classroom Activities: ***simulations, optimization*

- (All levels) Play a simplified game of Bad Santa similar to Gilmore’s model. Form groups of 10 students. For the gift pool, take 10 cards from a regular card deck. Higher numbered cards represent better gifts. Choose a set of rules from the article and play the game. Students should feel free to use strategies discussed in the article as well.
- (Mid level) Discuss the results of the Bad Santa card game.
- Who feels their strategy worked, and why?
- Whose strategy did
*not*work, and why? - Who felt that they had no control over their win or loss and why?

- (Mid level) Change the rules and repeat. If the results are different, explain why you think the rules helped or hurt.

*—Max Levy*

- Ada Lovelace’s skills with language, music and needlepoint contributed to her pioneering work in computing

*The Conversation,*December 8, 2022 - 6 Marvelous Math Stories from 2022

*Scientific American,*December 12, 2022 - The Brilliance and Weirdness of ChatGPT

*The New York Times*, December 5, 2022 - Who was George Dantzig, UC Berkeley’s Real Life Will Hunting?

*The Grunge*, December 10, 2022 - G.H. Hardy: Remembering Britain’s eccentric and brilliant mathematician

*El País,*December 10, 2022

- Math education in
*The New Yorker* - The geometry of logical arguments
- Number theory breakthrough, in
*Nature*

The *New Yorker* staff writer Jay Caspian Kang has two recent pieces on the magazine’s website about mathematics education in the United States. How Math Became an Object of the Culture Wars (Nov. 15, 2022) and What Do We Really Know about Teaching Kids Math? (Nov. 18).

In the first installment, Kang starts in 1915 and follows the ebb and flow of various progressive math education movements. Initially it seemed that Euclid would be jettisoned along with Caesar and Cicero as abstract material once deemed healthy for developing critical thought, but irrelevant to modern life. Then, during and after World War II, technology (computers, for example) showed mathematical skills to be important after all; the problem was how to implant them in the population. In response, the “new math” movement, started in the early 1950’s (Robert Hayden’s *A history of the “new math” movement in the United States* covers its genesis in detail) and greatly accelerated by the intellectual panic following the Sputnik launch, emerged as an attempted solution. As Kang observes, “The same fight has repeated itself on several occasions since then.”

In the current iteration, the discussion has expanded from pedagogy to include equity, because of the realization that the American student body is not homogeneous. Changes in curriculum and delivery will impact students from different segments of society in different, and sometimes inadvertently harmful, ways. For example, the draft plans for California’s elementary and secondary math education guidelines were “criticized by the usual suspects … but also many equity-focussed educators who worry that the program may be seen as a slackening of expectations for minority and low-income students.”

The impetus for Kang’s second piece was the announcement, dated October 18 by the Gates Foundation of an ambitious program to improve K-12 mathematics education, starting, according to *Education Week* with a $1.1 billion 4-year investment. (The program has an explicit equity component.)

The Gates Foundation plans to use technology and experimentation to identify education techniques that work. But there’s a more fundamental core to the problem of improving U. S. mathematics education. Kang evokes a collision between two uncontroversial facts: “The first is that a society has a duty to educate all of its citizens, regardless of race, socioeconomic background, or whatever else. The second is that parents will almost always do what they think is best for their children.”

It is disappointing that in this discussion the word “math” is a completely opaque token: nothing is said at all about how mathematics is different from other subjects, or why so many find it hard to learn.

The website *DailyNous* (“News for & about the Philosophy profession”) posted “The Artful Geometry of Logic” by Justin Weinberg (South Carolina), on November 11, 2022. Weinberg is bringing to our attention the online availability of The Leuven Ontology for Aristotelian Diagrams Database, a searchable collection of some 3200 logical diagrams. He shows some examples, including the *Square of Opposition* as rendered in a manuscript copy of *Peri Hermeneia* (“On interpretation” — the title is Greek but the text is Latin) by Apuleius of Madaura (born c. 124 CE). The copy dates from about 1000 CE.

The concepts in this diagram go back to Aristotle (384-382 BCE), but as far as we know Apuleius was the first to present them in a 2-dimensional arrangement. (He did not leave us an actual picture, but instructions: the square form, what should be on the top line, what should be on the bottom line, and so forth).

The Square of Opposition is part of the Aristotelian study of *syllogisms*. These are three-part arguments like the sequence of propositions “Socrates is human; all humans are mortal; therefore Socrates is mortal” (the traditional example). The propositions occurring in a syllogism can be in one of four logical forms:

*Universal Affirmation:*“all $A$ are $B$”*Universal Negation:*“all $A$ are not $B$”*Particular Affirmation:*“some $A$ are $B$”*Particular Negation:*“some $A$ are not $B$”

The Square of Opposition shows the logical relations between the four forms. In this translation, Apuleius’s example for each form is shown in italics and the names of the relations between the forms appear in blue.

The names of the relations can be deciphered as follows, following the Stanford Encyclopedia of Philosophy. Propositions are *contrary *if they cannot both be true but can both be false; *subcontrary *if they can both be true but cannot both be false; *contradictory *if they cannot both be true and cannot both be false. Finally, proposition $Q$ is a *subaltern* of proposition $P$ if $Q$ must be true if $P$ is true, and $P$ must be false if $Q$ is false.

A news item in the November 24, 2022 issue has Davide Castelvecchi commenting on a recent ArXiV posting by Yitang Zhang (University of California, Santa Barbara), which would prove a weakened version of the Landau-Siegel conjecture. Terence Tao has observed, as quoted in John Baez’s Twitter feed, that the as yet unrefereed (111-page) article has typographical errors and missing pieces which make it impossible to evaluate; Tao recommends patience.

Meanwhile, what is this conjecture and why does it matter? The Laudau-Siegel conjecture derives its importance in part from its proximity to the Riemann hypothesis. That hypothesis concerns the (Riemann) zeta-function, defined for real $s>1$ by

$$\zeta(s) = \sum_{n=1}^{\infty}\frac{1}{n^s}$$

and extended to the complex plane by analytic continuation. It can be shown (here, for example) that $\zeta(-2), \zeta(-4)$, etc. are all equal to zero. These are the “trivial zeroes” of the zeta-function. It is also known that all the other zeroes must lie in the *critical strip* of complex numbers with real part strictly between 0 and 1. The Riemann hypothesis states that these other zeroes must in fact have real part exactly $\frac{1}{2}$ (the dashed line in this diagram).

The *Generalized Riemann hypothesis* is about a larger class of zeta-type-functions $L(s, \chi)$ (which includes $\zeta(s)$), and asserts that all their non-trivial zeroes (defined as before) lie on the dashed line where the real part of $s$ is $\frac{1}{2}$. This class includes a family of generalized zeta-functions, $L(s,\chi_D)$, that depend on an integer $D$. A *Siegel zero* would be a zero for one of these functions $L(s,\chi_D)$ located on the real line, strictly between $\frac{1}{2}$ and 1, so it would be a counterexample to the generalized Riemann hypothesis.

The Landau-Siegel conjecture, in one formulation from Wikipedia, states that there exists a constant $\delta$ such that no zero of $L(s, \chi_D)$ can occur in the interval $(1-\frac{\delta}{\log D}, 1)$. Zhang’s Theorem 2 states that there exists a constant $\delta$ such that no zero of $L(s, \chi_D)$ can occur in the interval $(1-\frac{\delta}{(\log D)^{2024}}, 1)$. This is a much smaller interval! Presumably that large exponent can be whittled away, as happened with an earlier estimate of his.

]]>*Big Think*, November 7 2022

This month, one lucky person won a record-setting \$2.04 billion in the Powerball lottery, beating unthinkable odds of 292 million to 1. Surely that \$2 billion winner feels their \$2 lottery ticket was a good idea. But would a mathematician have advised they buy the ticket, not knowing they’d win big? There’s more to the math of Powerball than just the odds of matching five white balls numbered 1 through 69 and one red ball from 1 to 26. “That probability doesn’t factor in the cost of a ticket versus how much you can expect to actually take home: the mathematical definition of ‘expectation value’,” writes Ethan Siegel, in an article for *Big Think. *In this article, Siegel breaks down the math behind the question of *when *exactly playing Powerball is “worth it” mathematically.

