- Galileo was wrong about mathematics?
- Updates on the shape of the universe
- Topology and schizophrenia

In 1623 Galileo published *Il Saggiatore* (“The Assayer”), a salvo in his polemic war against the Jesuit astronomer Orazio Grassi, an apologist for a mixed geo/helio-centric solar system model. In *Il Saggiatore*, he compares the universe to a huge book which one can’t understand without first understanding the language and the letters in which it is written: “[That book] is written in the language of mathematics, and the letters are triangles, circles and other geometric figures.” The prize-winning British science writer Philip Ball disagrees. “Galileo was, to put it bluntly, wrong: maths is not the language of nature, but a tool with which we are able to make quantitative predictions about some aspects of nature,” Ball wrote in an April 24 opinion piece in *ChemistryWorld*.

Ball, who writes mostly about the life sciences, is actually taking Galileo out of context. Galileo was talking about astronomy and physics (“the universe”) without addressing medicine or the life sciences of his day, as the word “nature” might suggest. On the other hand, leaving Galileo out of the picture, Ball has a point. As he puts it, “There is a profound difference between using the maths and understanding the science.” That is, we need pictures and stories (“causal narrative”) for the equations to make sense; too often, Ball says, “maths can even be something to hide behind: if you can crank the handle and get results, you can disguise (for a while) the fact that you don’t quite grasp the underlying science.”

But why does science have to make sense? Ball starts his piece, in fact, with a 1930 quotation from Werner Heisenberg, the discoverer of quantum mechanics, who wrote: “It is not surprising that our language should be incapable of describing the processes occurring within the atoms, for it was invented to describe the experiences of daily life, and these consist only of processes involving exceedingly large numbers of atoms.” More generally, why should our minds be able to comprehend the fundamental structure of the universe or, for that matter, of the human brain itself? Luckily, as Heisenberg also remarked, we have mathematics at our disposal, which can lead us safely beyond what we can comfortably “understand.” There may be parallel universes where the speed of light is different, but there can’t be one in which 47 is not a prime. We should be grateful that we have this reliable guide, and not complain when it sees better than we can.

If our three-dimensional universe is not infinite, then what does it look like? Presumably, it doesn’t stop at a wall, so it goes on being 3-dimensional everywhere; it’s what we call a 3-dimensional manifold. But just as a 2-dimensional finite surface with no boundary can have different *topologies—*it can be a sphere, a torus, a 2-holed torus, etc.—there are many possible topologies in dimension 3.

Some 3-dimensional topologies have been excluded as possibilities for our universe because they don’t match with current observations. For example, by current consensus the universe is geometrically *flat*, so that in any region we can examine the overall geometry is essentially like 3-dimensional Euclidean space. This rules out the simplest model for a finite topology, which would be the 3-dimensional sphere since the sphere requires overall positive curvature. “Promise of Future Searches for Cosmic Topology”, published in *Physical Review Letters* on April 26, 2024, reviews the remaining options. One possibility is the 3-dimensional torus, the space obtained by identifying opposite sides of a cube. This image of what that space would look like inside was used in a *Physics Magazine* commentary on the *PRL* article.

The authors of the *PRL* article (the 15 members of the international “COMPACT collaboration”) explain that the place to look for information on the topology of the universe is in the map of the cosmic microwave background (CMB). If topology is the explanation for anomalous features of this map, then it must contain detectable topological information. The most obvious sign of a topology like the 3-torus would be recurring images of features like galaxies, or, more generally, what the authors describe as “pairs of circles with matched temperature (and polarization) … visible in different parts of the sky,” but there is no sign of these.

To show that “no matched circles” does not rule out an interesting topology for the universe, the authors examine several flat models and conditions under which they would not be expected to show such circles. They remind us that there are 17 non-equivalent flat possibilities $E_1, \dots, E_{17}$ and concentrate on the first three: $E_1$, the 3-torus, where the faces to be identified can be parallelograms, or *rhombi*; $E_2$ (one of the rhombic faces has been rotated by $\pi$ before the identification) and $E_3$ (a square face has been rotated by $\pi/2$ before the identification). In these examples they work out a range of parameters (e.g. relative sizes of faces) for which the topology would not manifest matched circles, and so which are still viable candidates as models of a finite universe.

An application of graph theory to the diagnosis of mental disease is reported in the *Schizophrenia Bulletin*, April 26, 2024. In the article, Topological Perturbations in the Functional Connectome Support the Deficit/Non-deficit Distinction in Antipsychotic Medication-Naïve First Episode Psychosis Patients, by Matheus Teles, Jose Omar Maximo, Adrienne Carol Lahti (all at the University of Alabama, Birmingham) and Nina Vanessa Kraguljac (Ohio State), the authors seek physiological support for the understanding that deficit syndrome (DS, also called Deficit Schizophrenia) is actually a distinct disease, and not just a peculiarly severe manifestation of schizophrenia. Deficit Syndrome, as its name implies, is marked by enduring negative symptoms, e.g., restricted affect, poverty of speech, diminished sense of purpose.

The team worked with 61 patients, 18 DS and the others diagnosed with non-deficit schizophrenia (NDS), comparing them with 72 healthy controls (HC). Their hypothesis was that DS is special in that it involves disruptions in the connectivity of the workings of the brain. To test it, they defined 246 regions of interest in the brain, and considered these regions as the nodes of a graph. They used functional magnetic resonance imaging to infer the strength of connections between the regions by correlating their activity, and based on these strengths, assigned weights to edges between the corresponding nodes. This yielded a weighted graph for each subject. The graphs were evaluated with graph-theoretic criteria including:

**Global efficiency**. For an unweighted graph, global efficiency is defined as the average of the reciprocal of the length of the shortest path between any two different nodes (so longer paths contribute less). For a connected graph $G$ with $N$ nodes, with $d_{ij}$ the length of the shortest path between node $i$ and node $j$, the formula for the global efficiency $E(G)$ is $$E(G)=\frac{1}{N(N-1)}\sum_{i\neq j}\frac{1}{d_{ij}}.$$**Global shortest path length**here refers to the average length of the shortest path between any two nodes.**Local efficiency**at a node $i$ is the global efficiency $E(G_i)$ of the graph $G_i$ made up of node $i$, all the nodes directly connected to $i$, and all the edges between them.**Global local efficiency**of $G$ is the average of $E(G_i)$ taken over all the nodes of $G$.

The findings show that the global efficiency, global shortest path length, and global local efficiency differed—in a statistically significant way—between the deficit syndrome groups and the others. (In their conclusion, the authors remark that the topological metrics showed no statistically significant difference between the controls and the NDS (schizophrenia without deficit syndrome) subjects.) Put together, this is evidence that deficit syndrome really is a distinct disease.

*—Tony Phillips, Stony Brook University*

*Vox*, April 10, 2024.

When people say that “a picture is worth a thousand words,” they usually mean that one image can relay a lot of information. However, you could also conclude that one word doesn’t say much. In this article, *Vox*’s Oshan Jarow writes about how mathematics might be better suited than words for describing our conscious experience. “Words could offer you a poem about the feeling of standing on a sidewalk when a car driving by splashes a puddle from last night’s rain directly into your face,” Jarow writes. “A mathematical structure, on the other hand, could create an interactive 3D model of that experience, showing how all the different sensations — the smell of wet concrete, the maddening sound of the car fleeing the scene of its crime, the viscous drip of dirty water down your face — relate to one another.”

**Classroom Activities:** *math in language, data analysis*

- (All levels) The limitation of language, according to the article, is that an adjective like delicious mainly exists in one dimension. A chocolate mousse can be delicious and so can a burrito. The word tells you
*something*about food, but not enough to distinguish between them. Answer the following questions.- Describe the last thing that you ate with one adjective.
- Now describe the last thing you ate with four adjectives.
- Find a partner and describe your food with the four adjectives you chose
*without saying what the food is.*Try to guess each other’s food. (Avoid words like “cheesy” which reveal the ingredients. A word like melty might be better, for example.)

- (Mid level) List five pairs of adjectives (e.g., “salty/unsalted,” “dry/moist,” and “fragrant/odorless”) that you can use to describe a variety of different foods by assigning a score of 1 to 10 for each pair. (For example, a saltine cracker might be a [8,1,3] meaning very salty, very dry, and fairly odorless).
- Assign scores to the following foods: apple, chocolate bar, raw broccoli, parmesan cheese, mozzarella cheese.
- Tabulate the numbers for each of the foods.
- Write five observations based on the data, such as “XX is similar to YY in ZZ way but different in VV way.”
- In what ways is this system useful for comparing and contrasting the experiences of eating different foods? In what ways is it insufficient?

- (Mid level) Explain how the above exercises relate to the mathematics of consciousness described in the article.
- Explain what “ineffability” refers to in your own words.
- Upload this image of Earth to this online tool for pixelating images. Pixelate the image with a block size of 1, 50, and then 100.
- Describe how you would apply the idea of ineffability to this example. What happens when ineffability is low versus high?
- Is there a block size limit where the image no longer looks like anything?

*—Max Levy*

*New Atlas*, April 15, 2024.

When you look closely at a fractal, patterns that are obvious in the zoomed-out view repeat infinitely on smaller and smaller scales. Fractals are present all over nature, from snowflakes and trees to mountains and coastlines. However, they’ve never been observed in proteins, the building blocks of life, until now. “We stumbled on this structure completely by accident,” Franziska Sendker, a microbiologist who recently discovered a fractal molecule, said in an interview with *New Atlas. *Sendker was studying a protein isolated from cyanobacteria when she noticed its strange shape. Its atoms were arranged in countless triangles with triangular holes in their center. The pattern repeated at different sizes, “totally unlike any protein assembly we’ve ever seen before,” according to Sendker. This article describes the research and the specific type of fractal observed.

**Classroom Activities:** *fractals, Sierpiński triangle *

- (All levels) Read the description of the Sierpiński triangle observed in Sendker’s research and the instructions on this worksheet from the National Oceanic and Atmospheric Administration.
- Complete the worksheet for 6 iterations of the Sierpiński triangle.
- How many white triangles do you count at steps 3, 5, and 6? Describe the pattern.

- (Mid level) Follow this activity from the Fractal Foundation, “Fractal Trees,” where you calculate the ratios of branch length.
- (All levels) For more, refer to previous Math in Media digests about fractals from 2021 and 2022.

*—Max Levy*

*Ars Technica*, April 10, 2024.

