*The New York Times, *September 11, 2021

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

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

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

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

*—Scott Hershberger*

*The Guardian*, September 12, 2021

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

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

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

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

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

*—Leila Sloman*

*The Guardian*, September 22, 2021

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

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

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

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

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

*—Max Levy*

*Quanta Magazine,* September 23, 2021

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

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

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

*—Leila Sloman*

*ZME Science, *September 10, 2021

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

**Classroom Activities: ***symmetry, geometry, functions*

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

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

*—Max Levy*

Some more of this month’s math headlines:

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

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

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

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

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

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

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

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

*The Washington Post,*September 3, 2021

]]>

- 5D topology in photonic metamaterials
- Dynamics of topological defects on cell membranes
- Ingrid Daubechies in the
*N. Y. Times Magazine*

Magnetic monopoles (e.g. a positive pole with no negative) have been studied seriously as theoretical objects since 1931 when Paul Dirac proved their link to the quantization of electric charge. They are of great mathematical interest because when restricted to a small sphere about the monopole, the magnetic vector potential forces the set of phases of charged particles on the sphere to have the structure of a *circle bundle* with (first) Chern number $C_1=\pm 1$, the sign depending on orientations. This bundle, discovered by Heinz Hopf (also in 1931) and now called the “Hopf Bundle,” is a foundational object in algebraic and differential topology. Its total space (the union of its fibers) turns out to be a 3-dimensional sphere.

In 1978 the physicist C. N. Yang published a “generalization of [the magnetic] monopole to $SU_2$ gauge fields.” The idea was to substitute for the 1-dimensional circle group relating phases the 3-dimensional symmetry group $SU_2$. (The complex analogue of the circle, this is is the group of rotations of *complex* 2-dimensional space.) The “Yang monopole,” as it came to be called, lives in 5-dimensional space.

This year *Mirage* news service and *Science Daily* picked up a press release (PDF) from the University of Hong Kong, “HKU Physicists and collaborators co-observe a higher-dimensional topological state with metamaterials,” dated August 26. The research in question was published in *Science *on July 30; the authors are a team of eight led by Shuang Zhang of the HKU Physics and Electric and Electronic Engineering Departments. As they report in their abstract, “We constructed a system possessing Yang monopoles and Weyl surfaces based on metamaterials with engineered electromagnetic properties, leading to the observation of several intriguing bulk and surface phenomena, such as linking of Weyl surfaces and surface Weyl arcs, via selected three-dimensional subspaces.”

An analogy with the magnetic monopole: “This nonzero linking number between the projected ${\mathcal M}_1$ and ${\mathcal M}_2$ reveals the nontrivial $C_2$ of the Weyl surfaces in the 5D space.” Just as circle bundles over a surface are characterized by their first Chern number $C_1$, principal $SU_2$-bundles over a 4-dimensional surface are characterized by their second Chern number $C_2$. The Yang monopole has $C_2=\pm 1$ depending on orientations, corresponding to the simple linking observed between the Weyl surfaces.

“Topological braiding and virtual particles on the cell membrane” appeared in *PNAS* on August 20, 2021. As the authors, a team of seven from MIT, Harvard, and the Flatiron Institute, explain at the start, “Combining direct experimental observations with mathematical modeling and chemical perturbations, we investigate the dynamics of spiral wave defects on the surfaces of starfish egg cells.” They continue: “To investigate the braiding dynamics of biochemical spiral waves in living cells, we compared here experimental observations of Rho-GTP activation waves on starfish oocyte membranes with predictions of a generic continuum theory. Rho-GTP is a highly conserved signaling protein pivotal in regulating cellular division and mechanics across a wide variety of eukaryotic species.”

The authors report: “Topological defects in the phase field are singular points with winding number +1 or −1 corresponding to counterclockwise or clockwise rotating centers of propagating spiral waves. These phase defects are created and annihilated in pairs, conserving the total topological charge. By tracking the 2 + 1-dimensional world lines of both defect types, we observed complex creation, annihilation, and braiding dynamics, similar to those in Bose−Einstein condensates. … In addition to short-lived loops which dominate at high activity, low-activity states exhibit a large number of long-lived defect world lines that undergo spontaneous braiding dynamics. Space−time braiding of spiral cores is indicative of chaotic dynamics of the Rho-GTP signaling patterns.”

“As a topological measure of complexity in dynamical systems, braiding analysis has the advantage that it is well grounded in group theory. Mathematically, a sequence of braiding history between particles can be treated as a series of sequentially multiplied generators, where each generator denotes the direction of ‘crossing’ between one particle and its neighbors projected onto a reference line at an instantaneous time. Analyzing such product of generators as a function of time then gives a measurement of complexity growth in the system.” Specifically, they represent each of the generators by an $(n-1)\times(n-1)$ matrix and take $\Sigma_n(t)$ as the product of the matrices corresponding to the crossings up to time $t$. “The matrix product $\Sigma_n(t)$ records the braiding history of particles and therefore contains information about the system dynamics. One important piece of information is the magnitude of its largest eigenvalue, $E_n(t)$, often termed as the braiding factor. In random matrix theory, the exponential growth rate of $E_n(t)$ at long time limit approximates the Lyapunov exponent of a chaotic system, which has also been verified in numerical experiments. Such exponential growth rate is therefore termed as the braiding exponent,

$$\lambda(n)=\lim_{t\rightarrow\infty}\frac{1}{t}\ln|E_n(t)|.$$

For our two-dimensional defect trajectories, the braiding factor calculated from taking the average of all reference line projections … displayed consistently positive braiding exponents $\lambda(n)$.”

Siobhan Roberts contributed a long profile of the mathematician Ingrid Daubechies to the September 19, 2021 edition of the *New York Times Magazine*. “A professor at Duke University, in Durham, N.C., Daubechies’ métier is figuring out optimal ways to represent and analyze images and information. The great mathematical discovery of her early career, made in 1987 when she was 33, was the ‘Daubechies wavelet.’ Her work, together with further wavelet developments, was instrumental to the invention of image-compression algorithms, like the JPEG2000, that pervade the digital age.”

What are wavelets? Roberts tries to give us some idea. First, waves. “Daubechies says […] ‘You can build almost anything by combining, in clever ways, waves of different wavelengths.’ This idea dates back two centuries: In 1822, the French physicist and mathematician Joseph Fourier […] proposed that all periodic functions — all periodic phenomena — could be understood as sums of sine and cosine waves. […] But this approach had its limitations: It couldn’t efficiently handle signals with abrupt changes, like spoken language or pictures with sharp edges and sudden transitions in luminosity.”