**Classroom Activities:** *probability, expected value*

- (Mid Level) Based on this article, if a lottery ticket costs \$5 and you have a greater than 50% chance of winning \$9, is playing worth it?
- What if you have a greater than 1% chance of winning \$500?

- (High Level) Suppose the lottery drawing consists of two white balls numbered from 1-50, and one red ball numbered 1-20.
- What is the new probability of matching all three balls for the jackpot?
- What is the new probability of matching one white ball and the red ball?
- How big must the jackpot prize be for the overall expected value to surpass the \$2 ticket cost?

*—Max Levy*

*New York Times*, November 10, 2022

What’s the point of learning math? Your teachers, parents, and peers may have instilled in you the value of learning math—to adjust baking recipes, to design or create art, to solve problems in science and engineering. But do you feel that your higher-level math education in algebra, geometry, statistics, or calculus is worth the effort? A recent article for the *New York Times* shares how students around the country responded to this question. Some answers are positive: “Math is timing, it’s logic, it’s precision, it’s structure, and it’s the way most of the physical world works.” Others are more negative: “Math could shape the world if it were taught differently.” Reading the spectrum of responses can help us find ways to appreciate math education, as well as ideas for how to improve it.

**Classroom Activities: ** *algebra, geometry, calculus, education, math anxiety*

- (All levels) Discuss the writing prompts from this survey in groups of three or four. Based on your discussion, write a paragraph individually answering the question: Is it important to learn math in school? Why or why not?
- (All levels) Several students responded to the prompt with negative responses about math. One wrote, “I think math is a waste of my time because I don’t think I will ever get it.” Respond to the following prompts:
- Can you remember a time that you struggled with a subject or activity you enjoy? What helped you overcome that obstacle?
- What would you change about how math is taught at your school?
- Math is essential for understanding concepts in science and engineering. Is it useful for people who don’t want to work in these fields? Why or why not?

*—Max Levy*

*Alaska Beacon*, November 5, 2022

After the death of their Representative in the US House, Alaska held a special election in August. The results were surprisingly interesting. Mathematicians Adam Graham-Squire and David McCune showed that if certain voters had switched from a losing candidate (Sarah Palin) to the winner (Mary Peltola), that influx of votes would have made Peltola lose, as James Brooks writes for *Alaska Beacon*. In other words, more votes need not be better. This counterintuitive outcome, called the monotonicity paradox, was possible because the election used ranked-choice voting. In ranked-choice voting, voters rank candidates in order of preference. The candidate with the fewest first-place votes is eliminated, their votes are reassigned to those voters’ second-favorite choices, and the process repeats until someone has a majority of votes. If some of Palin’s votes were transferred to Peltola, then Palin would have been eliminated in the first round. Peltola would have then lost in a head-to-head comparison with the third candidate, Nick Begich III.

**Classroom Activities:*** voting, elections*

- (All levels) Suppose that Candidate A, Candidate B, and Candidate C are running in an election, and that 40 voters prefer A, then B, then C; 35 voters prefer C, then B, then A; and 25 voters prefer B, then C, then A.
- Who would win if the election were awarded to the candidate with the most first-place votes?
- Who would win in the ranked-choice voting system described above?
- If Candidate C dropped out of the race, who would win in a race between just A and B? Who would win in a race between just A and C, or a race between just B and C? (A candidate that would win in a direct comparison with any other candidate is called a
*Condorcet winner*. In the Alaska election, Begich was a Condorcet winner but lost the election.) - For more information about different voting systems and how they can lead to different outcomes, watch this video from PBS Infinite Series.

- (All levels) The United States Electoral College is another example of a nonstandard voting system. Read the article The Electoral College, According to a Math Teacher by Ben Orlin. Try exercises 1-10 under the heading “Electors Per Capita: What Does It Tell Us?”
- Pair up with another student and compare your answers, especially for problems 5, 8, 9, and 10.
- Now, discuss the article as a class. Share your thoughts on the following questions:
- What did you learn about the Electoral College that you didn’t know before?
- Did the article and exercises change your opinion on the Electoral College? If so, how?
- Is the Electoral College fair? Why or why not?

*—Tamar Lichter Blanks*

*Discover Magazine*, November 3, 2022.

Ada Lovelace was a nineteenth-century mathematician known especially for her collaboration with Charles Babbage on early prototypes of computers. Some of her most groundbreaking work appeared in a paper she wrote about a machine Babbage had conceived of, called the Analytical Engine. Babbage envisioned that people would give the Analytical Engine instructions about formulas and values to compute, and it would print out the requested information. In her paper, Lovelace laid out a procedure for calculating Bernoulli numbers on the Analytical Engine. But some have held onto suspicions that Lovelace’s work was truly original. For *Discover Magazine*, Emilie le Beau Lucchesi details Ada Lovelace’s accomplishments, as well as the sexist barriers that sprang up both in life and after her death.

**Classroom Activities: ***programming, recursive sequences*

- The Bernoulli numbers form a recursive sequence, meaning they form an infinite list, and each number in the list depends on the previous items.
- (Mid level) Try Khan Academy’s lessons on recursive sequences and do the practice problems.
- (Mid level) Think of a recursive formula of your own, and calculate the first 5 terms in your sequence.
- (High level) Read this Project Lovelace page describing Ada Lovelace’s Bernoulli number algorithm. After learning what Bernoulli numbers are and how to calculate them, calculate the first 5 Bernoulli numbers.

- (High level, programming knowledge required) Write a computer program that, when given a number
*n*, outputs the*n*th element of the recursive sequence you came up with in the last exercise.

*—Leila Sloman*

*Popular Mechanics, *November 4, 2022.

A large group of fireflies flashing in the night can be delightful to watch, especially if you’re not used to seeing them. But sometimes something even cooler happens: The entire group coordinates their flashing. This isn’t common, writes Tim Newcomb for *Popular Mechanics*, but it was mysterious enough for mathematicians to explore further. The flashing fireflies form a dynamical system, a system that evolves over time, and their coordination is a dynamical phenomenon called “synchronization”. The researchers (Madeline McCrea, Bard Ermentrout, and Jonathan Rubin) simulated flashing fireflies in a mathematical model, which they refined until they observed the synchronization they were looking for.

**Classroom Activities: ***dynamics, differential equations*

- (All levels) At the start of this talk, mathematician Steven Strogatz gives several real-world examples of synchronization similar to what the fireflies exhibit. Watch starting at time 1:47 until minute 14:00 to see him describe all of the examples.
- (High level, Calculus) When modeling dynamical systems, mathematicians often use
*differential equations*. These are equations that involve a function*f*and its derivatives. Typically, a “solution” to a differential equation means a function*f*(*x*) that satisfies the equation. For instance,*f*(*x*) = 2*x*satisfies the differential equation*f’*(*x*) = 2. In the following exercises, show that*f*(*x*) solves the given differential equation.- (Integral Calculus) Find a solution
*f*(*x*) to . (Hint: Use u-substitution.)

- For more on differential equations, and some harder examples, check out Paul’s Online Notes.

*—Leila Sloman*

- 4 mind-bending math experiments that explain infinity

*Nova PBS,*November 29, 2022 - Fortress of Logic

*The Nation,*November 28, 2022 - Moral mathematics

*Aeon,*November 28, 2022 - The Simple Geometry Behind Brownie Bake Offs and Equal Areas

*Quanta Magazine*, November 21, 2022 - How many yottabytes in a quettabyte? Extreme numbers get new names

*Nature News,*November 18, 2022 - Mathematician who solved prime-number riddle claims new breakthrough

*Nature News,*November 14, 2022

- “Against Algebra” in
*The Atlantic* - Symmetry groups, curvature and biological cell polarization
- Mathematical singularities and optical vortices

Temple Grandin, a professor of animal science at Colorado State, is famous for being a high-functioning autist who used her extra sensitivity to intuit how animals react to stimuli and made our handling of livestock more humane and more efficient. In her “Against Algebra” article in *The Atlantic* (October 6, 2022) she laments the way elementary and high-school education has evolved towards material that can be tested with numerical scores (preferably the product of machine grading) and away from more hands-on activities — she mentions how when she went to school, shop class was “the highlight of [her] day.”