On April 10, the Association for Computing Machinery announced the winner of the 2023 Turing award, one of the most prestigious awards in computer science. It went to Avi Wigderson of the Institute for Advanced Study in Princeton, New Jersey. In this article for *Ars Technica*, Jennifer Ouellette describes some of Wigderson’s accomplishments, including work showing that the advantages of random algorithms could be reproduced without randomness and on zero-knowledge proofs.

**Classroom Activities: ***randomized algorithms, algebra*

- (High level) In Richard Karp’s 1991 survey, he gives several examples of randomized algorithms. One is for testing whether a polynomial equation is valid or not.
- Read Section 4.1 of the survey. Describe in your own words the randomized algorithm for testing whether a polynomial $f$ is zero or not.
- Graph the polynomial $f(x,y) = x^3-5x^2y + 7xy^2 -3y^3$ using an online 3D plotter like WolframAlpha. Is $f(x,y) = 0$?
- Evaluate $f(x,y)$ at the points $(0,0)$, $(3,1)$, and $(2,2)$. Then use a random number generator to choose 3 coordinate pairs from the set ${-3, -2, -1, 0, 1, 2, 3}^2$ and evaluate $f(x,y)$ again. Interpret the results.
- Based on Theorem 4.1, what are the chances that you got $0$ for all 3 of your choices in the last exercise? Why do you think it’s important that the variables $a_1,dots,a_n$ are chosen randomly?
- Come up with your own secret polynomial, and have a classmate try to guess whether it’s zero using the randomized algorithm.

- (High level) Read the
*Ars Technica*article and state, in your own words, what Wigderson showed about randomized algorithms like the one you just used.- Come up with a non-random algorithm for figuring out whether a polynomial of the form $ax^2 + bx + c$ is zero, by testing it on various values of $x$.

- Check out some activities on zero-knowledge proofs from our October 2022 digests.

*—Leila Sloman*

*The Guardian*, April 29, 2024.

Every two weeks, Alex Bellos publishes a mathematical puzzle for readers of *The Guardian*. On April 29, Bellos had readers come up with a grid whose tiles were colored in black and white according to some constraints. The problem is reminiscent of mathematical concepts like map coloring.

**Classroom Activities: ***graph theory, coloring*

- (All levels) Solve Bellos’ puzzle. Write a short paragraph describing how you came up with your solution.
- (Mid level) Bellos’ puzzle, along with the map coloring problem, can be represented on mathematical objects called
**graphs**. Complete this lesson on graph theory from the Park School in Baltimore, up through Exercise 17.- (Mid level) Rewrite Bellos’ puzzle as a problem about graphs. Do you find it easier or harder to solve now? Why?

- (High level) Read about the map coloring problem in the first section titled “Map Colorings” here.
- Come up with a graph that cannot be colored with only 4 colors. Describe why your graph is not a planar graph.
- Read the proof of the simpler six-color theorem given in the notes. Does the proof help you understand the examples you’ve worked with?

*—Leila Sloman*

*Scientific American*, April 18, 2024.

In mathematics, the concept of **measure **is used to quantify the length of sets of real numbers. For example, the measure of the interval $[0,1]$ is 1; the measure of the interval $[0,0.01]$ is 0.01. More complicated sets can be measured, too. The measure of the natural numbers is 0, while if you take each natural number $n$ and surround it by an interval of length $10^{-n}$, the total measure of all those intervals is $frac{1}{9}$. But in 1905, Giuseppe Vitali showed that there are sets of real numbers that can’t be measured. In this article for *Scientific American*, Manon Bischoff explains the concept of measure and Vitali’s proof. (Note: The proof given that Vitali’s set is non-measurable contains some minor errors. The reason $mu(V^*) = sum_p mu(V_p)$ is because the sets $V_p$ are disjoint, and there are uncountable sets with measure zero.)

**Classroom Activities: ***measure*

- (Mid level) Read the first section of the article. Based on the information provided, guess the measure of the following sets, with a brief justification for your answer:
- The interval $[0,4]$
- The union $[0,1] cup [2,3]$
- The union $[0,1] cup [1/2,1]$
- The set of all real numbers bigger than 1
- The set containing only 0

- (All levels) In dimension 1, the measure described by Bischoff represents “length.” The concept of measure can be generalized to two-dimensional sets, three-dimensional sets, and higher. Brainstorm as a class how this generalization should work before moving on to the next activity.
- (Mid level) Usually, the simplest measures in two-dimensional space represent
**area**and the simplest measures in three-dimensional space represent**volume**. Guess the measure of the following sets, with a brief justification for your answer:- The inside of a $1 times 1$ square in the $xy$-plane
- The inside of a circle of radius 1 in the $xy$-plane
- The line $y = 2x$ in the $xy$-plane
- A sphere of radius 2 in 3D space
- The surface of the sphere of radius 2 in 3D space

*—Leila Sloman** *

*Yahoo Finance*, April 13, 2024.

Usually, sellers are happy when the price of what they’re selling is abnormally high. Homes are different. Most homeowners who sell would need to purchase another home — either upgrading to a more expensive house or moving “laterally” to a similar one — and right now, the math is not in their favor. They’d wind up paying far more on their monthly loan than they currently do. “That gap is convincing many American homeowners to stay put, stunting the number of homes for sale and buoying prices on the paltry supply that exists,” writes Janna Herron in *Yahoo Finance*. In this article, Herron explains the math behind the mounting pressure on buyers and sellers in the housing market.

**Classroom Activities:** *mortgage math, data analysis *

- (All levels) Most people purchase homes with a long-term loan called a mortgage, which requires monthly payments based on the total loan amount, plus interest. Read this interactive guide about mortgage math, and test the mortgage calculator with different numbers to answer the following questions:
- What would be the monthly payment on a $500,000 loan with 7% interest paid over 30 years? What total amount will be paid by the end of the 30 year period? How much of that is interest?
- How about a $500,000 loan with 7% interest paid over 20 years?
- A $600,000 loan with 5.5% interest paid over 30 years?

- (Mid level) Using a spreadsheet such as Google Sheets or Excel, create your own mortgage payment calculator.
- What would be the initial monthly payment on a $700,000 loan with interest of 4.5% paid over 30 years? What total amount will be paid by the end of the 30 year period?

- (Mid level) Read the example given in the article comparing the “payment shock” of two homeowners A and B.
- Suppose you and your neighbor each own homes worth $400,000. You have $100,000 in home equity, and your neighbor has $300,000 of equity. If each of you decides to sell and use your equities for a down payment on $500,000 homes, what will each of your monthly payments be? (Assume 5% interest and a 30-year loan)
- Define the payment shock as the percent change in the mortgage balance after selling your home and buying a new one. Calculate the payment shock for you and your neighbor.

*—Max Levy*

- Female code-crackers get belated recognition

*Taipei Times*, April 24, 2024. - Mathematicians Gave a Billiard Ball a Brain—and It Led to Something Unbelievable

*Popular Mechanics*, April 22, 2024. - Mathematicians Marvel at ‘Crazy’ Cuts Through Four Dimensions

*Quanta Magazine*, April 22, 2024. - Particles move in beautiful patterns when they have ‘spatial memory’

*New Scientist*, April 18, 2024. - Why AIs that tackle complex maths could be the next big breakthrough

*New Scientist*, April 10, 2024. - Can You View a Round Solar Eclipse Through a Square Hole?

*Wired*, April 5, 2024.

Drawings of mathematical problems predict their resolution was a University of Geneva press release picked up by *Phys.org* on March 7, 2024. It concerns research published in *Memory & Cognition* on February 14. That article, by Hippolyte Gros, Jean-Pierre Thibaut, and Emmanuel Sander, focuses on the role mental representations play in how people solve word problems involving whole numbers.

As the authors encapsulated their findings, “*What* we count determines *how* we count.” According to them, there are two different mental models for arithmetic, with different problem-solving strategies, and the one we choose is often determined by the context of the problem. They call the first model the *cardinal representation*. In a cardinal representation, a number stands for how many elements are in a certain set. The other model is an *ordinal representation*, in which a number refers to a certain position on the number line. (This second usage is related to, but different from, the way the term “ordinal” usually occurs in mathematics.)

Here is an example of a cardinal problem, taken from the list used in their experiment:

A bag of pears weighs 8 kilograms. It is weighed with a whole cheese. In total, the weighing scale indicates 12 kilograms. The same cheese is weighed with a milk carton. The milk carton weighs 3 kilograms less than the bag of pears. How much does the weighing scale indicate?

In an earlier paper the authors had established that subjects faced with a cardinal problem of this type, and asked to use as few operations as possible, would get to the solution in three steps. Using the terms from this example,

- The cheese weighs 12 – 8 = 4kg.
- The milk carton weighs 8 – 3 = 5kg.
- The weighing scale will indicate 4 + 5 = 9 kg.

Now an example of an ordinal problem, *with the same arithmetic structure:*

Obelix’s statue is 8 meters tall. It is placed on a pedestal. Once on the pedestal, it reaches 12 meters. Asterix’s statue is placed on the same pedestal as Obelix’s. Asterix’s statue is 3 meters shorter than Obelix’s. What height does Asterix’s statue reach when placed on the pedestal?

(Asterix and Obelix are cartoon characters, as familiar to the French as Tom and Jerry are here.)

Subjects with the ordinal problem and the same instructions would not be distracted by the equivalent of calculating the weight of the cheese (which would be the 4m height of the pedestal), and solve the problem in one step:

- Asterix’s statue reaches 12 – 3 = 9 meters.

As they describe it, problems involving weight, price and collection tend to lead to a cardinal analysis, “due to these quantities usually describing unordered entities.” On the other hand, problems involving height and duration are more often treated as ordinal problems “due to daily-life knowledge underlining the intrinsic order of the entities they mention.”

In this new paper, the authors investigate what mental activity leads to these different strategies. To this end, they asked subjects to give a solution to the problem, as well as a *diagram *of the problem. The authors worked with 111 subjects (59 adults and 52 fifth graders). Each was given a booklet of 12 problems, a mix of 6 cardinal and 6 ordinal (the $x, y, z$ in these tables were substituted with numbers in the range $z < 4 < x < y < 15$ in the tests).