Then wavelets, a 20th-century innovation. “Sometimes Daubechies gives a fancifully impractical musical metaphor to describe the difference. For Fourier analysis, she envisions a room full of thousands of idealized tuning forks, each sustaining a uniquely assigned note indefinitely. […] Wavelets, by contrast, are a more sophisticated symphony orchestra of tuning forks that each ring for a shorter time. They can, in a manner of speaking, read and convey all the information contained in the musical score: information about tempo and note duration, and about even more granular nuances of musicality, like […] the attack at the start of a note, or the purity of tone held for bars at a time. ‘With wavelets you can decompose all that in an efficient way,’ Daubechies says.”

Mathematically speaking, Daubechies wavelets are a family of functions db1, db2, etc. that play a role (as suggested above) similar in some ways to the families $\sin x, \sin 2x$, etc. and $1, \cos x, \cos 2x$, etc., which can be combined, with the right coefficients, to represent any function of period $2\pi$. Here are the first few:

Each of the wavelets generates a bi-infinite family of its own through horizontal scaling by powers of 2 and shifting by integer lengths. All these functions together are orthogonal (the product of any two integrates to zero), just like the sines and cosines, and a judicious finite linear combination of them can give an excellent and efficient approximation of of one-dimensional signals as diverse as seismograph records or speech (at this level of explanation, the signal needs to have first been adjusted to have average value zero). Their extension to two dimensions has become essential in compressing photographic images and movies so that they can be stored and transmitted efficiently.

Roberts continues, “Daubechies is most famous as a pioneer of wavelets, but more broadly, her scientific contributions over the last three decades have rippled out in all directions from the field of ‘signal processing.’ […] Jordan Ellenberg, a mathematician at the University of Wisconsin-Madison […], points out that signal processing ‘makes up a huge proportion of applied math now, since so much of applied math is about the geometry of information as opposed to the geometry of motion and force.'” Roberts mentions in particular Daubechies’s recent participation in the restoration of *The Adoration of the Mystic Lamb* (the Ghent Altarpiece—closed and open), “a 15th-century polyptych attributed to Hubert and Jan van Eyck, arguably among the most important paintings in history.”

Much of the profile covers Daubechies’s life and personality. One striking detail: Daubechies had her first baby in 1988. “It was an unsettling and disorienting period, because she lost her ability to do research-level mathematics for several months postpartum. ‘Mathematical ideas wouldn’t come,’ she says. That frightened her. She told no one, not even her husband, until gradually her creative motivation returned. On occasion, she has since warned younger female mathematicians about the baby-brain effect, and they have been grateful for the tip.”

Plans for the future? “Machine learning’s success […] is something that Daubechies believes mathematicians and mathematically inclined scientists should attend to more. ‘Machine learning works very well, and we don’t know why it works so well,’ she says. ‘I consider that a challenge for mathematicians, to understand it better.’ […] Usually, the argument is that beautiful, pure mathematics eventually — in a year, in a century — produces compelling applications. Daubechies believes that the cycle also turns in the opposite direction, that successful applications can lead to beautiful, pure mathematics. Machine learning is a promising example. ‘You can’t argue with success,’ she says. ‘I believe if something works, there is a reason. We have to find the reason.'”

]]>The story was picked up by The Guardian, the Smithsonian’s SmartNews, Science News, Arab News and Popular Mechanics. Donna Lu’s coverage in *The Guardian* (August 4, 2021) has the headline “Australian mathematician discovers applied geometry engraved on 3,700-year-old tablet”, with subhead “Old Babylonian tablet likely used for surveying uses Pythagorean triples at least 1,000 years before Pythagoras.”

Lu begins: “Known as Si.427, the tablet bears a field plan measuring the boundaries of some land. The tablet dates from the Old Babylonian period between 1900 and 1600 BCE and was discovered in the late 19th century in what is now Iraq. It had been housed in the Istanbul Archaeological Museum before Dr Daniel Mansfield from the University of New South Wales tracked it down.”

First note that the area surveyed in the tablet has been partitioned into simple geometrical shapes (rectangles, right triangles, trapezoids) for which the Babylonians knew how to calculate the area from the linear dimensions.

For example, the rectangle $ABIH$ has sides marked 50 and 22;40 (rods; a BabylonianMansfield noticed the presence of two or maybe three Pythagorean triangles (right triangles with whole-number sides) in the diagram, including the triangle $ADE$ with sides $7;30$ and 18 ($\frac{3}{2}$ of the 5, 12, 13 right triangle) and the rectangle $GHML$, equal to two copies of $ADE$. He argues that measuring out Pythagorean triangles in the terrain (a practice still in use today) was a way to guarantee right angles and accurate area computations.

Davide Castelvecchi’s contribution to *Nature* for June 18, 2021 was “Mathematicians welcome computer-assisted proof in ‘grand unification’ theory,” with sub-head “Proof-assistant software handles an abstract concept at the cutting edge of research, revealing a bigger role for software in mathematics.” He tells us how Peter Scholze (Bonn; “considered one of mathematics’ brightest stars and has a track record of introducing revolutionary concepts”) and his collaborator Dustin Clausen (Copenhagen) have set forth an “ambitious plan,” which they call *condensed mathematics: *“they say it promises to bring new insights and connections between fields ranging from geometry to number theory. … Until now, much of that vision rested on a technical proof so involved that even Scholze and Clausen couldn’t be sure it was correct. But earlier this month, Scholze announced that a project to check the heart of the proof using specialized computer software had been successful.”

The specialized software in question is a *proof assistant*. Castelvecchi: “Proof assistants … force the user to lay out the logic of their arguments in a rigorous way, and they fill in simpler steps that human mathematicians had consciously or unconsciously skipped. … In this way, proof assistants can help to verify mathematical proofs that would otherwise be time-consuming and difficult, perhaps even practically impossible, for a human to check.”

Castelvecchi goes on to tell us something about condensed mathematics, and the ensuing ‘grand unification’ project; finally: “Around 2018, Scholze and Clausen began to realize that the conventional approach to the concept of topology led to incompatibilities between [the] three mathematical universes — geometry, functional analysis and $p$-adic numbers — but that alternative foundations could bridge those gaps.”