Grandin reminds us that people don’t all think the same way. Some are “visual thinkers” and some are more language-oriented. And among the visual thinkers she distinguishes the “object visualizers,” who process raw visual data (these people end up as designers or mechanical engineers) from the “spatial visualizers” whose perception filters through patterns and abstractions (here we have musicians, computer people and mathematicians). The problem she sees in education today, and particularly in math education, is that it privileges the last group and has little to offer the others.

The main culprit is Algebra. A typical curriculum for Algebra I is almost completely abstract. According to Grandin, this is both unnecessary and inappropriate. Unnecessary because many of the courses or career paths that require competence in Algebra as a prerequisite don’t actually need it. As she puts it: “I teach veterinarians, but I couldn’t get into veterinary school myself, because I couldn’t do the math.” Inappropriate because the cognitive skills necessary for handling abstract reasoning often mature later in adolescence.

Grandin remarks that in 2017 (well before the COVID-19 pandemic) two thirds of students entering community college and one third of those entering four-year schools required remedial math. She suggests that “maybe the decline in performance points to a deficiency not so much in how well students master material, but in what we are asking them to master.”

I think a better approach would have been “Against how Algebra is taught in schools today.” Grandin teaches veterinarians, but she is not responsible for calculating, for example, correct dosage based on an animal’s weight and age. Students need to learn how to understand and manipulate mathematical expressions. On the other hand, the “one size fits all” curriculum and teaching methodology that she decries is clearly not working for many of them. The waste of human potential is indeed scandalous, but remedies will be difficult and expensive.

*—TP*

“Forced and spontaneous symmetry breaking in cell polarization” ran in the August 2022 issue of *Nature Computational Science*. As authors Pearson Miller, Daniel Fortunato and Stanislav Shvatsman of the Flatiron Institute, NYC and their collaborators explain, cell polarity — the presence of a directional axis in each cell — is essential, for example, in establishing the direction in which a cell will move or grow. In this article the authors explain how in some cases, a cell’s polarization can be determined by its *shape*. For computational efficiency, they restrict themselves to shapes that are *axisymmetric*, i.e., are invariant under rotation about an axis.

They model the process with a solid shape enclosed by a boundary, where a single chemical species is located on the boundary or in the bulk. They posit that this chemical undergoes chemical reactions and also diffuses throughout the shape in a reaction-diffusion process (like those identified by Turing and that lead to stripes and spots on animal’s coats). Among other requirements, the diffusion is substantially faster in the interior than on the boundary. This leads to a set of equations that they solve numerically to determine steady-state solutions for various cell shapes. The results of the numerical experiments are summarized in this image, adapted from the article.

**Symmetry groups**. Symmetry groups allow us to quantify exactly how symmetrical an object or a configuration is. The *symmetry group* of an object is the set of all rigid motions (rotations and/or reflections) that leave it looking the same. It is a group because the *composition* of two rigid motions (performing one after the other) is again a rigid motion. This illustration uses standard mathematical symbols for the symmetry groups that appear. Shown in the center, a round sphere is symmetric under any origin-preserving rotation or reflection of 3-space. The group of these rigid motions is the 3-dimensional *orthogonal group* O$(3)$. The final distribution of the chemical has polarized the sphere so that only rigid motions that preserve the vertical axis leave the configuration unchanged. The broken symmetry group is the 2-dimensional *orthogonal group* O$(2)$.

If instead the cell has the vertically stretched (“prolate”) shape A, then its symmetries are the axis-preserving O$(2)$ plus up-down flips. Since flipping twice gets you back to where you started, the flips can be represented by the group $Z_2$ of integers mod 2, where $1+1=0$; since the flips and O$(2)$ operate independently, the complete symmetry group of the prolate sphere is the direct sum O$(2)\oplus Z_2$. Here the final distribution of the chemical occurs at one of the geometrical poles, so the flip symmetry is broken.

If the sphere is vertically squashed (“oblate”) the new shape (B) again has symmetry group O$(2)\oplus Z_2$, but the final chemical distribution will be centered about a point on the equator. This configuration is not preserved by rotations, but by up-down flips and front-back flips. These form the symmetry group $D_2$ of a non-square rectangle.

Pinching the sphere in its center (shape C) yields another solid with symmetry group O$(2)\oplus Z_2$; in this case there are two possible final distributions of the chemical, one of which admits the full symmetry group of the solid. Also, if the sphere is deformed to the cone-like shape D, there are two possible outcomes; one has the axis-preserving symmetry group O$(2)$, but the other only allows back-to-front flips, hence $Z_2$.

In the patterns shown in the figure above, the high-concentration area centers about a point (or points) where the Gaussian curvature has a local maximum, in the sense that it is greater than or equal to the curvature at neighboring points.

But if the surface also has areas of negative curvature, it is a different story.

“Spontaneous generation and active manipulation of real-space optical vortices” by Dongha Kim, Min-kyo Seo and collaborators at the Korea Advanced Institute of Science and Technology ran in *Nature *on October 12, 2022. Optical vortices are “beams of light that carry angular momentum.” Kim and collaborators demonstrate a way of producing oppositely-oriented pairs of vortices using reflection from a multilayered sandwich of materials called a *gradient thickness optical cavity* (GTOC). The vortices occur in the space just above the sandwich; they are due to interference between light rays reflected off the top of the nickel layer (shown dark blue in the diagram below) and rays which traverse that layer and are reflected off the aluminum mirror at the bottom. That interference pattern varies from point to point across the surface because the thickness of the upper silicon dioxide layer (shown light blue) increases from front to back, while that of the lower SiO$_2$ layer increases from right to left. Vortices only occur when the *optical thickness* of the nickel layer (which is controlled by an exterior magnetic field) is in a certain range, and then they appear and disappear as that thickness is varied.

Some details: the sandwich is about 2cm square; the gradients of $h_1$ and $h_2$ are on the order of one in a million. The nickel layer is only 5 nm thick, but varying an external magnetic field can give it optical thickness up to 20nm. The phenomenon described in the article occurs only when that thickness is in the range approximately 6 to 13 nm. In general, the reflective point-by-point response of the sandwich to the external magnetic field repeats 2-dimensionally (the period depends on the wavelength of the incident light and on the refractive index of SiO$_2$).

As the authors explain in the abstract, “the vortex–antivortex pairs present in the light reflected by our device are generated through mathematical singularities in the generalized parameter space of the top and bottom silicon dioxide layers.” The structure of the combined reflected rays is in fact characterized by the *complex reflection coefficient function*. Mathematically speaking, this is a complex-valued function defined for points on the surface of the sandwich (and therefore for pairs $h_1, h_2$ of layer depths), and vortices correspond to singularities of this function. The behavior of the complex reflection coefficient function near a positive minimum in its absolute value (“trivial” case) and near a pair of singularities (“non-trivial” case) is represented in the inset graphs in the illustrations of the experiment, above. A complex function would require a 4-dimensional graph; writing a complex number in polar notation as $re^{i\theta}$ the authors have represented the function by two ordinary graphs, one for the absolute value $r$, and one for the argument $\theta$. Imagine tracking the phase of the combined reflected beam around a loop enclosing the minimum; this is its argument when its intensity is written as a complex number. The lengths of the optical paths traversed by the two reflected beams vary from point to point, and consequently so does the phase of their sum. For special values of the optical thickness of the nickel layer, at the end of the loop the accumulated phase change is exactly $2\pi$. This spiraling phenomenon is an optical vortex.