The authors then developed a protocol for evaluating the extent to which a diagram represented cardinal or ordinal thought. Identifiable clusters of objects and graphically rendered set inclusion were cues to the presence of cardinal thought, while ordinal thought would manifest itself as axes, graduations, and intervals. Analyzing the results, the authors found that the cardinality of the diagrams predicted the use of the three-step strategy, and the ordinality of the diagrams predicted the one-step, regardless of whether the problem itself was of cardinal or ordinal type. This held for both children and adults.

The awarding of the 2024 Abel Prize (sometimes described as the Nobel Prize of mathematics) to the French probability theorist Michel Talagrand was reported by Kenneth Chang in the *New York Times*, March 20, 2024. Chang largely followed the Abel Committee announcement in citing three different areas of Talagrand’s work.

**Stochastic processes.**These are phenomena subject to random variation. Chang mentions the water level in a river—let’s call it $L$. Characteristics of such a process are its average value $\mu$ and its standard deviation $\sigma$. The standard deviation is a measurement of how often and how much $L$ deviates from $\mu$. A simple example of Talagrand’s work is an estimate for the*maximum*of the water level $L$ over a long period of time. The estimate calculates the probability $P$ of $L$ exceeding its mean by a certain amount; it has the form (I’m following Bodhisattva Sen’s 2022 lecture notes at Columbia, p. 101)

$$P[L\geq \mu + f(\mu, \sigma, x)] \leq e^{-x}$$

where $f$ is a specific function devised by Talagrand.**Concentration of measure.**As Chang describes it, in this second area Talagrand “helped show that there is a measure of predictability within random processes.” Here we have an expository lecture, “A new look at independence,” by the master himself. He describes the phenomenon as follows: “A random variable that depends (in a ‘smooth’ way) on the influence of many independent variables (but not too much on any of them) is essentially constant.” The setting for his estimate is much more general, but he starts his lecture with tossing a coin $N$ times, counting $+1$ for heads and $-1$ for tails and tallying the sum $S_N$. He estimates, for any $t\geq 0$, the probability that $S_N$ is greater than $t$ in absolute value:

$$P[|S_N| \geq t] \leq 2\exp\left(-\frac{t^2}{2N}\right)$$

for any $t\geq 0$. So the probability of $t = 550$ or more heads or tails in $N=1000$ tosses is $P[|S_{1000}|\geq 100]\leq .014$, and the probability that the number of heads is between 450 and 550 is at least $1-.014=.986$. (Besides the point for this exposition, a much sharper estimate is possible using the Chernoff inequality. This number, $.997$, is the one reported by Chang, presumably working from Talagrand’s 2019 Shaw Prize citation.)**Spin glasses.**A spin glass is a mathematical model for a special kind of matter in which atoms are arranged “amorphically” (not regularly as in a crystal) and in which their interactions are not confined to nearest-neighbors (unlike, for example, the Ising Model.) To investigate what configurations give local minima for energy, physicists have made mathematical models such as the Sherrington-Kirkpatrick (SK) model. This model dates back to 1975 and is described allegorically in this 2014 article by Dmitry Panchenko, in terms of the*Dean’s problem.*With this notation, in a perfect solution the product $g_{ij}\sigma_i\sigma_j$ is positive for every pair $i,j$: if students $i$ and $j$ like each other ($g_{ij}>0$), they get assigned to the same dorm ($\sigma_i\sigma_j =1$) and if they don’t ($g_{ij}<0$) they will be in different dorms ($\sigma_i\sigma_j =-1$). This is often too much to hope for—say 2 is BFFs with 4 and 5 while 4 and 5 cannot stand each other. But the Dean can make everybody as happy as possible by

*maximizing*the sum $\sum_{i,j}g_{ij}\sigma_i\sigma_j$. Formally, this is the same problem as minimizing the energy in the SK model of a spin glass.When $N$ is large, solving the problem is very hard. Around 1982, the Italian physicist Giorgio Parisi found a way to package the configurations to make the computation possible. He derived his equation using his powerful physical intuition. This work, in part, earned him the 2021 Nobel Prize. But mathematically speaking, it remained a conjecture. Chang quotes Talagrand: “For a mathematician, this doesn’t make any sense whatsoever.” Nevertheless, Talagrand developed the mathematical machinery necessary for a rigorous proof in 2006.

Michel Talagrand’s Abel Prize was also covered by Davide Castelvecchi in *Nature* on March 20. Both reporters mention Talagrand’s “unconventional” career among French mathematicians: he did not attend the *École Normale Supérieure* in Paris, their traditional spawning ground, and went instead through the regular university system, in Lyon. But he was not some nobody from the provinces. At age 22 he took the *Agrégation* examination in mathematics, the same national test that all the math *normaliens* have to pass, and came in first.

*—Tony Phillips, **Stony Brook University*

*WMDT*, March 14, 2024.

The concept of $\pi$ is essential. How could you study geometry without marveling at how the circumference of a circle is always a factor of $\pi$ times larger than its diameter exactly? To approximate its value, students usually prefer 3.14, while others prefer 22/7, said mathematician Dawn Lott. “I tend to use ten digits, which is 3.14159265358.” This story from Leila Weah at WMDT shares the history of calculating $\pi$ and the differences in how people use and represent it.

**Classroom Activities:** *pi*,* algorithms*

- (All levels) Read this Twitter/X thread from mathematician Alex Kontorovich about how to demonstrate $\pi$ using pizza crusts.
- Identify and explain at least 3 limitations of this exercise that would lead to falsely high or low estimations of $\pi$.

- (Mid level) Design your own pizza-based demonstration of $\pi$ that only uses pizzas but involves no cutting. (Think about other ways you might measure the circumference!)
- Identify and explain at least three limitations of your exercise.
- Write a concise step-by-step protocol so that someone not in this class could carry out the demonstration and understand what it says about $\pi$.
- Compare and discuss your idea with other students.

*NOTE: The article states that $\pi$ contains trillions of digits. The decimals of $\pi$ extend infinitely. The record for most decimal digits calculated by a human is 100 trillion.*

*—Max Levy*

*CBS News*, March 20, 2024.

*ESPN*, March 19, 2024.

During the yearly college basketball tournament known as March Madness, fans often fill out a “bracket” to try to predict the outcomes of each of the 63 games. What are the odds of getting the entire bracket right? Around 1 in 120 billion, according to *CBS News*. But 5 years ago, Gregg Nigl of Columbus, Ohio made history by predicting the outcomes of 49 March Madness games in a row. An *ESPN *article by Ryan Hockensmith tells Nigl’s story.

**Classroom Activities: ***probability, statistics*

- (Mid level) There are 63 games in the main March Madness bracket. Assuming every team has an equal chance at winning each game, calculate the following:
- The number of possible brackets.
- The probability of guessing a bracket perfectly.
- Nigl’s odds of finishing the bracket perfectly, once he had correctly predicted the first 48 games.
- The probability of correctly guessing 31 out of the 32 games in the first round.
- According to this article, a top-of-the-line prediction model is accurate 75% of the time. Repeat your calculations from above with 75% as the chance of guessing a game’s winner correctly.

- (All levels) Simulate a March Madness competition in class by breaking into 8 teams and running a tournament. The game you choose for the tournament should have outcomes that are somewhat, but not totally, random.
- Ahead of the tournament, have students fill out brackets. Did anyone guess the entire tournament correctly?
- By studying the class predictions, estimate the chances of correctly predicting the winner of each game.

*—Leila Sloman*

*Forbes*, March 13, 2024.

More than a century ago, a brewer from the Guinness company named William Sealy Gosset invented a far-reaching statistical method known as the $t$-distribution. Gosset was helping a fellow brewer who suspected hops with higher levels of a chemical called resin would improve the beer. Gosset’s colleague sought to prove his hunch with the scientific method: collect samples, measure resin concentration, and record his observations. But where was the line between “more resin in these samples” and “*significantly* more resin?” Gosset came up with the t-distribution to delineate between these two scenarios. “There is no doubt as to the significance of the method. The ‘$t$-distribution’ is featured in statistics textbooks and used in all fields utilizing stats from medicine to agriculture and much more,” writes Erik Ofgang for *Forbes*. In this article, Ofgang tells the story of Gosset’s work and influence.

**Classroom Activities:** *T-distribution; scientific analysis *

- (Mid level) Read this guide on the Student’s $t$-distribution from
*Wolfram MathWorld*. Then read Sections 3, 3.1, and 3.2 of Penn State’s “Statistics Online” textbook. Now suppose that the brewers collected the following data:

Batch |
Resin level |
Beer Quality |

1 | 3 | 20 |

2 | 3 | 13 |

3 | 10 | 20 |

4 | 12 | 20 |

5 | 15 | 29 |

6 | 16 | 31 |

7 | 17 | 25 |

8 | 19 | 21 |

9 | 23 | 27 |

10 | 24 | 32 |

11 | 32 | 16 |

- Calculate the $t$-score by hand.
- Calculate the degrees of freedom and determine a $p$-value for the hypothesis that more resin improves the beer.
- (High level) Read this
*WIRED*article about a potential new treatment for Lyme disease and follow the link to the press release.- Describe what the scientists were testing specifically in 2 or 3 sentences, using your own words.
- By looking through the technical report, describe how the scientists used the $t$-distribution test, also known as the “$t$ test” or “$p$-value test.”
- What $p$-values does the scientific team report in their study? Does this seem relatively more conservative or less conservative as a threshold for significance?

- (High level) Complete this calculator-based t-distribution activity and worksheet from TI Instruments.

*—Max Levy*

*BBC*, March 30, 2024.

Last month, Netflix premiered a new show featuring an old and famous scientific problem: The three-body problem. In the show, a group of aliens lives on a planet which orbits three suns at once. As physicists and mathematicians know, the movement of three objects acting on one another via gravity is nearly impossible to predict. Thus, the future of the planet is uncertain, and its inhabitants want to take over Earth instead. “The three-body problem is, then, the root cause of all the drama that plays out throughout the rest of the series,” writes Kit Yates for *BBC*.

**Classroom Activities: ***physics, algebra*

- (All levels) Read the article. Devise your own premise for a book, movie, or TV show centered around a mathematical or scientific problem you have studied in class. Describe how the details of the problem and its solution lead to a high-stakes plot. For more activities on math-related stories, check out our September digest on graphic novels.
- (High level) A two-body system follows paths described by conic sections. The equation $x^2 + y^2 = z^2$ describes a cone.
- A conic section is the curve you get when you intersect a cone with a plane of the form $ax + by + cz = d$, where $a$, $b$, $c$, and $d$ are constants. Find and draw the following conic sections:
- $x^2 + y^2 = z^2$ intersected with the plane $y = ½$
- $x^2 + y^2 = z^2$ intersected with the plane $z = 1$
- $x^2 + y^2 = z^2$ intersected with the plane $x – z = 1$

- On your own, come up with a few more planes to intersect the cone with, and draw the results. Write down any observations you have.
- A conic section is one of the following types of curves: An ellipse, a hyperbola, or a parabola. Derive the equations of an ellipse, hyperbola, and parabola in polar coordinates.
- For more on conic sections, read this section from Lumen Learning and try the Section Review Exercises.