“There was one catch, however: to show that geometry fits into this picture, Scholze and Clausen had to prove one highly technical theorem about the set of ordinary real numbers.” Scholze found a proof, but is was so novel and complex “that Scholze himself worried there could be some subtle gap that invalidated the whole enterprise.” Here is where Kevin Buzzard (Imperial College, London) and Johan Commelin (Freiburg), along with a team of other experts in the proof-assistant package Lean, enter the picture. “By early June, the team had fully translated the heart of Scholze’s proof — the part that worried him the most — into Lean. And it all checked out — the software was able to verify this part of the proof.”

On June 18, 2021, *SciTechDaily* posted “A New Bridge Between the Geometry of Fractals and the Dynamics of Partial Synchronization”, a press release from Universitat Pompeu Fabra, Barcelona. It refers to “Chimeras confined in fractal boundaries in the complex plane,” published in *Chaos* May 3, 2021 by UPF Professor Ralph Andrzejak. “The work generalizes the Mandelbrot set for four quadratic equations.”

The *Mandelbrot set* comes up in the study of the iterates of the complex function function $f_c(z)=z^2+c$. It is the set of complex numbers $c$ for which the sequence $0, c, c^2 + c, (c^2+c)^2 + c, \dots$ is bounded (this is the sequence defined by $z_0=0$ and $z_{n+1}=f_c(z_n)=z_n^2 + c$). The well-known intricacy of the Mandelbrot set reflects how delicately this criterion depends on $c$. The *SciTechDaily* posting quotes Andrzejak’s analogy: “the Mandelbrot set is based on one equation with one parameter and one variable. We can imagine this variable as a small ball moving on the surface of a large round table. What happens to this ball depends on the parameter of the equation. For some values of this parameter, the ball moves and is always on the table. The set of all these parameter values for which the ball remains on the table is what defines the Mandelbrot set. On the contrary, for the remaining parameter values, the ball falls from the table at some point in time.”

Andrzejak studies four coupled copies of the iteration, with the same value of $c$. The analogy continues: “one might think that the four equations we are using describe the movement of not only one, but four balls on the table surface. Since the equations are connected, the balls cannot move freely. However, they attract each other, like the sun, Earth and moon attract each other through gravity.” More explicitly, Andrzejak studies a network $F_c$ of four functions in two pairs: $u, v$ and $p, q$, where $u$ is strongly coupled to $v$ and weakly coupled to $p$ and $q$, etc. There are four equations, for example:

$$u_{n+1} = f_c(u_n) + K[f_c(v_n)-f_c(u_n)] +k[(f_c(p_n)-f_c(u_n))+(f_c(q_n)-f_c(u_n))]$$ with $K=0.01$ and $k=0.0025$ the strong and weak coupling constants, respectively.

The behavior of the network depends, delicately of course, on $c$. Among the phenomena Andrzejak studies is *synchronization*. For example, he calls the $u$ and $v$ defined with parameter value $c$ synchronized, and writes $U_c=V_c$, if they have the same limit as $n\rightarrow \infty$ (up to appropriate tolerances for the duration of the simulation, $5\times 10^5$ iterations), and $U_c \neq V_c$ if they do not. This figure, which appears in the *SciTechDaily* posting, gives an idea of how the synchronization regime varies with $c$. The points are color-coded:

Color |
Type |
Formula |

Light Blue | Full synchronization | $U_c=V_c=P_c=Q_c$ |

Orange | Within-pair synchronization | $U_c=V_c\neq P_c=Q_c$ |

Green | Across-pair synchronization | $U_c=P_c\neq V_c=Q_c$, or $U_c=Q_c\neq V_c=P_c$ |

Magenta | Chimera state | $U_c=V_c$ but $P_c\neq V_c$, or $P_c=Q_c$ but $U_c\neq V_c$ |

Red | Full desynchronization | $A_c\neq B_c$ for $A\neq B = U, V, P, Q$ |

The *SciTechDaily* posting ends with the question, “whether the mathematical model in question can be relevant to the dynamics of the real world.” They give Andrzejak’s answer: “Yes. Absolutely. The best example is the brain. … Our brain can only work properly if some neurons synchronize while other neurons remain out of sync. … If we study the basic mechanisms of partial synchronization in very simple models, this can help understand how it is established and how it can be kept stable in such complex systems as the human brain.”

*The Guardian*, August 17, 2021

Pi is something more than just a number. Pi has infinitely many digits after its decimal point and no observable pattern. It’s a classic example of so-called “transcendental numbers,” which can’t be calculated from any combination of ratios, powers, and roots of whole numbers. You may know $\pi$ as 3.14 or 3.14159, but in a new study, Swiss researchers used a supercomputer to calculate a world record 62.8 *trillion* digits. The new estimate—yep, it’s still technically an estimate—surpasses the previous record of 50 trillion digits with a calculation more than three times as fast. “It’s an impressive and time-consuming feat that prompts the question: why?” writes Donna Lu in *The Guardian. *Lu’s article explores the history and motivations of humans’ obsession with $\pi$.

**Classroom Activities: ***geometry, pi, circles, accuracy vs precision, transcendental numbers*

- Pi appears in nature as the fundamental ratio between any circle’s circumference and its diameter. But the elusiveness of its exact value has baffled humans for thousands of years
*.*Why do precise estimates of $\pi$ matter?- (All ages) Ask students to look up the first 10 digits after the decimal point in $\pi$, then calculate the circumference of a circle with radius 10 meters based on each different estimate: (i.e. 3; then 3.1; then 3.14; and so on.) Discuss how the precision changes with each additional decimal. In what situations does this increased precision matter?
- (Middle school) Repeat the above activity, but this time calculating the circumference of Earth at the equator based on the different $\pi$ estimates. (The equatorial radius of the earth is 3,963.161 miles.) How many miles longer is the estimated circumference if you use 10 digits in your $\pi$ estimate instead of just one? How many digits of $\pi$ do you need before the difference between consecutive estimates is less than 1 mile? Less than 0.01 miles? Discuss in which situations these differences matter.
- (High school) Repeat the above activity, now with different levels of precision for both $\pi$ and the equatorial radius. How do the values compare? What does this reveal about the level of precision needed in calculations? (If students are familiar with significant figures, this can also be a chance to practice the concept.)