*Washington Post*, October 5, 2022

What makes eating chocolate so satisfying? You may like the sweetness and creaminess of a milk chocolate. Maybe you like the sharp, rich bitterness of dark chocolate. To physicist Corentin Coulais, it’s all about the crunch. Coulais is a physicist who has conducted research on how the shape of a chocolate bar increases its *brittleness*, which in turn improves its quality. In a recent experiment, Coulais 3D-printed chocolate in various zig-zagging and swirling shapes. The most zig-zagging spiral shape was the most crunchy—and the one most preferred by Coulais’ ten volunteers. “A pleasurable eating experience doesn’t only take place in the mouth, but can be affected by the noises in your skull,” writes Galadriel Watson for the *Washington Post*. In this article, Watson describes the research project and why it may one day change the shape of the chocolate you can buy.

**Classroom Activities:** *brittle geometry, chocolate math*

- (Low level) Watch this Infinite Chocolate math riddle for another example of tricky geometry with math. Discuss why this trick works.
- (Mid level) Based on this image of the 3D printed chocolates, rank the shapes from most crunchy to least crunchy. Describe why you have ranked them this way.
- Do you expect that an S-shaped chocolate would be more crunchy if it were thicker (more chocolate) or less thick? Why?
- (Mid level) Geometry factors into the design of food in other ways as well. Imagine you are a chocolatier making a spicy chocolate entirely coated in chili powder. You make 1-ounce chocolates in three different shapes: a sphere, a flat bar, and a zig-zag (like Coulais’ printed treat). Each of these shapes has the same volume and weight of chocolate. Which will be the spiciest? Discuss what factor from its geometry most influences the overall spiciness of its chili coating.

*—Max Levy*

*BBC Future*, October 19, 2022

In a friendly game of Go Fish, it might not matter how thoroughly you mix your deck of cards. But an incomplete shuffle can make all the difference in a high-stakes casino game. The question of how to randomize a deck so that gamblers can’t take advantage of patterns in the cards is practical, but also mathematical. In 1992, mathematicians Dave Bayer and Persi Diaconis proved that it takes seven riffle shuffles to fully mix a deck. With six or fewer rounds of shuffling, the deck is still biased, but seven or more shuffles basically randomize it. Diaconis, who was a professional magician before pursuing statistics, has continued to work on the math of mixing cards: he even traveled to Las Vegas to assess a new card-shuffling machine, as Shane Keating describes in an article for *BBC Future*.

**Classroom Activities:** *probability, statistics, card games, computers*

- (All levels) Watch Persi Diaconis explain how to shuffle a deck in a video for Numberphile.
- (Probability/Statistics) In the video, Diaconis shows that if you take a standard deck of cards, mix it well, and ask someone to guess the identity of the top card, then do that for each card in the deck (where you reveal the top remaining card after each guess), you should expect the guesser to make about 4.5 correct guesses on average. With a partner, try playing this game with a set of 10 cards, such as the ace through 10 of a single suit. Record the number of cards you guessed correctly, then discuss your results with other students. Find the class average of the number of correct guesses from a deck of 10 cards.
- (Probability/Statistics) Now, try a smaller version of Diaconis’s probability computation from the video: in a well-mixed deck of just 10 cards, what is the expected average number of correct guesses? Hint: Diaconis starts describing the 52-card version of this computation at time 2:17.
- (All levels) If you watched the video above and are interested in learning more about the seven shuffles, watch this Numberphile2 follow-up video with more details.

- (Probability/Statistics; Advanced) Some magic tricks use a special kind of shuffling, called the perfect shuffle or faro shuffle, in which the deck is cut exactly in half and the cards are interspersed one by one so that the two halves alternate perfectly. Read the explanation of perfect shuffles from Jim Wilson at the University of Georgia, then try to answer as many parts of Questions 1, 2, and 4 as possible. To model the shuffles, try experimenting with Excel spreadsheet linked to on the webpage or building your own spreadsheet.

*—Tamar Lichter Blanks*

*Short Wave From NPR, *October 18, 2022.

In early 2020, mathematician Pamela Harris and some of her students posted a new paper online about parking functions, which let you assign a collection of cars to parking spots along a one-way street. The paper, called “Parking Functions: Choose Your Own Adventure”, was structured as an interactive experience. At various choice points, readers can decide what kinds of parking functions they want to study — and at the end of their adventure, they’re rewarded with an unanswered question about parking functions. In this episode of *Short Wave*, Regina Barber interviews Harris about what inspired her and her students to write this unusual paper and the basics of the math involved.

**Classroom Activities: ***combinatorics*

- (Mid level) Harris calls combinatorics “the art of counting”, and gives an example of counting the number of ways to make 37 cents out of change. For a slightly easier exercise, count the number of ways you can make 17 cents out of change. Write out your logic.
- (All levels) For a more thorough introduction to combinatorics, check out the Art of Problem Solving’s online videos.
- (Mid level) Read this description of the parking problem, adapted from Harris et. al.’s paper:Parking spots are lined up along one side of a one-way street, and cars arrive one at a time to park. A list of numbers represents the cars’ parking preferences: If there are 3 parking spots and 3 cars, the list (2, 2, 1) means the first car arriving would like to park in the second spot it encounters, the second car would also like to park in the second spot, and the last car would like to park in the first spot. Cars drive up their favorite spot; if it’s full, they keep driving until they find an empty spot. They can’t turn around, so that once they’ve passed a spot, they can no longer park there. If they pass all the spots without finding a place to park, they drive off and never return. The list of numbers is a parking function if every car is able to find a spot.
- Harris and her students write that if there are 5 spots and 5 cars, then (1, 2, 4, 2, 2) is a parking function, while (1, 2, 2, 5, 5) isn’t. Can you convince yourself if this is true? If so, explain why.
- For 3 spots and 3 cars, is (2, 2, 1) a parking function?
- For 3 spots and 3 cars, is (2, 2, 2) a parking function?
- For 3 spots and 3 cars, is (1, 1, 1) a parking function?

*—Leila Sloman*

*Nature News*, October 5, 2022

Think of the hardest math problem you’ve ever seen. Perhaps it’s a gnarly long division problem or a quadratic equation that seems impossible to factor. Maybe it’s a long, repetitive matrix multiplication, where you multiply large grids of numbers. Whatever the challenge, the sequence of steps you take to solve it are *algorithms. *Mathematicians and scientists have realized that they can rely on computers and artificial intelligence to carry out complicated algorithms for their hardest problems. In this article, Matthew Hutson describes a new AI for matrix multiplication that goes one step further. Rather than just fly through the standard algorithm for matrix multiplication, this AI discovers faster, better algorithms itself. And now, the math world is eager to find out which other math problems it can tackle.

**Classroom Activities: ***matrix multiplication, algorithms*

- (Low level) Follow a real-life algorithm to make a paper plane (from code.org). Discuss what you think would happen to your paper plane if you skipped or messed up one of the steps?
- (Mid level) Create your own algorithm for drawing a picture of your choice. You can come up with any picture you like, but make sure your algorithm takes no more than 10 steps, and that others can follow it easily.
- (Mid level) Read more about how to perform matrix multiplication on this Math is Fun tutorial. Solve the example about selling pies, and do questions 1, 2, 3, and 8.
- (Mid level) The week after this AI discovery, two people beat its record. Read more here.

*—Max Levy*

*Science News, *September 29, 2022.

Researchers have shown that it’s possible to create a tool that allows different people access to different information, while keeping data they don’t need securely locked away. That concept, called indistinguishability obfuscation (or iO), contrasts with conventional digital security, in which those possessing a key or password have free reign over all the protected information. Elizabeth Quill covers the accomplishment — by Huijia Lin, Amit Sahai, and Aayush Jain — for *Science News*.