- A conic section is the curve you get when you intersect a cone with a plane of the form $ax + by + cz = d$, where $a$, $b$, $c$, and $d$ are constants. Find and draw the following conic sections:

*—Leila Sloman*

*Quanta Magazine*, March 13, 2024.

To Claire Voisin, mathematics is something you feel, imagine, and meditate on while walking through Paris. Voisin is an award-winning French mathematician. Voisin appreciated math from a young age, thanks to its elegant proofs and definitions, but lost interest as a teen when math instruction felt more like a game. She turned to the creativity and structure of poetry, philosophy, and painting. Eventually, however, Voisin found the same depth in math that she loved in her creative pursuits and began a four-decade-long career. In this article, Voisin spoke with *Quanta Magazine*’s Jordana Cepelewicz about her work and inspiration.

**Classroom Activities:** *creativity in math*

- (Mid level) Based on the reading, describe what “deeper” things that Voisin eventually found in math. What do you think it has in common with the depth she found in philosophy, painting, and painting?
- (All levels) Use this mathematical art resource to draw the “three bugs problem.”
- How does the problem compare and contrast when you begin with a square versus a hexagon?
- Now, draw a shape with a concave region and repeat the exercise. Discuss how this compares to the square and hexagon versions.
- Next, follow the directions for “curves of pursuit” and sketch one that begins with a pentagon.

- (High level) Voisin works in Hodge theory, a subfield of topology. Complete the Day 1 and Day 2 exercises from this topology introduction for high school students, which cover differences between topology and geometry. (Note for teachers: An introduction and more difficult lessons are available within the document. #6 of the Day 1 exit slip is incorrect.)

*—Max Levy*

- Why 2024 Abel prize winner Michel Talagrand became a mathematician

*New Scientist*, March 21, 2024. - Mathematicians plan computer proof of Fermat’s last theorem

*New Scientist*, March 18, 2024. - Pi Day: How One Irrational Number Made Us Modern

*The New York Times*, March 15, 2024. - Pi calculated to 105 trillion digits, smashing world record

*Live Science*, March 15, 2024. - A Black mathematical history

*Nature*, March 14, 2024. - This 200-Year-Old Law Of Heat Has A Blind Spot. It Could Change Engineering.

*Yahoo! News*, March 14, 2024. - This Pi Day Let’s Celebrate The First Black Woman To Earn Her PhD In Mathematics

*Essence*, March 14, 2024. - Applied maths to the rescue: the Jack Powers story

*Cosmos*, March 8, 2024. - Have You Heard The One About The Mathematician…

*Discover*, March 7, 2024. - New Breakthrough Brings Matrix Multiplication Closer to Ideal

*Quanta Magazine*, March 7, 2024. - When a Math Museum Moves, Geometry Helps

*The New York Times*, March 4, 2024.

“Decimal fractional numeration and the decimal point in 15th-century Italy”, by Glen Van Brummelen (Trinity Western University) was posted online by *Historia Mathematica* on February 17, 2024, and picked up two days later by Jo Marchant in a *Nature* news item whose headline proclaimed “The decimal point is 150 years older than historians thought.”

Before Van Brummelen’s paper, historians thought the first appearance of the decimal point was in a table published by the Jesuit German astronomer Christopher Clavius in 1593. Now, we’ve learned from Van Brummelen that it can be traced back to trigonometric tables written by Giovanni Bianchini, a Venetian mathematician, around 1440. Bianchini’s tables have been digitized by the Jagellonian Library in Kraków; the decimal points Van Brummelen noticed are on the page numbered 108.

For some perspective, the “Treviso Arithmetic,” published in 1478 (the first known printed, dated arithmetic book) has no trace of this notation.

Below, you can see an excerpt from Bianchini’s table. The first column gives angles in degrees and minutes. The second gives corresponding values of the tangent function multiplied by 10,000 [my calculator gives $\tan$(68$^\circ$) = 2.47508]. The third column, where the decimal points occur, gives the increment corresponding to an additional minute of arc. [Since 21.2 = (25,387 – 24,751)/30, etc., this sets up linear interpolation between the whole-degree and half-degree values]. So for 68$^\circ$5′ the table would give 24,751 + 5 $\times$ 21.2 = 24,857. My calculator ($\times$ 10,000) gives 24,855.

Besides illustrating this early occurrence of the decimal point, Van Brummelen’s article points to the paradoxical situation of calculation in Europe in the early 15th century. While merchants and bankers were struggling with the transition from Roman numerals to Indo-Arabic notation, and choosing the best way to do long division, astronomers and mathematicians were computing trigonometric tables with four or five significant digits of accuracy.

A team in Japan has used concepts from graph topology to probe the way in which our brains synchronize with those of people we are interacting with. The experiment, reported in “The topology of interpersonal neural network in weak social ties” (*Nature Scientific Reports*, February 29, 2024), focused on the difference between interaction with a stranger and with an acquaintance. The authors, Yuto Kurihara, Toru Takahashi and Rieko Osu of Waseda University, explain that each run of the experiment involved a pair of participants. In total, there were 14 pairs of strangers and 13 pairs of acquaintances. Each participant wore headphones and a wireless 29-channel electroencephalogram (EEG) headset; they faced away from each other; each controlled a computer mouse which when clicked produced a tone audible to both of them. They were instructed to listen to a sequence of eight equally-spaced tones produced by a metronome and then to continue the sequence in alternation: participant 1, then participant 2, then participant 1, up to 300 clicks (150 each), matching the metronome’s inter-tap interval (ITI) as closely as possible.

The experiment was run for each pair under four conditions, including slow tap (ITI = 0.5s) and fast tap (ITI = 0.25s). The EEG records were filtered into three bands: waves of low (theta), medium (alpha) and high (beta) frequency, which were analyzed separately. In each case a 58$\times$58 matrix of inter/intra-brain synchronization was set up with an entry for each pair of monitored EEG channels (there are 29 + 29 = 58 channels in all), including pairs from the same participant. After processing to remove background effects, the matrix entries were reduced to 1 if the synchrony between the channels was significantly higher than the background and 0 otherwise. These data were used to construct a graph with one vertex for each of the 58 channels and links between the pairs corresponding to 1s in the matrix.

The team report that for the “fast tap” condition the theta band graph shows a significant difference between the performance of stranger pairs and acquaintance pairs of subjects. The graph-theoretic criterion used for this distinction is the *local efficiency*, first introduced in a *Physical Review Letters* paper in 2001. This is the average of the local efficiency at a particular vertex $v$, which is calculated as follows: Look at the neighbors of $v$, which form a subgraph $G_v$. For each pair of vertices in $G_v$, calculate one over the length of the shortest path in $G_v$. (Notice that when it is easy to get from one vertex to another, there will be short paths, making this number high.) The local efficiency at $v$ is the average of this number over all pairs in $G_v$. The local efficiency overall is the average local efficiency over all $v$.

In their Discussion section, the authors cite previous research showing a correlation between interpersonal interactions and activity at lower EEG frequencies. They further remark that interacting with a stranger, a more novel experience, will require more processing and more retention than interacting with an acquaintance. The question why exactly local efficiency should be the criterion that exhibits that difference is not, to my understanding, addressed.

Edward Frenkel (University of California, Berkeley) posted “Maths, like quantum physics, has observer problems” on the Institute for Art and Ideas website, February 6, 2024. Frenkel starts by reminding us how in the quantum world, different experimental setups can lead to seemingly contradictory results, with electrons sometimes looking like particles and at other times like waves.

How could something like this happen in mathematics? The answer is that the mathematics we do depends ultimately on the *axioms* we choose as a base for our logic, and that there is no way to check that we have a completely satisfactory set of axioms.

Frenkel focuses on *set theory*, the system that provides the language of mathematics. It has its axioms. The current set used by most of us is called ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) but a minority (Frenkel estimates 1%) reject one of the axioms: the *axiom of infinity*. The axiom of infinity states that the set 1, 2, 3, … of natural numbers exists; as Frenkel explains, this is much stronger than the statement that for every natural number there is a bigger number–there, you are always talking about some finite number. Accepting the natural numbers as a set means that there can be a set with infinitely many elements and some, the “finitists,” find this disturbing and refuse to do it. (It is disturbing, if you stop to think about it.) And as Frenkel explains, Gödel’s Second Incompleteness Theorem means it is impossible to know how the axiom of infinity affects the soundness of mathematics.

Why does this matter? It turns out that there are some theorems *about finite sets* that cannot be proved without the axiom of infinity. One of them (see below) is a strengthened general version of the “Cocktail Party theorem” we discussed last November. In 1977 Jeff Paris and Leo Harrington proved that this was a theorem, but that its proof made essential use of the axiom of infinity. Using the language of quantum physics, in one “setup” the statement is true while in the other it is unprovable. Whether the “observer” is a finitist or not makes a difference.

The theorem in question is the *strengthened finite Ramsey theorem*. To start working towards the statement, let’s go back to the “Cocktail Party theorem” for a moment. It is based on the following special case of the (unstrengthened) finite Ramsey theorem: suppose $N$ points are joined two by two by lines colored either red or blue. Then if $N$ is large enough any such configuration must include a red triangle or a blue triangle. (In this special case $N=6$ is large enough, and that’s the “Cocktail Party theorem.”) The finite Ramsey theorem is about sets, but it is convenient for us to continue to “visualize” it in terms of geometric objects, which here are $k$-*simplexes*. A $0$-simplex is a point, a $1$-simplex is a line segment, a $2$-simplex is a triangle, a $3$-simplex is a tetrahedron. In general a $k$-simplex consists of $k+1$ vertices and all the line segments, triangles, tetrahedra, etc. spanned by any pair, triple, 4-tuple, etc. of those points. Those sub-simplexes are called the *faces* of the $k$-simplex: 0-faces (vertices), 1-faces (edges), 2-faces (triangular faces), etc. For the general statement of the theorem we start with three numbers $n,m$ and $c$ with $m\geq n$, and an $N$-simplex we’ll call ${\bf K}$. We start by coloring every $n$-face of ${\bf K}$ with one of the $c$ colors. The finite Ramsey theorem states that if $N$ is large enough then ${\bf K}$ has a face ${\bf Y}$ of dimension at least $m$, all of whose $n$-faces are of the same color. (In the context of the “Cocktail Party theorem,” $n=1$ and $m=c=2$. If $\bf{K}$ has at least $6$ points, then there is a $2$-face all of whose $1$-faces—interpreted as edges of the triangle—are the same color.)