- (High school/Algebra/Geometry) Amelia is a pilot. Her small plane holds 50 gallons of fuel. Each gallon can take her 20 miles. If Amelia flies around the earth along the equator, how many times will she need to stop for more fuel? Solve the problem using $\pi$ with two digits after the decimal, and again with 10 digits after the decimal. Does accounting for her altitude above the earth (say, 10,000 feet) significantly change the answer?
- (High school) Learn about transcendental numbers, and why it’s so fascinating that some numbers (most numbers!) can’t be produced using algebra, by watching this Numberphile video.
*—Max Levy*

#### Israeli data: How can efficacy vs. severe disease be strong when 60% of hospitalized are vaccinated?

*Covid-19 Data Science*, August 17, 2021In places with high COVID-19 vaccination rates, like Israel, a high proportion of people hospitalized with the disease are vaccinated. Some commentators have cited that fact to incorrectly claim that the COVID-19 vaccines are not effective. The first mistake in this claim is the use of raw counts ($X$ people) rather than rates ($Y$ people per 100,000). The second mistake stems from a statistical phenomenon called Simpson’s paradox: In some data sets, a certain trend is present when the data are put into groups but reverses when the data are combined. In a blog post (and accompanying Twitter thread) drawing upon detailed Israeli data, statistical data scientist Jeffrey Morris of the University of Pennsylvania explains how the data actually show that the vaccines are highly effective in preventing severe COVID-19. (A

*Washington Post*article by Jordan Ellenberg also covers the topic.)**Classroom Activities:***Simpson’s paradox, statistics*- (Introductory statistics) Introduce Simpson’s paradox using this MinutePhysics video. For more examples of the phenomenon, see this article from
*Towards Data Science*. - (Introductory statistics) Once students understand the basics of Simpson’s paradox, walk them through the reasoning in the COVID-19 article. Present each data table in the article one at a time, asking students to explain what they see. Conclude with a discussion of the caveats that Morris mentions.

*—Scott Hershberger*

**Mathematicians are deploying algorithms to stop gerrymandering***MIT Technology Review*, August 12, 2021In an ideal democracy, everyone’s vote counts equally. But that’s not always the case—especially when politicians can draw electoral districts to their party’s advantage. This practice is called gerrymandering, a reference to the salamander-shaped district drawn by the administration of Massachusetts governor Elbridge Gerry in 1812. But gerrymandering isn’t always that obvious. A map may look perfectly normal, yet still produce election results that don’t match the overall will of the citizens of the region. To deal with this, mathematicians from all over the country have been working on software that compares redistricting maps with randomly generated “samples.” If a map produces wildly different electoral results than the samples, that’s a clue that it may have been drawn to intentionally benefit one party over another. With this software in hand, mathematicians can help the courts and the public identify biased maps, and perhaps help make elections fairer for everyone.

**Classroom activities:***politics, gerrymandering, probability, law of large numbers, central limit theorem*- (Middle school) Teach about gerrymandering with this lesson from KQED (PDF).
- (Introductory statistics) The redistricting approaches described in the article implicitly use probabilistic ideas like the law of large numbers and the central limit theorem. Teach the law of large numbers through a hands-on simulation as a homework or in-class assignment.
- Give each student a six-sided die (or a similar “random number generator”). They will each roll their die 10-20 times, and the class will pool their data (a total of $N$ rolls) to find \[N(i)=\text{number of times } i \text{ was rolled, } i=1,2,3,4,5,6.\]
- Before rolling the dice, ask students to predict $N(i)/N$ for each $i$ and write down a confidence interval. This should be the
**narrowest**interval that they think will match the data. - Once the data is in, you will probably find that $N(i)/N \approx 1/6$ with some small variation. This is the expected outcome due to the law of large numbers.
- Have students write a paragraph relating this activity to the redistricting software described in Roberts’ article.

- (Introductory statistics) Introduce Simpson’s paradox using this MinutePhysics video. For more examples of the phenomenon, see this article from

*—Leila Sloman*

*NPR*, August 11, 2021

Although the delta variant of SARS-CoV-2 is one of the most contagious respiratory viruses that we know of, it’s still not as contagious as chickenpox. In this story for *NPR,* Michaeleen Doucleff uses the concept of $R_0$, the average number of people that a sick person will infect when the entire population is vulnerable to the virus, to compare the transmission rate of the delta variant with that of several other viral infections. For the original coronavirus, $R_0$ was between 2 and 3. Recent estimates place the $R_0$ of the delta variant between 6 and 7. That increase makes a huge difference in how fast the virus spreads due to the mathematics of exponential growth.

**Classroom Activities: ***exponential growth, exponential decay*

- (Middle school) To bring to life how fast exponential growth is, make a large $x$-$y$ plane on the floor and have students map out an exponential curve with their feet.
- (High school) The National Council of Teachers of Mathematics offers a set of resources on the math of the pandemic. In particular, use this interactive tool to simulate the spread of a virus.
- Have students predict how the spread depends on the number of days a person is contagious, the chance of contracting the virus per contagious contact, and the number of contacts per day.
- How do those three numbers relate to $R_0$?

- (High school) This lesson from the
*New York Times*introduces exponential growth and exponential decay using data from the coronavirus pandemic. The lesson also includes links to*NY Times*activities on herd immunity, vaccine efficacy, and vaccine hesitancy.

*—Scott Hershberger*

*Quanta Magazine, August 9, 2021*

1 minus 1 seems like the easiest math problem in the world. But the brain processes required to solve it reveal a messy world where the borders between mathematics, biology, and psychology become hazy. Monkeys, baby birds, and even bees can do arithmetic. These animals understand the relative positions of numbers on a number line (e.g. two bananas are greater than one banana)—a concept called numerosity. But can they understand where *zero* falls on the number line? Zero is special—it’s not a quantity, it’s *an absence*. “Even humans struggle with zero,” writes Jordana Cepelewicz*. *Humans only began acknowledging zero in the seventh century. Its complexity led many to think that only humans are in the know. But new research on animal cognition is proving that assumption wrong. Cepelewicz writes about numerosity in animal brains and how crows can, remarkably, understand zero.

**Classroom Activities: ***What is zero?, number systems*

- (All levels) Challenge students to explain why 0 falls on a number line before 1
*without*referencing the fact 1 minus 1 is 0. There are no real “right” answers here; this should just be a way of reinforcing the notion of zero being weird. - (All levels) Although other animals have a sense of numerosity, humans are unique in having a “symbolic” understanding of mathematics, using characters like 0 and 1 to express numbers abstractly. This simplifies our communication and allows us to reach a deeper understanding of mathematics.
- (Middle school) Ask students to invent their own set of symbols to replace our standard digits 0-9. Then have them teach their new systems to each other. Discuss the challenges of learning these symbolic values, and what you think humans gain from this ability.
- (High school) Ask students to invent their own number system, in base 10 or another base (or another system entirely). Show them the number systems used by the ancient Romans, Mayans, and/or Babylonians for inspiration. Then have the students teach their new systems to each other. Discuss the challenges of learning these symbolic values, and what you think humans gain from this ability.