**Classroom Activities: ***cryptography, zero-knowledge proofs*

- (Mid level) Quill writes that Lin was taken with cryptography as a graduate student, especially zero-knowledge proofs. Embedded in the article is a video of Amit Sahai explaining what a zero-knowledge proof is to 5 different people. Click here to watch the section where Sahai explains to a teen.
- In the clip, Sahai illustrates the idea of a zero-knowledge proof by reading out Daila’s secret which she placed in a locked box. Come up with your own example of a zero-knowledge proof. Then partner up with a classmate and use your example to convince your classmate you know something, without revealing what it is.
- After trading your zero-knowledge proofs, answer these questions about your partner’s example:
- Did they convince you?
- Did they reveal any information they didn’t mean to? If so, what?

- Read your partner’s feedback and improve your proof if necessary. Now, change partners and repeat the exercise again.

- (High level) For students interested in learning more about cryptography, Khan Academy has a cryptography unit online. After learning about the Caesar cipher and frequency analysis, encode a secret message with it using the Caesar cipher exploration tool. Then trade messages with a partner and see if they can crack it using this frequency analysis tool.

*—Leila Sloman*

- Statistics Are Being Abused, but Mathematicians Are Fighting Back

*Scientific American,*September 30, 2022 - Princeton mathematics professor June Huh and Melanie Matchett Wood GS ’09 named 2022 MacArthur Fellows

*The Daily Princetonian,*October 13, 2022 - Ada Lovelace Day: who was the mathematician and what is she known for?

*Evening Standard*, October 11, 2022 - Mathematician Yitang Zhang’s Pursuit of the Landau-Siegel Zeros Conjecture

*Pandaily*, October 18, 2022

- Interview with Dennis Sullivan
- “Math is the Great Secret” in the
*New York Times.* - Topology and mutations in the COVID spike protein

Dennis Sullivan, recipient of the 2022 Abel Prize, was interviewed by Mihai Andrei for the web newsletter ZME Science (posted on July 26, 2022). Dennis (my friend and colleague at Stony Brook; also at the CUNY Graduate Center) spends some time on generalities about being a mathematician. His passion is simplicity: everything in mathematics reduces to space and number. He tells Mihai that these are exactly the parts of our surroundings that every toddler starts to explore. But in school, mathematics becomes a chore, made even more onerous by the idea that you have to be clever and fast to succeed. Dennis tells a story from his own college days with the moral: “if you really want to understand something, you have to keep plugging at it.”

The interview gets into some detailed mathematics with a discussion of the limitations of calculus in describing the world, in particular fluid motion. The problem is that calculus relies on limits of ratios of lengths, as the lengths involved get closer and closer to zero. But in the real world, arbitrarily small lengths don’t exist. As Dennis puts it, “physics doesn’t make sense below thirty-three zeroes.” Mihai interprets for us: he’s referring to the Planck length, $10^{-33}$cm, below which measurements have no physical meaning. Back to general statements about the profession: You don’t have to be a math genius to succeed, but love of understanding is key. His last words: “So try to understand, don’t try to learn a lot, try to understand.”

Alec Wilkinson is back, proclaiming his new-found fascination with mathematics to an even larger audience, on the *Times*

Op-Ed page, September 25, 2022. Along with the story of how he struggled with mathematics as a teenager but reattacked at age 65, he brings in some new mathematical details: the prime numbers; as he reminds us, these are the numbers like 2, 3, 5, 7, 11 and 13 which have only 1 and themselves as divisors. The context is the age-old question about whether mathematics is discovered or invented. If we claim that humans invented numbers and counting, how can we account for the prime numbers [there are infinitely many of them, as we have known since Euclid], “that have attributes no one gave them?” Wilkinson alludes poetically to the realm where mathematics exists: “It is the timeless nowhere that never has and never will exist anywhere but that nevertheless is.”

Quenisha Baldwin (Tuskegee), Bobby Sumpter (Oak Ridge) and Eleni Panagiotou (UT Chattanooga, now at Arizona State) published “The Local Topological Free Energy of the SARS-CoV-2 Spike Protein” in *Polymers*, July 26, 2022. As they explain, the spike protein is part of the mechanism by which the COVID virus attaches itself to, and then penetrates, a target cell. Furthermore, many of the variants of COVID come along with mutations at sites along the *backbone* of this particular protein. (The CoSARS-CoV-2spike protein is a linear chain of 1273 amino acids (the *residues*), linked along a backbone of carbon atoms). But which sites? The authors give a characterization of the sites most likely to be loci of mutations, in terms of the nearby topology-geometry of the backbone, considered as a 1273-node polygonal curve in 3-space.

The criterion they define is the *Local Topological Free Energy*: using the free energy formalism from statistical mechanics, they measure the LTE by comparing the twistedness (in a specific sense—see below—borrowed from knot theory) of each 4-node subunit with the average twistedness of such subunits all along the molecule.

The authors refer to two different measures of twistedness, although in the main part of the article they concentrate on the first. *Writhe*, as they describe it, “is a measure of the number of times a chain winds around itself.” For closed, smooth curves in 3-space (most interestingly, knots) it can be defined in terms of numbers and signs of crossings in a planar projection of the curve (here is an example); recently the definition has been extended to open curves, including polygonal ones. *Torsion* (“describes how much [a chain] deviates from being planar”) was also initially defined for smooth curves. For polygonal curves, it is simpler to define than the writhe: if the curve has $n$ segments its torsion is $(1/2\pi)$ times the sum of the $n-2$ dihedral angles between the planes spanned by consecutive segments. (The *dihedral angle* between two intersecting planes is the angle between the lines they determine in a third plane perpendicular to their intersection line).

As the authors report, “most mutations of concern are either in or in the vicinity of high local topological free energy conformations, suggesting that high local topological free energy conformations could be targets for mutations with significant impact of protein function.” An example of the current relevance of this research: they mention that “the recently discovered omicron variant has two new mutations … at two conserved high LTE residues.”

]]>*Scientific American*, September 22, 2022

When you shop online with a credit card or send an email to a friend, your information is packaged, encrypted, and sent through systems that were designed using math and physics—but not quantum physics. A fundamentally different kind of computer, called a quantum computer, could use properties of quantum mechanics to efficiently solve problems that are impractical for ordinary computers. Quantum computers, which exist today but are small and not yet powerful, could completely change how information is communicated and protected. In an article for *Scientific American*, Daniel Garisto reports on the latest Breakthrough Prize in Fundamental Physics, awarded to Charles H. Bennett, Gilles Brassard, David Deutsch, and Peter Shor for “foundational work in quantum information.” The prize was for trailblazing research on the ideas behind the machines: they developed much of the conceptual framework for quantum computing and quantum cryptography.

**Classroom activities**: *quantum computing, quantum physics*

- (All levels) Play the Quantum Circuits card game from the American Physical Society PhysicsQuest 2021 kit, following the game’s pictorial rules for quantum gates. (There is also a teacher’s guide here.) Note that this game has cards that need to be printed out in advance!
- (All levels) Play through the puzzles in the Quantum Chess game.
- (High level) Come up with your own example of a chess puzzle using superposition and measurement or entanglement.
- (High level) Read this article about quantum superposition. Compare this to the way that superposition is represented in the chess game.

*—Tamar Lichter Blanks*

Netflix, September 26, 2022

*IAI News, *September 23, 2022

No matter what level you’re at, infinity is indispensable to how we do mathematics today. Even elementary school students implicitly rely on it when they add and subtract using integer numbers—a fact duly acknowledged by any child who, in a contest to name the biggest number, has triumphantly yelled “infinity plus one!” But does infinity ever enter into the physical world? Both Peter Cameron’s article and this 80-minute Netflix documentary discuss the mathematics of infinity as well as its role in physics.

**Classroom activity: ***infinity, integers, real numbers*

- (Mid level) If students are not already familiar with the concept of rational and irrational numbers, use this online lesson before doing the other activities.
- (Middle school and up) Use this lesson plan, “Teaching the Mathematics of Infinity”, to study different kinds of infinity in class.
- (High level) Watch this video from Numberphile explaining countability and Cantor’s diagonal argument.
- Ask students if they think the following sets are countable: The points on an
*xy*-plane, the set of prime numbers, the set of odd numbers, the set of line segments of any length.