The finite Ramsey theorem can be proved without using the axiom of infinity. The *strengthened* version cannot. It involves identifying the $N+1$ vertices of ${\bf K}$ with the integers $1, 2, \dots, N+1$, and it reads the same as above except that ${\bf Y}$ can be required to have the additional property that its dimension is greater than the smallest of the integers corresponding to its vertices.

*The Conversation*, February 28, 2024.

In this article, education professor Leah McCoy argues that the game of Tetris is intrinsically linked to an area of mathematics known as *transformational geometry*. Introduced in middle school, transformational geometry involves linking two distinct mathematical objects through transformations like translation or reflection. What’s more, transformational geometry is used by a wide variety of professions, including animation and architecture. “There’s far more to Tetris than the elusive promise of winning,” McCoy writes.

**Classroom Activities: ***geometry, optimization*

- (All levels) Have each student play five minutes of Tetris on their own. Then, teach the definitions of the four basic geometric transformations: Translation, reflection, rotation and dilation. Ask each student to answer the following questions:
- Which transformations show up in Tetris?
- How and when do they show up as you play? Be specific. For example, in what direction, by how much, and in what context are the objects transformed?
- Which types of transformations do not show up in Tetris?

- (All levels) Have students come up with their own variation on Tetris by changing at least one of the types of transformations used in the game. (Students cannot just add a new transformation to the game—they must “delete” at least one existing one. Other features of the game, such as the way new shapes appear, or the shapes that are included, can be changed.) Students will create prototypes of their game using posterboard and construction paper shapes, and present to the class.
- (All levels) Play a game of “Tetris musical chairs.” Have students play Tetris in class. At random time intervals, alert them that it’s time to pause the game. Based on their paused screen, students will write down the transformations needed to get their piece where they want it to go.

*—Leila Sloman** *

*New Scientist*, February 2, 2024.

To physicist Suman Kulkarni, music is more than just sound, rhythm, and notes. She sees the greatest works of Western classical music as complex networks of information. Kulkarni recently analyzed the works of revered composer Johann Sebastian Bach with tools from the field of information theory, which describes data based on its individual parts and the connections between them. It applies to cryptology just as well as it applies to linguistics. Kulkarni wanted to identify *why* Bach’s music is so revered. Bach “produced an enormous number of pieces with many different structures, including religious hymns called chorales and fast-paced, virtuosic toccatas,” writes Karmela Padavic-Callaghan for *New Scientist*. In this article, Padavic-Callaghan describes Kulkarni’s experiment and analysis.

**Classroom Activities:** *information theory, graph theory*

- (Mid level) The article describes Kulkarni parsing Bach’s compositions into a graph of “nodes” connected by “edges.” Watch this short TED-Ed video about the origins of graph theory for more context and to answer the following questions.
- Based on the article and video, describe what the “nodes” and “edges” are in the Bach graphs. What would it mean when two nodes are connected by an edge?
- Can one node be directly connected to more than two other nodes? If so, what would this mean in the Bach piece?
- How would you expect the graphs of a Bach Toccata to differ from a Bach chorale, based on the information in the article?

- (High level) Learn more about graph theory methods, such as depth-first search (DFS), with this free TeachEngineering lesson. Complete the Making the Connection lesson worksheet, which asks you to define terms and analyze a graph. (An alternative to the “friends in class” criteria could be “students with more than one class together” or “students who share an extracurricular activity.”)

*—Max Levy*

*SFGate*, February 5, 2024.

In 1876, physicist Edward Bouchet became the first African American to earn a doctoral degree at a university in the United States. That long-awaited feat was delayed not by a lack of willing students, but by policies barring those students from equal rights. A new documentary titled “Journeys of Black Mathematicians: Forging Resilience” tells the story of similar hurdles in even more recent memory. “African Americans are not only underrepresented in the field, their achievements have been overlooked because the public perceives the road to excellence for African Americans is limited to sports and the arts,” writes Francine Brevetti for *SFGate*. In this article, Brevetti speaks about the stories he uncovered by speaking with Black mathematicians.

**Classroom Activities:** *equity, math history*

- (All levels) Watch the trailer (or rent the full film on Vimeo) and note the names of interviewees in the documentary.
- Research at least two of these mathematicians, and list five facts about their work or life stories.
- Describe the research or recent work of at least one of these mathematicians, in your words.
- Choose one mathematician and present what you’ve learned about them in class.

- (All levels) Read this profile of Christine Darden from
*Quanta Magazine*: The NASA Engineer Who’s a Mathematician at Heart. Discuss what barriers Darden mentions existing in her career. How have things changed or not changed since Darden worked at NASA?

*—Max Levy*

*Scientific American*, February 17, 2024.

In 1938, mathematicians proved a principle called the Ham Sandwich theorem. The theorem states that you can always “cut” $n$ objects in half simultaneously under certain conditions: If those objects are $n$-dimensional, and if your cut has $n-1$ dimensions. For example, if you have 2 circles in a 2-dimensional plane, there exists a 1-dimensional line that bisects both circles simultaneously. And they don’t have to be circles; they can be *any* shape, including discontinuous shapes like scatters of random points and blobs. “Contemplate the bizarre implications here,” writes Jack Murtagh for *Scientific American*. “You can draw a line across the U.S. so that exactly half of the nation’s skunks and half of its Twix bars lie above the line.” Murtagh’s article explains the Ham Sandwich theorem and how it complicates efforts to prevent political gerrymandering, a geographic trick often used to devalue the votes of minority groups or to favor one party.

**Classroom Activities: ***coordinate systems, geometry*

- (All levels) For each example below, draw a coordinate grid spanning $x=[0,8]$ and $y=[0,10]$ using blank or graph paper. Draw the two shapes/sets and find the line that bisects both. (All squares are upright; that is, their sides are perfectly horizontal and vertical, rather than tilted.)
- One circle with a radius of 8 centered at (1,9); one circle of $r = 2$ centered at (4,5)
- One square with side lengths of 2 centered at (4,9); one oval with a width and height of 2 and 1, respectively, centered at (7,2)
- One set of 4 squares, each with areas of 1, centered at (2,2), (3,3), (4,1), and (6,5); one circle of $r = 2$, centered at (2,7)
- What is easiest and most challenging about these examples?

- (Mid level) Murtagh writes “with sufficiently many voters, any percentage edge that one party has over another (say 50.01 percent purple vs. 49.99 percent yellow) can be exploited to win every district.” Demonstrate how this happens with the following exercise based on the images with purple and yellow dots in the article:
- Describe the tally of purple and yellow votes in each district before and after the straight lines shown in the article. Which group is being advantaged and disadvantaged?
- Starting from the second image, draw four more bisecting lines to create 8 “districts.”
- Is it possible to use the “Ham Sandwich” lines to
*increase*the power of the minority group?

*—Max Levy*

*Science News Explores*, February 29, 2024.

In this article, Lakshmi Chandrasekaran recounts several ways that geometry bleeds into research. One group that she covers, at George Mason University’s Experimental Geometry Lab, modeled four-dimensional shapes by 3D-printed projections of the shapes. Other examples were more practical. Laura Schaposnik at the University of Illinois Chicago is studying the possible geometries of viruses, which can help researchers get ahead of as-yet-undiscovered viruses. “Many people may find it hard to see the appeal or everyday uses of such math,” writes Chandrasekaran. “But modern geometry is full of problems both beautiful and useful.”

**Classroom Activities: ***geometry, tiling*

- (All levels) Read the article. In class, brainstorm other possible applications of geometry. As homework, write a paragraph about one of these possible applications. Be creative—your paragraph can be hypothetical, it can incorporate research about what related geometric work has been done, or it can involve your own experiments, but it must be specific about how geometry shows up.
- (All levels) The article describes projection in terms of shadows. Try to work out what the following shadows would look like, then check your work with physical objects:
- A sphere
- A cube oriented upright
- A cube balancing on one corner, with the opposing corner directly above it
- A rectangular prism balancing on one corner, with the opposing corner directly above it
- (High level, Linear Algebra) Now work backwards by calculating the projections explicitly. Assume light is coming from straight above your objects.

- (All levels) Explore tiling with this lesson plan from PBS. For more on tiling, check out the April digests.

*—Leila Sloman*

- The mathematical muddle created by leap years

*BBC*, February 28, 2024. - ‘Entropy Bagels’ and Other Complex Structures Emerge From Simple Rules

*Quanta Magazine*, February 27, 2024. - How two outsiders tackled the mystery of arithmetic progressions

*Science News*, February 26, 2024. - Mathematicians discover ‘soft cell’ shapes behind the natural world

*New Scientist*, February 26, 2024. - What’s the Best Place to Watch the Solar Eclipse? This Simulator Can Help You Plan

*Wired*, February 24, 2024. - Never-Repeating Tiles Can Safeguard Quantum Information

*Quanta Magazine*, February 23, 2024. - Marvel at Hundreds of Mathematician Max Brückner’s Remarkably Precise Models of Polyhedra

*Colossal*, February 22, 2024. - Can you solve it? The magical maths that keeps your data safe

*The Guardian*, February 19, 2024. - Google’s Chess Experiments Reveal How to Boost the Power of AI

*Wired*, February 18, 2024. - How String Theory Solved Math’s Monstrous Moonshine Problem

*Scientific American*, February 5, 2024. - Can you solve it? Are you smarter than a 12-year-old?

*The Guardian*, February 5, 2024.