*—Max Levy*

*Big Think*, August 4, 2021

In the age of computers, many people believe they never need to think about math. “The supermarket checkout totals the bill, sorts out the special meal deal, adds the sales tax,” writes Ian Stewart in this excerpt from his book *What’s the Use?: How Mathematics Shapes Everyday Life.* But, he explains, these technologies actually rely heavily on advanced mathematics. Computers are an invaluable tool to mathematicians, with their ability to do millions of calculations in an instant. And the algorithms that make this possible rely on all kinds of mathematical ideas, from linear algebra to topology.

**Classroom Activities: ***algorithms, programming, computer science*

- (Middle school) Stewart mentions Google’s PageRank algorithm as a prime example of the reliance of modern life on mathematics. Introduce the concept of algorithms with Teach Engineering’s lesson plan, and explore the PageRank algorithm further by playing the game in the associated Acting Like an Algorithm activity.
- (Introductory programming) One way to take advantage of computers’ ability to do thousands of calculations per second is by programming loops.
- Teach for and while loops in Python with this online tutorial. Have students write a loop that calculates $f(x)$ for 10 different choices of $x$ and their choice of the function $f$. Use the
`timeit.timeit()`

function to time how long their code takes to run. - Time students doing the same calculation by hand. How long does it take?

- Teach for and while loops in Python with this online tutorial. Have students write a loop that calculates $f(x)$ for 10 different choices of $x$ and their choice of the function $f$. Use the

*—Leila Sloman*

**Everyone maps numbers in space. But why don’t we all use the same directions?**

*Science News,*August 23, 2021**Why You May Have More Friends Than Your Friends Do**

*Nautilus*, August 20, 2021**Just four colors are enough for any map. Why?**

*Big Think*, August 20, 2021**A Deep Math Dive into Why Some Infinities Are Bigger Than Others**

*Scientific American,*August 16, 2021**Modern Mathematics Confronts Its White, Patriarchal Past**

*Scientific American,*August 12, 2021**The twisted math of knot theory can help you tell an overhand knot from an unknot**

*Massive Science*, August 11, 2021**Meet Melba Roy Mouton, the Space Race mathematician and keeper of orbiting satellites**

*Massive Science,*August 5, 2021**Galois Groups and the Symmetries of Polynomials**

*Quanta Magazine*, August 3, 2021**Meet Evelyn Boyd Granville, the mathematician who mass-produced computers and shot Apollo into space**

*Massive Science,*August 3, 2021**Bob Moses: Civil rights activist who used maths to fight inequality**

*The Independent,*August 2, 2021**The tangled physics of knots, one of our simplest and oldest technologies**

*Massive Science,*August 1, 2021

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*The Conversation*, July 15, 2021

Emmy Noether is one of the most famous female mathematicians of the twentieth century. In a celebration of the 100th anniversary of some of her greatest work, Tamar Lichter Blanks recounts the difficulties Noether faced as a woman and Jew in a world that discriminated against both groups. Lichter Blanks also describes Noether’s most famous result concerning rings, a set in which one can multiply, add, and subtract.

**Classroom Activities: ***diversity and inclusion, ring theory*

- Read Lichter Blanks’ article in class. Discuss the discrimination Noether faced and how the modern world is similar or different.
- Introduce rings (see Wolfram MathWorld or Academic Kids for definitions and examples) and assign the following exercises:
- Prove that the set of integers ($\{ \dots, -2, -1, 0, 1, 2, \dots\}$) forms a ring.
- Does the set of even integers form a ring? Which axioms does it satisfy?
- Does the set of odd integers form a ring? Which axioms does it satisfy?

*—Leila Sloman*

*Quanta Magazine, *July 8, 2021

Scorch a magnetic bar of iron with enough heat, and the solid mass will lose its magnetism. We can observe this type of “phase transition” without a microscope. But zooming into the neighborhood of atoms, we notice *why *the transition occurs: swaths of atoms that previously aligned magnetically are now in disarray. So-called critical points—where the look and feel of a system teeters between two different states—appear everywhere in nature, from boiling water to crystallizing minerals. “Mathematicians try to bottle this magic in simplified models,” writes Allison Whitten for *Quanta Magazine*. Whitten’s article describes new mathematics research showing that phase transitions contain rotational invariance—a circular symmetry where the model has the same physical properties regardless of how it’s rotated. A “mathematical elegance” sets in at the critical point, Whitten writes. The finding inches researchers closer to proving that phase transitions have *conformal* invariance, strong “overall” symmetry.

**Classroom Activities: ***symmetry, fractals, Penrose tiles*

- Discuss the different kinds of symmetry which make up conformal invariance: rotational, translational, and scale symmetry. Which symmetry/symmetries is/are present in a tiling of identical squares? What about a series of concentric circles?
- Explore fractals, which have scale symmetry, and talk about some examples of fractal-like behavior in nature. (Also see the Wolfram Demonstrations Project on Fractals.)
- Introduce the Penrose tiles (see this Veritasium video), which do not have translational symmetry and yet
*do*have rare fivefold rotational symmetry like a pentagon. As you zoom out further and further, Penrose tiles have a fractal-like behavior.

*—Max Levy*

*The Atlantic, *July 2, 2021

How can you tell when good luck is *too* good? Last year, a popular gamer named “Dream” performed so well in the game Minecraft that his audience took notice—something seemed fishy. “He was the equivalent of a roulette player who gets their color 50 times in a row,” writes Stuart Ritchie in *The Atlantic. *“You don’t just marvel at the good fortune; you check underneath the table.” Dream’s drama concluded when a diffuse group of gamers published a robust mathematical analysis of his dubious performance. They compared his luck receiving useful items in the game with the probability of getting those items by chance. Then, they used statistical methods to deduce that Dream’s luck was “unfathomably” unlikely. The article closes with a shift to academic science. Mathematical analysis like this one in gaming, Ritchie writes, can maintain the integrity of science by weeding out cheating and fraud there too.