- Ask students if they think the following sets are countable: The points on an

*—Leila Sloman*

*Horizon Magazine*, September 1, 2022

Conflicts between humans are a lot like games. You can win, or you can lose. You can draw, or compromise to win some battles and accept defeat in others. If two people win the lottery, they simply split the winnings 50/50. But what’s the best compromise for conflicts that involve concepts more abstract than money? In math, “game theory” deals with finding the optimal strategies to resolve competing interests. Game theory can help win a game of poker, and it’s also used to understand global conflicts. Researchers recently created tools based on game theory to facilitate discussions about land use conflicts around the world, such as a conflict between farmers and geese conservation efforts in Scotland. In an article for *Horizon,* Gareth Willmer writes about this research and describes the role of math in messy human situations.

**Classroom activities:** *game theory*

- (Low level) Can you guess how your classmates will behave? Given a range of integers between 0 and 100, guess the whole number that is closest to
**two-thirds of the average**of all numbers guessed by your classmates. Collect all the individual guesses, calculate the average, then two-thirds of the average. Discuss how you did.- After playing, watch this TED Ed video about predicting human behavior with game theory, which explains the math behind this game.

- (Mid level) Play crops versus creatures, a game designed by the researchers to illustrate the math of settling complex conflicts. Note: Game play will be recorded and used in a study by researchers at the University of Stirling. Students must be 16 or older to fill out the consent form.
- (Algebra II) For more information on game theory and helpful games and examples, use this NSF-sponsored teaching tool by Cornell University.

—*Max Levy*

*Popular Mechanics*, August 23 2022

Good science is not a story of what happened one time in one lab—it is repeatable and significant. Scientists strive to draw broad conclusions based on experiments that let them study the effects of an “experimental variable” on a sample. For instance, clinical trials let scientists conclude that vaccines protected people from Covid-19 better than a placebo did. To measure the *statistical significance *of these effects, scientists calculate a “p-value.” A p-value represents the likelihood that observed effects can be explained by chance, rather than the variable. (Lower p-values mean a result is less likely to be due to chance.) The p-value is standard across science, but it’s not foolproof: Researchers can cleverly analyze data to get better p-values, thereby overstating weak results. Though sometimes harmless, writes Sarah Wells, “when this practice is used in medical trials, it can have much deadlier results.” In this article, Wells explains how some scientists are exploring ways to close those loopholes.

**Classroom activities: ***probability, statistics, p-hacking, fragility index*

- (All levels) Watch this brief TED-Ed video that explains how p-hacking works in more detail.
- (Statistics) Now watch this Khan Academy video about calculating p-values and answer the quiz questions here.

- (Mid level) According to the article, scientists hope to weed out significant yet weak results with a new measure called the fragility index. An experiment’s
*fragility*corresponds to how sensitive it is to small perturbations. For example, if a drug’s clinical trial data turns from significant (p < 0.05) to insignificant (p > 0.05) if one single participant would have had a weaker outcome, then it would be considered fragile. Discuss why you think it’s important to make such a distinction. - (Mid level) Suppose you are testing whether an antibiotic works better than a placebo to treat an infection. Your clinical trial has 200 total participants, who are all infected at the start of the experiment. Half receive the placebo. Half receive the drug. By the end of the study, 10 placebo recipients and 90 drug-recipients are infection-free.
- Calculate an experiment’s fragility index using this online calculator.
- Repeat this calculation assuming instead that 30 placebo recipients and 45 drug-recipients are infection-free.
- Discuss what each result means in terms of whether the antibiotic should be recommended as a treatment.

*—Max Levy*

*Science News Explores, *September 6, 2022

As a mathematician, I can’t count the number of people who have told me they hated math as a kid. It’s hard not to wonder how many of those people would feel differently if they’d been able to engage with math without timed tests or the “genius myth” hanging over their heads. Maria Temming’s article for *Science News Explores *looks at ways to deal with math anxiety, sharing articles and resources to help students practice math in a stress-free way.

**Classroom Activities: ***math anxiety, algebra, geometry, arithmetic*

- (All levels) Ask students to read some of the suggested articles under “Want to know more?”. After reading, ask students to reflect on the following discussion questions:
- Do math assignments or tests make you more or less anxious than assignments in other subjects? Do you think you experience math anxiety? Why or why not?
- Do you know anyone who you think experiences math anxiety? Write what you notice about this person’s relationship with math.
- In this 2017 article, Evelyn Lamb writes about how a “fixed mindset” and the “genius myth” may contribute to math anxiety. Do you agree or disagree that these concepts make math anxiety worse? Why?

- (All levels) Mathematical skills come up in many areas of our lives, sometimes when we’re not even aware of it. As a class, brainstorm some games and activities that use logical thinking or numbers. For homework, ask students to spend 20 minutes on an activity they enjoy that involves some mathematical skills, and answer the following questions:
- Did you feel more or less anxious during the activity than you normally are while doing math?
- How was the experience different or similar to your usual experience of math?
- Did you use skills that might help you in math class? If so, what?

—*Leila Sloman*

- The number that is too big for the universe

*New Scientist,*September 9, 2022 - W.Va. Week in History – Katherine Johnson

*The Register-Herald,*August 27, 2022 - Are All Brains Good at Math?

*Nautilus*, August 31, 2022 - Play puts mathematician Emmy Noether in the spotlight

*The Daily Progress*, September 7, 2022

*New Yorker* staff writer Alec Wilkinson takes us along with him on his quest to “penetrate the mysteries of mathematics” (July 8, 2022). As he explains, he was bad in math as a child, and at 65, decided to revisit the subject and to document his experience in a book: *A Divine Language: Learning Algebra, Geometry, and Calculus at the Edge of Old Age*, published this year by Farrar, Straus and Giroux. In the *New Yorker*, Wilkinson tells us that he had been turned off by math’s “smugness” and “self-satisfaction” in high school. But he found himself enjoying his exploration anyway. Here are some insights he brought back from the trip. These insights and questions are shared by most mathematicians, and they refute the notion that math is smug and self-satisfied. On the other hand, they are certainly not obvious to someone experiencing mathematics at the introductory level. It is welcome to hear them from a relative outsider, and to read them in a wide-circulation, general-audience medium like the *New Yorker*.

- Math is unfinished business: even though we have been at it for several thousand years, there are still “many speculations that are not capable of being settled.”
- “Math is both real and not real.” On the one hand, what a mathematician studies exists only in his or her own mind, but on the other mathematicians who have never met agree on the contents of this imaginary world.
- Is mathematics invented or discovered? In particular, the counting numbers: where do they come from? Creation myths usually do not cover numbers, but there they are, and they have their own behavior. “Someone who says that human beings created the operations of arithmetic cannot say that we created the results.” No matter what words or symbols you use for two, plus and four, two plus two will always be four.

Wilkinson’s book was reviewed in the *New York Times*, July 17: “Math Defeated Him in School. In His 60s, He Went Back for More.”

Rachel Crowell has an article on the *Scientific American* website (posted June 28, 2022) with the title “Mathematicians Are Trying to ‘Hear’ Shapes—And Reach Higher Dimensions.” This topic was the subject of one of our earliest “Feature Columns,” by Steve Weintraub back in June, 1997: “You Can’t Always Hear the Shape of a Drum.” Crowell’s article covers recent progress on the problem.

We are all familiar with a big bell sounding a lower note than a small one; similarly, many of us know that long strings vibrate more slowly than short ones. When the frequencies are in the audible range, we hear lower notes from the longer strings. So, all other properties being equal, one can tell by listening if two strings are the same length or not: in that sense, one can *hear the length* of a string.