- “DeepMind AI outdoes human mathematicians”
- AI takes on Geometry in the Math Olympiads
- $\ell$-adic numbers in
*Scientific American*

We have two items this month on the use of large language models (LLMs) in mathematics, both reported in recent *Nature *articles. In one item, the LLM was able to push forward a research-level problem in combinatorics; in the other, it solved geometry questions from the International Mathematical Olympiad, a math competition for high schoolers. LLMs work by predicting, based on a huge number of samples, the next word in a sentence, or, as in these examples, the next line of code in a program or the next step in a proof. They work amazingly well, but since they have no internal model of reality to work with, the predicted text may have no truth value, the predicted code may not work, and the predicted step may lead nowhere. In each instance, the authors relied on more conventional computer mathematics to keep the LLMs on track.

The first of these applications of LLMs appeared in *Nature* back in December 2023, and was highlighted in a “News” piece by *Nature*‘s math commentator Davide Castelvecchi: “DeepMind AI outdoes human mathematicians on unsolved problem.” The journal article was written by a team of twelve, mostly from Google DeepMind in London, with leaders Bernardino Romera-Paredes, Pushmeet Kohli and Alhussein Fawzi.

The researchers used an LLM to attack a research question that Castelvecchi describes as a generalization of the card game “Set.” The “Set” deck has 81 cards, each bearing one, two or three figures. The figures on a single card are all identical. But between cards, the figures differ: They can be ovals, diamonds, or squiggles; can be colored red, purple, or green; and can be outlined, solid, or shaded. So one card could bear two solid purple diamonds, while another could have three shaded green ovals. There are 3$\times$ 3$\times$ 3$\times$ 3 = 81 possible combinations of number, shape, color, and shading, one on each of the 81 cards in the deck. A SET is three cards which in each of the four aspects are either all the same or all different.

Mathematicians have worked out that if you lay out 21 different Set cards, there must be at least one SET among them. But what happens if we complicate the game? Imagine that each figure has five properties, or six, or $n$. (Keep the number of options for each property fixed at three.) In general, the size of the *cap set*, the largest set of cards possible without allowing a SET, is unknown. In an equivalent geometrical formulation this is called the *cap set problem;* the authors of the *Nature* article tell us that Terence Tao (possibly the world’s leading combinatorialist) once called it “perhaps my favorite open question.”

The authors studied the cap set problem, along with other combinatorics problems, with an AI strategy they call “FunSearch.” (The Fun is for “function.”) Rather than directly look for solutions (like a new cap set), they look for a program that will in turn find solutions. The implementation involves a Large Language Model trained on around one million samples of code, an evaluator that applies the LLM output programs to the problem at hand and grades their performance, and a *genetic* programming component (it uses an analogue of natural selection) that “breeds” the best-rated programs to generate new ones, which are put back into the LLM. At any point, the team can stop the process and pick out the highest-functioning program.

When FunSearch was applied to the cap set problem with 8 “properties” (to continue with Castelvecchi’s Set-game formulation) it generated a program that constructed a 512-element cap set. Previously, 496 was the size of the largest cap set known for $n=8$. The authors remark that FunSearch was able to discover a genuinely new cap set “from scratch,” on its own, and by methods different from those that had previously been used.

The second LLM-related item, “Solving olympiad geometry without human demonstrations,” appeared in *Nature*, January 17, 2024. The team (He He from New York University and Trieu H. Trinh, Yuhuai Wu, Quoc V. Le, and Thang Luong from Google DeepMind in Mountain View, CA) reported on the prowess of their “geometry theorem prover” AlphaGeometry. They tested it on a set of 30 problems taken from the International Mathematical Olympiads since 2000.

The authors make it clear that they have not solved the general problem of how to communicate geometric data to a machine. As they say, this is a “separate and extremely challenging research topic.” Instead, they restrict themselves to problems which can be translated into the input language they use, one that works with “human-like non-degeneracy and topological assumptions,” as they explain it. They report that this language was suitable for 75% of the IMO geometry problems. Problems involving geometric inequalities and combinatorics were also set aside. Of the remaining 30 problems, AlphaGeometry solved 25. Compare to the average Silver medalist at the IMO, who solved 22.9 questions, while the average Gold medalist solved 25.9.

The authors distinguish between a geometry problem that can be solved using only data included in the problem statement, and a problem that requires one or more additional *constructions*: adding to the figure points, lines, etc. that are not specified in the premises. The latter type of problems require creativity on the part of the solver. AlphaGeometry combines a *symbolic deduction engine* (which can be used to derive all the possible consequences of a set of premises) and a *language model* which has been trained on nearly a billion problems and can be used to predict a relevant construction, much as ChatGPT can predict the next word in a sentence. That new material is fed back into the deduction engine, and the process cycles until one of the deduced consequences is the solution to the problem.

The authors give as an example of AlphaGeometry’s work its solution of Problem 3 from the 2015 Olympiad.

Note: The exhibited proof of IMO 2015 P3 uses a nonstandard notation that may confuse readers as it confused me. This is the convention of *directed angles* (thanks to Thang Luong for this link). Part of the convention is (a) the notation $\angle$XOY means the angle involved in turning the segment XO *counterclockwise* to match YO and (b) all angle measurements are taken *modulo* 180$^{\circ}$. In particular the statement $\angle$GMD = $\angle$GO$_2$D must be interpreted this way.

“Simple Math Creates Infinite and Bizarre Automorphic Numbers” by Manon Bischoff ran in *Scientific American*, January 11, 2024. It starts with automorphic numbers (see below) but it’s mostly about $\ell$-adic numbers: taking $\ell = 10$, a 10-adic number is an infinite sum like $5 + 2\cdot 10 + 6\cdot 10^2 + 3\cdot 10^3 + \dots$. Let’s not worry for now about convergence in the usual real-number structure, but accept it as an analogue of an infinite decimal like $a_0.a_1a_2a_3\dots$, which can be written as $a_0 + \frac{a_1}{10} + \frac{a_2}{10^2} + \frac{a_3}{10^3} + \dots$. Continuing the formal analogy, we can add and multiply 10-adic numbers exactly as we do the same operations with decimal expansions. This is the arithmetic of *10-adic numbers*.

10-adic arithmetic can be tricky: consider *automorphic numbers*. An automorphic number is a natural number which reappears at the end of its square. Bischoff explains that when you take the sequence of automorphic numbers $5, 25, 625, 90625, 890625, …,$ and extend it forever, you get a number $n$ which equals its square; it’s infinitely long, but we can calculate that it ends in $…8212890625$. So the 10-adic number $5 + 2\cdot 10 + 6\cdot 10^2 + 0\cdot 10^3 + \dots +$ satisfies $n^2 = n$. But then $(n-1) \times n = n^2 – n = 0$, even though $n-1$ and $n$ are both nonzero, exhibiting non-zero *zero divisors,* which is very bad for arithmetic. But, as Bischoff tells us, the problem goes away if we work with a prime number as our base instead of 10, and consider 2-adic numbers like $1 + 0\cdot 2 + 1\cdot 4 + 0\cdot 8 + \dots$, 3-adic numbers like $1 + 2\cdot 3 + 0\cdot 9 + 1\cdot 27 + \dots$, etc. The digits in a $p$-adic number take the values $0, 1, \dots, p-1$. Any calculation that can be done with real numbers can be duplicated with the $p$-adics—we even have $p$-adic analysis, a flourishing area of research extending calculus to the $p$-adics.

There is however an enormous difference in *topology*, the way the numbers fit together. In the $\ell$-adics two numbers are close if their difference is divisible by a high power of $\ell$. So, for example, the $10$-adic numbers $0$ and $2$ are far apart, while $10^{20}$ and $10^{30}$ are close together; whereas in the usual real-number topology, where two numbers are close if the absolute value of their difference is small, $10^{20}$ and $10^{30}$ are very distant. In particular, in the $\ell$-adics, the higher powers of $\ell$ themselves get closer and closer to 0. This is why a sequence like $5, 25, 625, 90625, \dots$ converges in the $10$-adic topology: the difference between successive terms accumulates more and more zeroes at its end, making it divisible by higher and higher powers of 10; it goes to zero.

*Wired*, January 26, 2024.

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

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

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

*—Leila Sloman*

*CNN*, January 26, 2024.

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

**Classroom Activities:** *prime numbers, data analysis*

- (All levels) Watch this Numberphile video about cicada math.
- Describe in your own words the mathematical reason why 13 and 17 may have benefited cicada evolution. (Note: “The jury’s still out” on whether predators are truly the motivation for the cicada’s life cycle length, writes Golembiewski.)
- What would be the next largest cycle length that offers the same evolutionary benefit?

- (Mid level) The US is home to 12 broods of 17-year cicadas, and three broods of 13-year cicadas. Look up the 15 broods and what years they emerge.
- On a spreadsheet, create a table showing when each brood (columns) emerges in a span of years (rows) from 2024 to 2224. When does the next overlap of broods occur?
- Based on the math, when will these same two broods overlap again?
- How many overlapping cicada emergences occur within 200 years?

*—Max Levy*

*Quanta Magazine*, January 9, 2024.

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

**Classroom Activities:** *error correction, exponentials*

- (All levels) Read this explainer about error-correcting code from
*Brilliant*, and familiarize yourself with the NATO phonetic alphabet table.- In groups of two, demonstrate the idea. Each person should write their own sequence of 16 random letters. Out loud, one person should communicate their secret message once, only using standard letters. Next, the other person communicates their own message using the NATO alphabet. Compare the success of communicating in each strategy.
- Explain in your own words why communicating in code with the phonetic alphabet is less prone to misinterpretation.

- (Mid level) As the number of bits $N$ to be relayed in a message grows, the number of bits in the error-correcting code grows exponentially in $N$. This is
**exponential complexity**. Other computer science problems are mathematically more efficient, exhibiting linear complexity, logarithmic complexity, and polynomial complexity.- Using your own online research, describe linear, logarithmic, polynomial, and exponential complexity, and suggest a real-world example for each.
- Write an example mathematical expression for each type of complexity.
- Rank each type of complexity in terms of efficiency.

*—Max Levy*

*Scientific American*, January 30, 2024.

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

**Classroom Activities:** *probability, statistics*

- (All levels) Read the first two sections of the article (through “Detecting Randomness”).
- Describe in your own words the randomness test that Lutsko explains in “Detecting Randomness.”
- Suppose you had a sequence of 10 whole numbers between 1 and 10. Come up with a procedure that applies Lutsko’s test to the numbers.
- Read the third section (“Other Tests”). What does the second type of test capture that the first does not? Is there any property of a random sequence that is not captured by the two types of tests described?

- (All levels) Come up with three ways to generate a sequence of 10 pseudorandom numbers between 1 and 5. (Each number should come up with equal probability $1/5$.)
- Generate the sequences and plot them on a number line. Do they look random?
- Apply the procedure you came up with in the first exercise. What did you learn about your sequences?