**Classroom Activities: ***probability, p-values, Benford’s Law*

- Tell half of your students to flip a coin 100 times and tally the results. Tell the other half to just make up a sequence of 100 coin flips. Afterward, ask everyone to count their longest streak of heads or tails. You’re likely to find that the real sequences of flips had longer streaks than the fake ones. Making up truly random data is hard! (See this Texas Instruments activity for a more in-depth exploration of such streaks.)
- Work through this classroom activity on statistical testing from Carleton College: Are Female Mallards Attracted To The Color Green? This worksheet walks students through forming a statistical hypothesis, gathering and analyzing evidence, and interpreting their analysis as a conclusion via a
*p*-value. - Discuss Benford’s Law, otherwise known as the law of anomalous numbers, which helps detect fraud in finance. Watch this video from Numberphile about using math to detect fraud with Benford’s Law.

*—Max Levy*

*Scientific American,* July 2021

The Fields Medal, the highest honor in mathematics, was initially conceived of in part as an equalizer. In a recent paper in *Humanities and Social Sciences Communications*, Feng Fu (Dartmouth College) and Ho-Chun Herbert Chang (University of Southern California) write “The award was intentionally given to individuals that would otherwise not receive any recognition, rather than the best young mathematician.” But by analyzing data from the Mathematics Genealogy Project, Fu and Chang find that this plan has fallen by the wayside—44 out of 60 Fields Medalists are academic descendants of either Jean le Rond d’Alembert or Gottfried Liebniz. Moreover, mathematicians of Arabic and African descent are underrepresented among medalists and within the elite community. The in-depth graphic by Clara Moskowitz and Shirley Wu in *Scientific American *shows the connections between Fields Medalists revealed by Fu and Chang.

**Classroom Activities: ***diversity and inclusion, data visualization*

- Fu and Chang use mathematicians’ names as a proxy for their “lingo-ethnic” identities. Discuss whether this is a good measure or not and what potential errors it might bring up.
- Have students interpret the graphs shown in Fu and Chang’s paper and reflect on the over/underrepresentation of certain groups in mathematics. Ask questions like:
- What effect might this have on mathematics research and the community?
- Do they notice connections to their own lives?
- Should the Fields Medal Committee return to its original practice of awarding individuals who would otherwise not be recognized, and why or why not?

If (or when!) students have different answers to these questions, have them read and respond to one another’s ideas.

- Discuss the way the data is visualized in the
*Scientific American*graphic. Is it effective or not? What specific elements of the graphic contribute to that effect? - Have students devise alternative ways to visualize the data on Fields Medalists, or have them devise a visualization of another data set. What aspects of the data do they emphasize, and why? (Teach Data Science offers further resources on teaching data visualization.)

*—Leila Sloman*

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*The New York Times*, June 25, 2021

What would a font based on Tetris look like? What about a Sudoku font, or a juggling font? Erik and Martin Demaine, a computer scientist and an artist, have designed a multitude of “algorithmic puzzle fonts,” mathematically inspired typefaces that double as puzzles. In this article, Siobhan Roberts details how the duo create their fonts, which are brought to life here in colorful images and animations. Some of the fonts, like the “Conveyer Belt Font,” even related to unsolved mathematical problems. Whether inspired by origami or checkers, the Demaines’ designs burst with playful curiosity and joyful exploration.

**Classroom Activities: ***origami, geometry, polygons*

- Read more about the fold-and-cut process that underlies the Fold & Cut font pictured at the top of the article. Have students create their names in the font with real paper to see how the process works.
- One font described in the article is made from “polyforms.” Explore Henri Picciotto’s classroom activities on polyforms, which include a set of virtual pentominoes.

*—Scott Hershberger*

*The Denver Post*, June 15, 2021

Millions of Americans have student loan debt, and many owe hundreds of thousands of dollars. A recent paper by mathematicians from Dublin City University and the University of Colorado, Boulder supplies an optimal repayment strategy. The strategy, covered by Elizabeth Hernandez for *The Denver Post*, is designed to minimize the total cost to the borrower. To do this, the researchers had to balance the rapidly rising compound interest against the possibility of eventual loan forgiveness. Their work has profound implications: Borrowers with the highest debt could save tens of thousands of dollars using the proposed strategy. And as Colorado student loan ombudsperson Kelsey Lesco told Hernandez, “People aren’t just in debt. They’re delaying marriage. They’re not able to have kids. They’re not able to pass a credit check to get a job. It’s a huge problem.”

**Classroom Activities: ***exponential growth, compound interest, finances*

- Explore exponential growth, discussing how it relates to the problem of compound interest. If you’d like to study compound interest specifically, try these word problems.
- Have students look up tuition and financial aid information at various institutions: community colleges, public four-year universities, and private liberal arts colleges. Were students surprised by their findings? What kind of college do they think they might want to attend?
- Engage in some financial planning with this free financial literacy lesson. Include the cost of college based on students’ answers to the previous activity.

*—Leila Sloman*

*Smithsonian Magazine*, June 11, 2021

Math helps us predict the future. When COVID-19 began spreading uncontrollably around the world last March, US public health experts depended on complex mathematical models to create policies to stifle disease transmission. A “model,” as *Smithsonian* writer Elizabeth Landau explains in her article, is a predictive tool that combines measurable data with *assumptions *of how those data relate to each other. Remember the campaigns pleading people to help “flatten the curve”? That *curve* was the steep anticipated rise in COVID deaths calculated from factors like active cases, hospital capacities, and evidence-based assumptions of what worsens the spread of disease. “Models are like ‘guardrails’ to give some sense of what the future may hold,” one expert told Landau. This story follows the research journeys of disease modelers throughout the pandemic. It discusses how experts refined their models and why abundant data helped policymakers adapt on the fly.

**Classroom Activities: ***m**odeling from data*

- Give students hypothetical $x, y$ data, and ask them to plot the data and arrive at a conclusion. For example, suppose the $x$ values are the set of integers between 0 and 10, representing the distance in feet between an unmasked infected person and an (imaginary) virus detector; and $y$ is a hypothetical “safety score” (where 0 is the least safe and 100 is the safest), calculated from the number of viral particles detected: $[1, 3, 4, 9, 17, 25, 30, 51, 68, 78, 98]$. In this case, students could notice that plotting the data will reveal approximately quadratic growth, $y=x^2$. Discuss what this means for disease risk. They may also notice that data don’t fit mathematical functions perfectly—some $y$ values are perfect squares, while others fall below or above $x^2$.
- Discuss the concept of
*weighted*models by listing what factors are important for disease spread (or any other problem), and assigning them weights—coefficients that denote relative importance. - Use data from The COVID Tracking Project to practice fitting data on spreadsheets with the trendline functions on Excel or Google Sheets. (Many online resources exist to guide them in this, including this one from Saint Louis University.)