Some more about strings. A string fixed at both its ends is limited in the ways it can vibrate. A few of the possibilities are shown in this animation:

Note from the animation that modes with shorter wavelength beat faster: in fact, for a string, the vibration speed of a mode is inversely proportional to its wavelength. As this image suggests, for a string of length $L$ the possible wavelengths are ${2L, L, \frac{2}{3}L, \frac{1}{2}L, \dots}$. So if $\nu$ is the frequency of the first mode, then the set of possible frequencies has to be ${\nu, 2\nu, 3\nu, 4\nu, \dots}$. This ordered list of frequencies is the *spectrum* of the string.

The head of a drum is the 2-dimensional generalization of a string fixed at both ends. Just as for a string the vibrations of such a membrane are sums of modes, each with its own frequency. For a circular drum, modes and frequencies can be identified explicitly.

The analysis shows that if mode $(0,1)$ has frequency $\nu$, the others have frequencies $1.593\nu$, $2.135\nu$ and $2.295\nu$, respectively (writing just the first three decimal places of each coefficient), and that in mode $(0,2)$ the inner circle’s radius is $0.436$ of the outer’s. Our numbers come from Russell’s Vibrational Modes of a Circular Membrane page, which has animations of these and several other modes.

Crowell’s survey reaches back to a 1966 work by Mark Kac: “Can one hear the shape of a drum?” Kac considers drums that are all cut from the same physical cloth, in that the combination of density, stiffness and tension that determines their mathematical behavior is the same for all of them. He proves for such an ideal drum that its area and perimeter can be “heard” (as above); moreover, he shows that any drum with a *spectrum* (its set of frequencies of vibration) of the same type as the disc, i.e. ${\nu, 1.593\nu, 2.135\nu, 2.295\nu, \dots }$ has to be circular.

So ideally one can tell by listening if a drum is circular or not. The question remained in general, if two ideal drums are *isospectral* (same spectrum, so they sound the same), beyond having the same area and perimeter, are they the same shape and size?

As Crowell tells us, it took more than twenty years for this question to be settled. “One Cannot Hear the Shape of a Drum” was published in 1992 by Carolyn Gordon, David Webb and Scott Wolpert. Here is one of the isospectral pairs the team exhibited in their paper.

Recent progress: as Crowell reports, it is now known for triangles and for quadrilaterals with two parallel faces (including parallelograms, rectangles and trapezoids) that sounding the same means having the same shape and size. Crowell goes on to examine isospectral problems in higher dimensions, but leaves us with two unanswered questions in the plane. Are there any *convex* examples of non-congruent, isospectral pairs? (Note that the Gordon-Webb-Wolpert isospectral pairs are very non-convex). And, quoting from a recent article, “Specifically, it is not yet certain whether the counterexamples [i.e., non-congruent, isospectral pairs] are the rule or the exception. So far, everything points towards the latter.” In other words, differently shaped drums will sound different *in general*. But this has to be proved.

Daniel Goldman (Georgia Tech) led a team of four publishing Coordinating tiny limbs and long bodies: Geometric mechanics of lizard terrestrial swimming in *PNAS*, June 27, 2022. They study the transition from four-footed ambulation in long-legged lizards to pure forward slithering in snakes in a sampling of reptiles with shorter and shorter legs. This work is part of a research direction that started some 45 years ago, and has roots in mathematics (topology and differential geometry) and in theoretical physics.

To recapitulate the history very briefly, it was discovered in the 1970s that two very different fields, topology and theoretical physics, had independently come up with the same concept. Topologists called this concept “connections in principal bundles”, and physicists called it “gauge fields”. The discovery led to important progress in both fields. It is also remarkable that not so long afterwards the same set of ideas turned out to be useful in *robotics,* in the study of how a mechanism could move forward over terrain or in water by periodically changing its shape (the basic Geometry of self-propulsion at low Reynolds number, by Alfred Shapere and Frank Wilczek, appeared in 1989). The next step was to apply these concepts, known as *geometric mechanics*, to natural history, in the investigation of how animals crawl and swim, which brings us to lizard terrestial swimming.

In this analysis, the standard use of the term *gait* to denote the different ways a horse (or other animal) moves forward (e.g., walk, trot, canter, gallop) is generalized to refer to any periodically repeating sequence of postures (in an animal) or configurations (in a robot) that results in locomotion. Mathematically a gait can be studied as a loop in the *shape space* of an animal or a mechanism.

How does geometric mechanics work? The basic process is easiest to visualize for a simple process like a snake’s slithering across a sandy surface, where the relevant part of shape space is 2-dimensional. It is encapsulated in the next image, taken with permission from an arXiv posting by Jennifer Rieser (also at Georgia Tech) and ten collaborators.

Roughly speaking, if the creature has taken a particular shape $s$, and starts to wiggle its body in a certain way, that will make it start to move in a certain direction. This change in position can be drawn as a vector in the copy of position space associated to the initial shape $s$. Now the creature has a new shape $s’$; it wiggles again, and produces a new change in position: a vector in the copy of position space associated to $s’$. As the creature continues to execute its gait, it will eventually return to its original position $s$. When these infinitesimal vectors are summed up around the closed loop (blue) corresponding to the gait, the integral gives the displacement resulting from one cycle.

Goldman and his team start from the observation that, just as Rieser *et al.* had observed for pure slithering, the periodic changes in reptilian body shape during locomotion can be approximated by a cycle of combinations of two modes, like the first two of the string analyzed above: one where the body oscillates between **(** and **)**, to use a graphical shorthand, and one where it oscillates between **S** and **Z**. Their shape space is consequently very much like Rieser’s, but with fore and hind limbs also taken into account.

They report: “… we find that body undulation in lizards with short limbs is a linear combination of a standing wave and a traveling wave and that the ratio of the amplitudes of these two components is inversely related to the degree of limb reduction and body elongation.” This surprisingly mathematical statement implies that for a long-legged lizard, the loop in shape space corresponding to its gait collapses to a straight-line path going back and forth from **(** to **)**; whereas for a limbless snake, the gait mixes the **(** **)** mode with the **S** **Z** mode in such a way as to produce a *traveling wave* moving down the creature’s body and making it progress forward.

For more background I recommend Geometric Phase and Dimensionality Reduction in Living (and Non-living) Locomoting Systems, a lecture given by Daniel Goldman in Trieste in 2019. It gives details about how the *connection form*—the mathematical name for the process illustrated in Rieser *et al.*‘s diagram—can be derived experimentally in each case. Towards the end of the lecture Goldman explains how a “height function” defined on shape space can be calculated for such a connection. This is the same computation that yields the Gaussian curvature in Riemannian geometry, and it allows optimal gaits to be designed for robots by choosing paths that enclose the largest amount of what would be positive curvature.

The link between connections and gauge theory has an elementary presentation in Fiber Bundles and Quantum Theory, an article Herb Bernstein and I wrote for *Scientific American* in July, 1981. Some historical details can be gleaned, among other topics, from a 2008 video interview of C. N. Yang and Jim Simons.

*New York Times*, July 5, 2022

Math helps us make the most out of space. Sometimes that involves familiar dimensions: Geometric rules can dictate how to stack oranges in a three-dimensional box or arrange tiles on a two-dimensional plane so that as much space as possible is used up. Other times, mathematicians answer these questions in higher dimensions. This summer, mathematician Maryna Viazovska of the Swiss Federal Institute in Zurich was awarded the Fields Medal—considered by many to be math’s highest honor—for proofs of sphere packing in space of dimensions 8 and 24. The work represents an extreme twist on a 400-year-old conjecture that stacking bowling balls in a pyramid fills nearly 75% of the available space. Viazovska is only the second woman to win a Fields Medal. “I feel sad that I’m only the second woman,” Viazovska told the Times. “I hope it will change in the future.” In this article for the *New York Times*, Kenneth Chang describes Viazovska’s work and gives more context on her momentous award.