- (Mid level) In past digests, we’ve analyzed statistics of coin flips. The same ideas apply here. Calculate:
- The probability that if you randomly generate 2 numbers between 1 and 5, you get the same number each time.
- The probability that if you randomly generate 3 numbers between 1 and 5, you get the same number each time.
- If your sequences were truly random, how many times should you see the same number twice in a row? Three times in a row? Compare to your pseudorandom sequences.

*—Leila Sloman*

*CDM*, January 4, 2024.

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

**Classroom Activities:** *resonance, frequency*

- (All levels) Watch the first video, “Why do I always spill my coffee?”
- Test Sophie Abrahams’ theory by walking with a full mug of water in three different ways. First, walk at a normal pace holding the mug by the handle. Then, walk at a slow pace holding the mug by the handle. Last, walk at a normal pace holding the mug in whatever way you expect will minimize spillage. Note your observations of the fluid movement and discuss in class.

- (Mid level) Watch the other videos referenced in the article. Define
**oscillation, amplitude, period, frequency,**and**resonance**in your own words as they relate to the following contexts:- Pushing a person on a swing
- Walking with coffee in a mug
- The Rubens tube
- Singing high notes by glassware

- (High level) We can use trigonometric functions, such as sine and cosine, to express model oscillations and resonance. For the following questions, assume that the function $\theta = \cos(t)$ describes the angle ($\theta$) of a swing from vertical at time ($t$).
- What is the frequency of the swinging function?
- Describe in your own words the motion of the swing. What do the variables $\theta$ and $t$ represent? Graph the function, and describe what it says about the swing’s motion.
- Which of the following terms can you add to the above equation to most accurately represent being pushed on a swing with resonance? Explain why and sketch the resulting graph.
- $\frac{1}{2\pi}$
- $\cos(t)$
- $t$
- $\frac{t}{4}\cos(t)$

*—Max Levy*

- Mathematicians Have Just Reversed the Sprinkler

*Gizmodo*, January 31, 2024. - This civil engineer turns to math to make energy more affordable

*Science News Explores*, January 29, 2024. - Lewis Carroll transported to Haryana

*Hindustan Times*, January 29, 2024. - The Quest to Decode the Mandelbrot Set, Math’s Famed Fractal

*Quanta Magazine*, January 26, 2024. - The Surprisingly Simple Math Behind Puzzling Matchups

*Quanta Magazine*, January 25, 2024. - A predicted quasicrystal is based on the ‘einstein’ tile known as the hat

*Science News*, January 25, 2024. - This Nomadic Eccentric Was the Most Prolific Mathematician in History

*Scientific American*, January 24, 2024. - A Wild Claim About the Powers of Pi Creates a Transcendental Mystery

*Scientific American*, January 24, 2024. - DeepMind AI solves geometry problems at star-student level

*Nature*, January 17, 2024. - Exploring The Intersection: Math And Machine Learning Forge A Two-Way Street Of Mutual Influence

*IndiaEducationDiary.com*, January 11, 2024. - What the mathematics of knots reveals about the shape of the universe

*New Scientist*, January 5, 2024. - Mathematicians Identify the Best Versions of Iconic Shapes

*Quanta Magazine*, January 5, 2024. - Here’s how much fruit you can take from a display before it collapses

*Science News*, January 4, 2024. - ‘I Got Here For a Reason’: Using Mathematics For Social Justice

*Holy Cross Magazine*.

- Progress in game-of-life studies
- Ominous shakeup for UWV mathematics
- Magnetic skyrmions and hopfions

The “Game of Life” was invented by the late mathematical genius John Conway around 1970, and brought to general attention in Martin Gardner’s “Mathematical Games” column in *Scientific American*, October 1970. Mathematically speaking, it is a *cellular automaton*, but it doesn’t hurt to think of it as a game. Working on a large grid, you mark a few squares in black. These are the live cells; the others are dead. That is Step 1. For Step 2, live cells are created or destroyed according to the following rules. (A cell’s *neighbors* are the eight closest cells).

- A live cell with 0 or 1 live neighbors dies.
- A live cell with 2 or 3 live neighbors persists.
- A live cell with 4 or more live neighbors dies.
- A dead cell with exactly 3 live neighbors comes to life.

Applying the rules to the new configuration leads to Step 3, etc. You can experiment at playgameoflife.com.

The Game of Life proved to be a mathematically fascinating universe, with many remarkable inhabitants, starting with the glider, the Herschel and the Gosper glider gun. Some are *periodic*, returning to their original shape after a certain number of steps. For example, three squares in a row has period 2. The question arose, can any positive integer occur as the period of a game-of-life configuration? That is, for any integer $n$, is there a shape in the Game of Life that returns to itself after precisely $n$ steps? Over the years it was shown that every period was possible except perhaps 19 and 41 and, as Matthew Sparkes reports in *New Scientist* (December 25, 2023), those two last numbers have just been accounted for (“Conway’s Game of Life is Omniperiodic”, posted on ArXiV, December 5, 2023).

The University of West Virginia is closing down its master’s and doctoral programs in mathematics, according to “An ‘Academic Transformation’ Takes on the Math Department,” by Oliver Whong, writing in the November 28, 2023 *New Yorker*. Whong’s story starts in December 2020, with WVU president Elwood Gordon Gee proclaiming that “the perceived value of higher education has diminished” and that WVU should “focus on market-driven majors, create areas of excellence, and be highly relevant to our students and their families.”

“Gee has long argued that land-grant universities, which were created in 1862 by an act of Congress, are meant to ‘prioritize their activities based on the needs of the communities they were designed to serve,’ as he puts it in the book ‘Land-Grant Universities for the Future,‘” writes Whong. The nation’s 105 public and 7 private land-grant institutions include Cornell, MIT, Wisconsin, the University of California … and WVU.

Last September, it was decided that the master’s and PhD programs in mathematics at WVU would be discontinued and a third of the department’s faculty positions eliminated. When Whong asked Gee about the driving force behind the changes, he was told that “it came down to assuring the public, which finances the school, that the university was serving the public’s needs.” A more specific diagnosis came from Maryanne Reed, the WVU provost. The overwhelming majority of math enrollments were in service courses (5000 versus 65 math majors and 23 grad students). Some of those service courses had many low grades: D’s and F’s, along with withdrawals. In an email, Reed told Whong that higher grades “drive first-year student retention and are a primary factor in students’ ability to complete their degrees in a timely fashion. The key point here is that we need to focus on what our students and their future employers want and need.”

Whong reports conversations with several WVU math faculty members, including John Goldwasser, who thinks that students’ mathematical talent can be awakened by a high-quality college program. “I’ve had students in my honors math classes who could be good students anywhere. And many of them didn’t know what their potential was before coming to my class,” Goldwasser told Whong. Goldwasser also mentions the essential belief, shared by many of us in the profession, that “a public university should help to create and shape values, not just reflect the things the majority of people already care about.”

Two kinds of nanoscale physical structures, magnetic skyrmions and hopfions, are attracting attention right now because of their possible application to data storage. Both structures exist inside magnets. We will focus on Hopfion rings in a cubic chiral magnet, contributed to *Nature* (November 22, 2023) by a team at Forschungszentrum Jülich (Fenshan Zheng, Nikolai Kiselev, Filipp Rubakov and collaborators), which reports the first realization of hopfions inside a single magnetic crystal. Getting an idea of what these experiments mean, and how they relate to topology, requires some background. Since skyrmions are simpler to describe, and are necessary ingredients and companions of hopfions, we’ll start there.

Inside a magnet, there is a magnetization vector at each point in space. A skyrmion is a localized 2-dimensional pattern of orientations of the magnetization vector; it is a kink in the magnetization vector field.

In this illustration the magnetization vector field points straight up everywhere, except within a disk. If you trace along the diameter of this disk, the magnetic field vector will gradually rotate clockwise, pointing straight down at the center and pointing up again when you get to the opposite edge. This kink cannot be removed by a continuous deformation –in physical terms, it would cost a lot of energy– and this stability is a topological phenomenon. To understand that, first imagine mapping the disc containing the skyrmion to a sphere about the origin in 3-space: each point of the disc maps to the vector representing the field at that point. Then following the vectors around concentric circles in the disc shows that each circle maps to a circle of latitude on the sphere.

You can imagine the map as stretching the disk over the sphere, with the center of the disk at the South Pole and the periphery at the North Pole. Topology teaches us that, holding the periphery fixed, this map cannot be continuously deformed so as not to cover the sphere. (In topological notation, it represents a non-trivial element of $\pi_2S^2 = Z$, the second homotopy group of the 2-sphere). The skyrmion is trapped in the field, just as a knot cannot be removed from a string if the ends are held fixed.

In a 3-dimensional magnetic solid, skyrmions usually appear aligned together in *skyrmion strings*.

Now for hopfions: a skyrmion string can be bent around to join itself and form a loop. If it is twisted one or more times in the process, the resulting 3-dimensional structure is a hopfion.

For the hopfion illustrated above it is necessary to use a skyrmion string more complicated than the one described by Koshibae and Nagaosa. In particular the magnetization vector field, instead of being constant along the boundary cylinder, rotates as the base point goes around a circumference. (The colors in this image encode the direction of the magnetic vector field at those points; note that the direction-color correspondence is completely different from the one used in the Koshibae-Nagaosa article). The same color-coding is used on the sides of the three skyrmion strings enclosed by a hopfion, and on the surface of the hopfion itself, in the left-hand image below.

In this case the piece of skyrmion string forming the hopfion has been given one complete twist. As a consequence, the colored stripes on the surface corresponding to different directions of the magnetic field are topologically linked, like consecutive links on an anchor chain. That makes the configuration stable under small deformations and guarantees the permanence of the hopfion structure. (The proof is by modern algebraic topology; the magnetization-vector map on the whole solid torus represents a nontrivial element of another homotopy group of $S^2$, this one 3-dimensional.)

]]>*/Film, *December 3, 2023.