*—Max Levy*

*The Atlantic*, June 1, 2021

In his new book *Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else*, mathematician Jordan Ellenberg explores the many surprising uses of geometry. Derek Thompson interviewed Ellenberg about *Shape* for *The Atlantic *this month. The article touches on issues from pizza to COVID-19 predictions, all of which have a surprising geometric side to them.

**Classroom Activities: ***geometry*

- In the Q&A, Thompson and Ellenberg discuss geometric metaphors in nonmathematical thinking, particularly the use of phrases like “on the one hand, … on the other hand” to evoke an image of an argument’s structure.
- Discuss other examples of geometric thinking entering a nonmathematical realm.
- Have students read a verbal argument such as a persuasive essay and look for appeals to visual or geometric thinking.

- Practice geometric thinking with these puzzles created by Catriona Shearer.

*—Leila Sloman*

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*Politifact, *May 28, 2021

Measuring the efficacy of a COVID-19 vaccine involves several different—and potentially confusing—statistics. When we say that the Pfizer vaccine is 95% effective, we’re talking about relative risk reduction. A recent Instagram post with thousands of likes conflated the concepts of relative risk reduction and absolute risk reduction, using numbers from a *Lancet Microbe *commentary to make the false claim that the vaccine is not effective. This *Politifact *article explains the difference between the two statistics, why they’re both important for policymaking, and why the Instagram post is misleading.

**Classroom Activity: ***relative risk vs absolute risk *

- Using the equations on page 4 of the appendix of the
*Lancet Microbe*commentary, have students calculate the relative risk reduction and absolute risk reduction for different values of $a$, $b$, $c$, $d$, $n_1$, and $n_2$. In which situations does focusing on just one measure give a misleading impression of the impact of a treatment or vaccine? - Have students look up the absolute risk and relative risk associated with other diseases and treatments, discussing how they are similar or different to the case of COVID-19 vaccines.

*—Scott Hershberger*

*Quanta Magazine*, May 6, 2021

For *Quanta Magazine*‘s series “Quantized Academy”, Patrick Honner introduces the grazing goat problem: If a goat is tied to the side of a barn by a length of rope, how much area can it graze? The answer depends on the shape of the barn, the length of rope, and whether the goat is inside or outside the barn. A more complex problem is to start with the size of the grazing area and deduce what length the rope is. Honner gives examples of goat grazing problems and shows how to solve them when possible. He also discusses situations in which the problem is not exactly solvable due to the appearance of transcendental equations.

**Classroom Activity: ***circles, quadratic equations, trigonometric functions *

- Complete the exercises at the end of the article.
- Consider solving for the length of rope $r$ which gives the goat access to half the area inside a square or circular barn. As discussed in the article, $r$ is very difficult or impossible to compute exactly. Have students estimate $r$
- In the case of the goat tied outside a square barn, the shape of the grazing area is a semicircle attached to two quarter-circles and we can compute the rope length $r$

*—Leila Sloman*

*The New York Times, *May 5, 2021

As it turns out, it’s easy to shrink a president—all you need is the right camera lens. President Joe Biden and First Lady Jill Biden recently met with their predecessors from 40 years prior, Jimmy and Rosalynn Carter. A White House photographer snapped a photo of the four in a small room, and something immediately looked odd. The Carters looked miniscule compared to the Bidens, who knelt beside them on the edge of the photo. “It was as if the hosts had been turned into Hobbits,” writes Heather Murphy in *The New York Times.* Photographers interviewed by Murphy explain that the lens causes this distorted perspective. “Wide-angle” lenses capture a complete view of small spaces, but at a cost: Objects close to the camera or near the edge of a frame get exaggerated. But did the Bidens get bigger or did the Carters shrink? It’s just a matter of *perspective*.

**Classroom Activity**: *focal length, projections*

- Use smartphone apps (or online images) of fisheye lenses, which are an example of an ultrawide-angle lens, to play with this type of distortion. Notice how the distortion changes with position in the frame and with distance from the camera.
- Revisit the math of focal length in ellipses (or introduce the math of focal length for lenses with this site and interactive calculators). Supplement this with an experiment with reflective spoons. Explore where the image on the concave side of the spoon inverts—the image should only appear upright once the object passes
*inside*its focal point. - Discuss other distortions and projections, such as the Mercator Projection of the globe into a 2D map, which makes landmasses near the edge of a map such as Greenland appear much larger than they truly are.

*—Max Levy*

*Scientific American, *May 2021

In the world of mathematical research, chalkboards provide a unique space for scratch work and collaboration. Jessica Wynne documents this in her recent book, *Do Not Erase: Mathematicians and Their Chalkboards*. The book includes photographs of mathematicians’ chalkboards along with essays by the mathematicians. The work shown in the photos spans many fields of math, from topology to statistics. For *Scientific American*, Clara Moskowitz gives a brief description of a few of the topics that show up, including branching waves, vertex models, symplectic dynamics, and more.

**Classroom Activity: ***math art, visualization, collaboration*

- Have students choose one of the topics described in the article to research and give a short presentation on. Can they connect the photo of the chalkboard shown to what they learned in their research?
- Introduce more collaborative exercises in class. Have students work at a chalkboard or whiteboard together rather than using paper or computers. Discuss how this affects the inquiry process. Was it easier to focus and collaborate at a chalkboard?

*—Leila Sloman*

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*Quanta Magazine*, April 12, 2021

When Trachette Jackson was a student, mathematics and biology remained largely separate disciplines. Now, she is a pioneer in mathematical biology, with two decades of experience in using math to improve cancer treatments. In this podcast interview with host Steven Strogatz, Jackson discusses how she entered the field and how she creates mathematical models of tumors and treatments. A key drawback of chemotherapy, she explains, is that it kills healthy cells as well as cancer cells. With differential equations and computer simulations, Jackson is working to develop multi-drug therapies that more effectively target cancer cells while leaving healthy cells unaffected.

**Classroom Activity: ***logistic functions*

Use data from the American Cancer Society to introduce students to logistic functions. Have them use graphing calculators to determine the parameters for the logistic model, then estimate a woman’s risk of developing breast cancer at various ages. Conclude with a discussion of the model’s limitations and ways to improve it.