**Classroom activities:** *stacking, geometry, higher dimensions*

- (All levels) In high-dimensional spaces, like those that Viazovska worked in, it’s tough to visualize geometric objects. For more information about math and geometry in higher dimensions, watch “A Journey into the 4th Dimension”.
- In the video we learn that we can imagine higher dimensions by picturing their projection onto a lower dimension. Can you draw a 2D projection of the following 3D shapes? 1) Sphere, 2) rectangular prism, 3) triangular prism from the top, 4) triangular prism from the side.

- (Mid level) Demonstrate a type of sphere packing with a small box (such as a shoebox) and ping pong balls. (The box should have straight sides and the balls should all have the same diameter.) Cover the bottom layer of the box with as many ping pong balls as possible.
- How many can you fit in that one layer? What is the area of the floor of the box? Looking from the top down, what fraction of the area is empty space? (Hint: calculate the total area covered by ping pong balls.)

- (High level) Now add a second layer of ping pong balls on top of the first. Place them such that you are maximizing the number of ping pong balls that will fit in two layers.
- What is the total volume of ping pong balls in that space? What total volume spans the bottom of the first layer to the top of the second? (Measure the height of the second layer from the bottom of the box.) What fraction of the volume is empty space?

- Look at Figure 1 of this article about sphere packing. Do your arrangements from the last exercise resemble the photos? Do the packing densities (the fraction of volume taken up by ping pong balls) match the numbers in Table 1 for dimensions
*n*= 1 and 2?- Figure 2 shows the densities of the best packings mathematicians have found in each dimension. What do you notice about the chart? Are you surprised by its appearance?

*—Max Levy*

*Irish Examiner, *August 24, 2022.

As the buzz surrounding Maryna Viazovska’s Fields Medal win demonstrated this summer, the granting of one of math’s most prestigious prizes to a woman remains, unfortunately, big news. But over a century ago, women like Sophie Bryant were already breaking barriers in mathematics: When Bryant earned her Ph.D. in 1884, no other women in the United Kingdom or Ireland had done so. In this article, Clodagh Finn sketches the accomplishments of Bryant, who tragically died in a hiking accident 100 years ago this August.

**Classroom Activities:*** geometry, trigonometry, calculus*

- (All levels) For a more in-depth look at Sophie Bryant and her mathematical work, read this 2017 biography by Patricia Rothman for
*Chalkdust Magazine*.- Now, choose another mathematician who broke gender barriers to research and write a 500-word report about. You may find this list of women mathematicians from Agnes Scott College helpful for getting started.

- Bryant’s 1884 paper, the first by a woman for the London Mathematical Society, studied the shapes of cells within honeycombs. Read this article by Philip Ball for
*Nautilus*on that same topic.- (Trigonometry) Ball writes: “Hexagonal cells require the least total length of wall, compared with triangles or squares of the same area.” Suppose you have a regular hexagon, an equilateral triangle, and a square, each with total area 1. Calculate their perimeters. Hint: A hexagon is made up of 6 equilateral triangles. Why?
- (Trigonometry) Now suppose you have a regular polygon with
*n*sides and total area 1. What’s the perimeter? - Calculate the perimeter of the area 1 10-gon, 100-gon, and 1000-gon. What shape of area 1 do you think has the smallest perimeter? (Calculus) Prove it.

—*Leila Sloman*

*Science News, *August 22, 2022.

Packing problems seem to be a theme in math news this month. A new paper in *Journal of the Royal Society Interface* shows that sea urchin skeletons resemble a mathematical packing pattern called a Voronoi pattern. These patterns look like a web or mesh whose holes, or “cells”, appear somewhat irregular. But the cell shapes actually obey strict rules. Within each cell is a “seed” that governs the cell’s shape, writes Rachel Crowell for *Science News*. Cells must hug their seeds tightly—if you sit down anywhere on the Voronoi pattern, the closest seed should be in the same cell as you. In this article, Crowell explains the rules of the Voronoi pattern, and some of the engineering benefits this structure may offer both the sea urchin and, potentially, human technology.

**Classroom Activities: ***geometry, **Voronoi patterns*

- (All levels) Try out computer scientist Alex Beutel’s interactive Voronoi diagram generator.
- (All levels) For more on Voronoi patterns, read this
*Scientific American*article by Susan D’Agostino.- Now, make your own Voronoi pattern: Draw three dots anywhere you like on a piece of paper, and treat these as the “seeds” for the cells. Try to figure out where the boundaries of the cells should be! Come up with a guess and write a brief justification.
- Once you have your guess, read this tutorial on making a Voronoi diagram from
*Plus Magazine*. What did you learn from their solution? Does it match your guess?

—*Leila Sloman*

*The Independent*, August 15, 2022

Here’s a question you might not know is contentious: how many holes are there in an ordinary drinking straw? You could say that a straw has two holes—one on each end—but you could also say that the empty space inside the straw is just one long hole. In an article for *The Independent*, mathematician Kit Yates examines the straw question through the lens of the mathematical field of topology. Yates compares topological shapes to objects made of dough, which can be stretched, pulled, or squished without being fundamentally changed. To topologists, a drinking glass is the same as a plate, a mug is the same as a donut, and a pair of binoculars is the same as a pair of glasses. And a straw can be compressed down into a ring—which has just one hole.

**Classroom activities:** *topology, geometry*

- (All levels) Look at some objects in the room and try to determine how many holes they have, topologically. Try, for example, to determine the number of holes in a book, a rubber band, a water bottle, and a pair of pants.
- Pair up with another student and discuss your answers. If you disagree about the number of holes in a particular object, work together to figure out the right number.
- Scavenger hunt: Find an object that has one hole, an object that has two holes, and an object that has three or more holes.

- (All levels) Read the section on cylinders in this article about topological tic-tac-toe, then play a few games of cylindrical tic-tac-toe with another student.
- (Advanced) Try some of the tic-tac-toe puzzles in the article. Note: the puzzles get progressively harder!

- (Geometry, Precalculus) Watch this video about using topology to solve problems by 3Blue1Brown.
- Draw an ellipse on a piece of paper and find an inscribed rectangle inside the ellipse.
- (Advanced) Draw a few different closed loops and try to find an inscribed rectangle in each one. Discuss: Why is this hard to do, even after watching the video? Notice that the video proves that there is always a solution to the inscribed rectangle problem, but doesn’t show how to construct a solution for any particular loop. (A proof that demonstrates that a solution exists without giving a method for finding it is sometimes called an “existence proof.”)

*—Tamar Lichter Blanks*

*The Guardian*, July 25, 2022

In a game show from the 1980s called *Blockbusters*, participants try to connect two sides of a map of tiles. Each tile on the map is an identical hexagon. Tiles on two opposing edges are blue (one team’s color), and the other opposing edges are white (for another team). When participants answer questions correctly, a tile between the edges becomes white or blue—a point for their team. The goal: connect your team’s two edges with a path of hexagons of the same color. In this article, puzzle expert Alex Bellos describes the Blockbusters game and imagines a diamond shaped map (shown below). Bellos poses a mathematical riddle. How many configurations of the map will contain a path connecting the two blue edges? The solution requires a little math, but it mostly uses a clever trick of logic.

**Classroom activities: ***probability, logic puzzles, percolation*

- (All levels) Assuming there are 100 non-edge tiles, solve the riddle. Compare your answer to the solution published here.
- (All levels) Play Blockbusters in class using this online version. (Find some default questions here.)
- (All levels) Demonstrate percolation as introduced in the article. What you’ll need: 1 strainer or sieve; 1 bowl or cup with an opening large enough to hold up the strainer; 3 materials of different size (coffee grounds, sand, small rocks, little rocks, marbles, or something similar). Fill the strainer with one material. Place on top of the cup/bowl. Pour a cup of water over the material in the strainer. Measure how long it takes for the entire fluid to percolate. Repeat for each material. What do you notice about the relationship between fluid flow and material size?

*—Max Levy*

- Octonions: The strange maths that could unite the laws of nature

*New Scientist,*August 16, 2022 - Here’s the quickest way to grill burgers, according to math

*Science News,*July 26, 2022