An episode of the animated sci-fi sitcom *Futurama* features a machine that allows any two characters to swap bodies. The catch: once two people switch, the machine won’t work for that same pair of bodies again. They can’t switch back. At least, not directly. During the chaos of the episode, nine characters hop around to different bodies. But can they return things to normal? The answer is yes—but it requires some serious math. The show’s writer Ken Keeler, who has a PhD in mathematics, proved that it is possible to return everyone to their own bodies, as long as you add in two new participants who have not used the machine before. The story of Keeler’s result, also known as the “Futurama Theorem,” is described by Witney Seibold in an article for */Film*. In the show, two members of the Globetrotters (a futuristic, math-genius version of the Harlem Globetrotters) discover the theorem, and return everyone to their original bodies to save the day.

**Classroom Activities:** *group theory, symmetric group*

- (All levels) At the beginning of the episode, two people swap bodies. Call them Person 1 and Person 2. Suppose that Person 3 and Person 4 agree to help out by participating in some body swaps. Find a sequence of swaps so that no individual swap is repeated and everyone ends up in their original body. (Keeler’s theorem proves that this is possible.) Hint: This can be achieved with a total of 5 swaps. To keep track of the swaps, make a table whose columns represent the swaps as well as the four bodies (1, 2, 3, and 4), and where each row keeps track of the mind in each body, as below.
- Suppose that only Person 3 agrees to help. Is there still a way to return everyone to their original bodies? Why or why not?
- (High level) The episode shows a sketch of Keeler’s mathematical proof, which is written using symmetric group notation. In this notation, (12) stands for the swap between Person 1 and Person 2, and a sequence of swaps is read right to left. For example, (23)(12) stands for the swap between 1 and 2 followed by the swap between 2 and 3. Try writing out your answer for four people using this notation. Then see if you can write down a solution for five people—that is, once three people have swapped around their bodies, find a way to fix things using two new participants.

- (High level) As a class, work through the Futurama Theorem activity by Cheryl Grood featured on mathcircles.org.

*—Tamar Lichter Blanks*

*The New York Times*, December 6, 2023.

At a United Nations climate conference in December, policymakers debated the future of fossil fuels. Nonrenewable sources contribute 80 percent of the world’s energy supplies, and around 75 percent of greenhouse gas emissions. Fossil fuel emissions must go down to halt climate change, but oil and gas producers contend that technology can capture and sequester the harmful gases. But this scientific solution isn’t as simple as it sounds. “It would be nearly impossible for countries to keep burning fossil fuels at current rates and capture or offset every last bit of carbon dioxide that goes into the air,” Brad Plumer and Nadja Popovich write. “The technology is expensive, and in many cases there are better alternatives.” In this article, Plumer and Popovich analyze the limits of carbon capture technology, and what those limits mean for the future of energy production.

**Classroom Activities:** *rates, data analysis, percentages*

- (All levels) Choose from these classroom activities about climate change and math from NRICH, including one that allows students to compare their carbon footprints, and another that asks students to design an efficient map for delivering energy by drawing a network that maximizes connections but minimizes overall length.
- (Mid level) Approximately 15% of greenhouse gas emissions comes from livestock production, 30% come from automobile emissions, and 40% come from burning coal for energy. Which of the following would reduce total emissions the most?
- Reducing automobile emissions by 20%
- Reducing livestock production by 95%
- Reducing coal use by 40%

- (Mid level) Suppose that the world emitted 50 billion metric tons of greenhouse gases (GHG) in 2023. Create a table based on the following to show how that number changes in the future.
- What will be the amount of annual emissions if the global emissions rate increases by 1.7% every year from 2023 through 2030?
- How many years would it take to get back to 50 billion metric tons per year if, starting in 2030, emissions reduce by a net 1% every year?
- Suppose that burning coal makes up 40% of total emissions and a new carbon capture technology promises to sequester GHG from coal emissions. However, in the first year, the technology can only sequester 0.5% of annual coal emissions. If in every subsequent year, the technology can capture 10% more than it did the previous year, how many years will it take for global emissions to decrease by 10 billion metric tons per year? (Begin your calculations from a total of 50 billion metric tons per year in 2023, and assume that all other emissions don’t change.)

*—Max Levy*

*New Scientist*, December 6, 2023.

In labs around the country, researchers are working on building computers unlike any you have ever used. Traditional computers store and process information as binary digits, or “bits,” 0 and 1. A quantum computer uses principles from quantum physics to carry out calculations at speeds far beyond traditional computers thanks to “qubits” which can be a combination of both 0 and 1 simultaneously. However, when researchers link together many qubits to create a more practical machine, computational errors become more common. In this *New Scientist *article*,* Karmela Padavic-Callaghan writes about a promising breakthrough that benefits from a special type of qubit. “Using thousands of rubidium atoms cooled to near absolute zero, they achieved a record-breaking 48 logical qubits simultaneously, over ten times the previous high,” Padavic-Callaghan writes. “This achievement marks a crucial step toward practical quantum computing.”

**Classroom Activities:** *quantum mechanics, probability*

- (Mid level) Watch this PBS video about the mathematics of quantum computers.
- Explain in your words how a quantum computer works.
- What is special about the mathematics of a qubit compared to a traditional bit?
- In quantum computing, what role does probability play in the measurement of qubits, and how does it affect the outcome of computations?
- If you have a quantum system with 3 qubits, how many different possible states can it represent in superposition?

- (Mid level) Read this article from
*IEEE Spectrum*. If two qubits are entangled, the first qubit can be in a superposition of states $|0\rangle$ and $|1\rangle$, and the second qubit will always have the opposite state of the first one due to entanglement, how many possible states of these entangled qubits exist? What are the possible states?

*—Max Levy*

*BBC*, December 31, 2023.

“Laura is a sprinter,” writes Kit Yates for *BBC.* “Her best time to run 100m (328ft) is 13 seconds, how long will it take her to run 1km (3,280ft)?” If you multiplied 13 seconds by 10 to answer this question, you’re not alone, says Yates. But view this as more than a multiplication problem with set dressing, and you might notice an issue with that strategy: While Laura’s 100-meter sprint time is well behind the times recorded in *World Athletics *list of top 100-meter times for women, a 130-second kilometer is unheard of. “The linear answer would see Laura utterly destroying the world record for running 1km,” writes Yates. That’s just one example of a scenario in which people may be inclined to assume, against the evidence, that things are linear. In this article, Yates argues that this “linearity bias” keeps us from viewing things accurately, with unwanted consequences.

**Classroom Activities: ***nonlinear equations, mathematical modeling, exponentials, statistics*

- (All levels) On the Desmos graphing calculator, let the horizontal axis measure race distance and the vertical axis measure race time. Using data from
*World Athletics*, plot the women’s world records for all races less than 1500 meters.- Using these instructions, find a line of best fit of the form $y = ax + b$. How close can you get $R^2$ to 1? (Hint: Adjust the slider settings so that the step size is 0.001, and $a$ ranges between 0 and 0.5.)
- Now try finding a quadratic curve of best fit, of the form $y = ax^2 + bx + c$. Now how close can you get $R^2$ to 1? Assuming this equation applies to Laura’s running time, how fast would she run the 1000-meter race?
- According to François Labelle, a better formula has the form $y = ax^b$. Using this form, how close can you get $R^2$ to 1? According to this formula, how fast would Laura run the 1000-meter race?
- Do you think these formulas would apply to Laura? Why or why not?
- (High level) Using least squares, calculate the linear and quadratic equations of best fit, and compare the $R^2$ values.

- (Mid level) Yates brings up the example of debt, which grows exponentially over time. Suppose you take out a $100 loan, and every year it accrues 20% interest.
- Without using a calculator or writing anything down, guess how much the debt will have grown after 10 years if you don’t pay anything back.
- How would you calculate how much you will owe after one year if you don’t pay back the loan? Find a general formula for how much you will owe after $n$ years.
- Using your formula, after 10 years, how much will you owe? Was your initial guess based on linearity bias?
- How long will it take for the debt to double?

*—Leila Sloman*

*Scientific American*, December 7, 2023.

In 1979, Adi Shamir came up with a technique which allows a group of people to crack a code, but only if they all work together. In this article, Manon Bischoff walks us through how it works, using a mother who wants to leave an inheritance to her five sons, but who also wants to make sure they split it fairly.

**Classroom Activities:** *finite fields, fitting curves*

- (Mid level) Bischoff explains how in a simpler scenario with two or three sons, the woman could use linear or quadratic curves to make sure the sons have to work together to unlock the combination to her safe. We’ll repeat her example, with a quadratic curve instead of a linear one.
- Have students read the article to understand how this technique works.
- Now students will fit a quadratic curve of the form $y = ax^2 + b + c$. The combination to the safe is the coefficients $(a, b, c)$. Give them two coordinate pairs: $(1, 2)$ and $(-1, 0)$ are both on the curve. What can students figure out about the coefficients based on this information?
- After a few minutes, give the third coordinate: $(0, -2)$. This should be enough to calculate the combination.
- This technique works for any curve with three parameters, as long as you can solve for the parameters. Have students come up with their own combination and parametrized curve, and trade problems with a friend.

- (Mid level) Bischoff notes that if you know the parameters of the curve are integers, that might make the technique less secure. Does this apply to the previous problem? Why or why not?
- (High level) Shamir’s technique uses finite fields. Learn about the modular arithmetic of finite fields using this article from NRICH. Repeat the previous exercise mod 5. How does the arithmetic change?

*—Leila Sloman*

- How magicians control flip of a coin

*Mint*, December 28, 2023. - The Year in Math

*Quanta Magazine*, December 22, 2023. - Asia’s Rising Scientists: Mayuko Yamashita

*Asian Scientist Magazine*, December 21, 2023. - A Close-Up View Reveals the ‘Melting’ Point of an Infinite Graph

*Quanta Magazine*, December 18, 2023. - The Secret Lives of Numbers — a book to make you love maths

*Financial Times*, December 15, 2023. - DeepMind AI outdoes human mathematicians on unsolved problem

*Nature*, December 14, 2023. - Mathematicians Prove the “Omniperiodicity” of Conway’s Game of Life

*Discover*, December 13, 2023. - Tiny balls fit best inside a sausage, physicists confirm

*New Scientist*, December 12, 2023. - A Triplet Tree Forms One of the Most Beautiful Structures in Math

*Quanta Magazine*, December 12, 2023. - How to befriend Maths? Get crafty & play with it

*Times of India*, December 11, 2023. - What Can You Do With an Einstein?

*The New York Times*, December 10, 2023. - How Quantum Math Theory Turned into a Jazz Concert

*Scientific American*, December 6, 2023. - Mathematician Answers Geometry Questions From Twitter

*WIRED*, December 5, 2023.