*—Scott Hershberger*

*FiveThirtyEight*, April 8, 2021

Conventional wisdom in the soccer world holds that players shouldn’t try to score when they’re far away from the goal. But does that advice hold up to scrutiny? Researchers at MIT studied this question using a Markov decision process. In this type of model, an agent in an environment (e.g., a soccer player on the field) makes decisions to maximize a reward (number of goals scored). The reward and the actions available are determined by the environment’s current state—in this case, the current location of the player. In this article, John Muller describes the conclusions of the paper and the implications for the game of soccer.

**Classroom Activity: ***Markov decision processes, probability*

- Use Van Roy et. al.’s interactive tool to explore how changing shooting behavior in a soccer game affects the number of goals.
- Discuss in class what the new paper means for soccer players. Should they change their shooting techniques? Why or why not?
- One example of a Markov process is the number of heads that have come up during a series of coin flips. Practice analyzing this process asking students to make bets (with candy, gambling chips, etc.) on the outcomes of a series of coin flips. How much are they willing to bet on a rare event, like ten heads in a row? How big does the payoff have to be? Analyze mathematically which bets are “worth it,” and discuss why. Discuss what this game has in common with the model in Van Roy et. al.’s paper.

*—Leila Sloman*

*Popular Mechanics, *April 2, 2021

A crumpled piece of paper may seem unremarkable. But when Harvard mathematicians looked closer, they found that the creases follow an elegant mathematical pattern. The research, which required hand-tracing thousands of creases, draws on the same mathematics that describes how rocks break down into smaller pieces. It provides a potential physical explanation for a surprising result: The total length of the creases increases logarithmically with the number of times the piece of paper has been crumpled. As writer Courtney Linder notes, understanding the “dynamics of squished paper” will also help researchers understand the folding of the Earth’s crust as well as engineer thin devices.

**Classroom Activity: ***logarithms*

Use this article as a fun example of where logarithms show up in everyday life. Discuss what it means that the total crease length increases logarithmically rather than linearly, quadratically, or exponentially with the number of crumples. Compare this with the total crease length that results from repeatedly folding a piece of paper in half or performing other simple folding patterns.

*—Scott Hershberger*

*Scientific American, *April 1, 2021

How well-connected does a community have to be before information flows easily among its members? Before an infectious disease spreads out of control? It turns out that if edges in an infinite network are randomly distributed, there is a precise level of connectedness that implies information or infectious disease will spread infinitely far. But real networks of human contacts, or networks that change over time, are far more complicated. In an article for *Scientific American,* Kelsey Houston-Edwards describes several real-world examples of these networks and the importance of understanding them.

**Classroom activity:** *network theory*

- Have students play with an interactive percolation simulator.
- Play an analog “percolation game”. Collaborate on drawing a network from real-world data. Discuss the network structure and whether or not it lends itself to percolation. Try asking questions like:
- How large are the clusters in this graph?
- Could we make the clusters disappear by adding or deleting a few nodes?
- Are the edges evenly distributed?
- How many edges are there, compared to the number of nodes?

*—Leila Sloman*

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On April 6, 2021 the Museum of Nature & Science in Las Cruces, NM presented a Zoom lecture by Rima Ajlouni (School of Architecture, University of Utah) with the title “Quasi-crystalline Geometry in Islamic Art and Architecture.” (The lecture is now available on Facebook).

This presentation is an attempt to reverse-engineer the remarkable non-periodic tilings found in medieval islamic monuments: how they were constructed with the the tools available at the time. In the process Prof. Ajlouni eleganty exhibits the connection between Penrose tilings and quasi-crystals, starting from the original April 8, 1982 electron diffraction scan that ultimately won Daniel Shechtman his Nobel prize:

The process starts by observing in Shechtman’s image a series of nested decagons, each of which (joining every third vertex) gives a 10-pointed star containing the next smaller one. In a series of five, the innermost partitions naturally into the union of ten Penrose tiles, here colored red and yellow. This union, together with the smaller group of five red tiles, forms the *seeds* of the tiling.

In the next step, one or the other of the seeds is planted at a symmetrical subset (the *hidden grid*) of the vertices of the outer two stars.

The remaining open space is filled with symmetrically placed copies of the two *connecting formations*

yielding a Penrose tiling with 5-fold symmetry:

Prof. Ajlouni shows how this tiling, together with a star-shaped subset, can be used as seeds, with the same hidden grid and two analogously chosen filling configurations, to give a “larger” tiling (assuming the tile size to be constant) and that this hierarchical process can be continued to make arbitrarily large Penrose tilings.

“How the geometry of cities determines urban scaling laws,” by Carlos Molinero and Stefan Thurner, was published on March 18, 2021 in the *Journal of the Royal Society Interface*. They begin: “One of the surprising findings in urban science is that many of the hundreds of quantities and variables that characterize the dynamics, functioning, and performance of a city exhibit power law relations. These are called *scaling laws,* meaning that a quantity $X$ depends on a variable $p$ (such as population) in a power-law fashion. In particular, this means that $X$ is related to the population of the city as

$$ X \propto p^{\gamma}$$

where $\gamma$ is the scaling exponent and $p$ represents population size.

“Scaling laws can often be explained directly from the geometry of the underlying structures of a system. Classic examples include Galileo’s understanding of the relation between the shape of animals and their body mass … or the scaling laws of river basins given by their fractal geometry. In the same spirit, we provide a simple and a direct geometrical explanation of urban scaling exponents, derived from the fractal geometry of cities.”

- This study involves the concept of
*fractal dimension*; one definition is the box dimension. Here is how the authors introduce it:In this example from the article, the street grid of a city has fractional dimension $d_i$ if its length $\ell$ is proportional to $L^{d_i}$, where $L$ is the linear size of the city (so, for example, the city area varies as $L^2$).

“Cities across countries, latitudes and cultures are different—and so is their geometry. How should cities that are significantly different in their geometry lead to similar scaling exponents? To answer this question we focus on the ratio of two geometric aspects of a city, the fractal dimension [$d_i$] of its infrastructure (street networks), and the fractal dimension [$d_p$] of the population … . The fractal of the population can be imagined as the cloud of people that is obtained by identifying the position of every person in three dimensions.” One of the results reported is that even though those two fractal dimensions depend on the size of the population, their ratio stabilizes as population grows.

In a related result, the authors explain earlier measurements from other sources: “scaling laws with respect to population size were found for GDP, the number of patents, walking speed or crime rates. The associated scaling exponent for these relations appears to be in a range of $\gamma \sim 1.1 – 1.2$ … ,” by arguing that for these parameters, $\gamma$ should be equal to $2-\gamma_{\rm sub}$.

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