*Scientific American*, April 27, 2023.

Groups of tiny aquatic worms known as California blackworms can cluster together to form big, entangled blobs. The worm balls are knotty and complex, and it takes several minutes for the worms to build the shape, yet if danger appears they can untangle in just milliseconds. In an article for *Scientific American*, Jack Tamisiea reports on a new paper in *Science* that analyzes the worms’ method for tangling and rapid untangling. The paper’s authors filmed the worms and used ultrasound imaging to see inside the knots. They then used topology and other mathematical methods to model the worm balls and the creatures’ skillful separation.

**Classroom Activities:** *topology, knot theory*

- (All levels) Watch the video in this article from Georgia Tech about the worm knots, and notice the “helical wave pattern” shown at time 0:38. One of the authors of the California blackworm study told
*Scientific American*:*“If it spends longer winding in one direction before switching to winding in the other direction, you get the tangling behavior. … If the worm rapidly switches between winding clockwise and anticlockwise, you get untangling behavior.”*- Discuss: Based on your own experiences with tangled materials, does this seem surprising to you? Why or why not?

- (All levels) Give each student two lengths of rope, string, yarn, or other knottable item, and a secret knot randomly assigned from this list. Ask them to tie the ropes into their assigned knot, then trade knots with a partner. Now, each partner will try to untangle the knot they’ve been given.
- Was it easier to make the tangle or to untangle it? What factors do you think might be involved in the difficulty of tangling and untangling? Does this affect the way you think about the worms tangling and untangling?

- (High level) Read the first three sections of the introduction to knot theory from the Oglethorpe University Knot Theory website. Then try the trefoil knot activity. You may want to refer back to Section 3 of the introduction to help you answer questions 1-3.

*—Tamar Lichter Blanks*

*The New York Times*, April 22, 2023.

Every winter, high school students wait anxiously to hear whether their college applications will bring good news or bad news. They try to glean some sense of how hard it is to get into a certain university by using metrics like the school’s acceptance rate, which quantifies what fraction of the total applications get accepted. This year, some colleges have reported record-low acceptance rates. Does this mean that getting into college is harder than ever? Not necessarily, writes Jessica Grose in an Op-Ed for the *New York Times*. This article uses college admissions statistics to describe how data literacy is important in everyday life.

**Classroom Activities:** *statistical literacy, data analysis *

- (All levels) Answer the following based on the article:
- List the reasons why the admissions data may be misleading.
- List the reasons why it may be harder than ever to get into college.
- Then, for each reason listed, come up with at least one source of data (e.g. “acceptance rates from the top 100 public universities”) that would confirm or quantify it. Which data sources are the most trustworthy? Form a hypothesis about whether college admissions have gotten harder.

- (Mid level) In small groups, consider the importance of most or all of the reasons listed.
- How would you weigh the importance of one reason compared to another?
- Which reasons are the most
*objective*? - Which are the most
*subjective*? - Describe what forms of evidence can support subjective versus objective reasons.
- (Hard) Now that you’ve studied all of your data, think about how you would prove or disprove your hypothesis. Discuss what challenges you would encounter when testing this hypothesis.

- (Mid level) The article notes that the fraction of qualified applicants has gone down since the pandemic. Fill in each blank on the table below, and write a one-sentence analysis of each case based on its hypothetical data. If the college accepts 10,000 qualified students each year, what are the chances of a qualified applicant in each case?

Total Applicants (2022) |
Qualified Applicants (2022) |
% Qualified |
Total Applicants (2023) |
Qualified Applicants (2023) |
% Qualified |

60,000 | 21,450 | _________ | 70,000 | 23,000 | _________ |

60,000 | 25,632 | _________ | 60,500 | 25,633 | _________ |

60,000 | 14,555 | _________ | 65,000 | 10,555 | _________ |

*—Max Levy*

*New Scientist*, May 10, 2023.

Mathematicians and physicists often rely on symmetry to simplify their theories and calculations. Symmetry might explain mathematicians’ proclivity for exploring simple shapes like spheres, or it might be much more fundamental — and abstract — than that. The Standard Model, the starting point for the behavior of all known particles in the universe, is based on symmetries, writes Michael Brooks in this article. Symmetry may or may not remain a backbone of future theories. But mathematician Marcus du Sautoy’s words reveal that symmetry contains wonders in its own right: “The understanding of symmetry I have now is so much deeper and stranger, and it gives me access to symmetries that are so much more exotic than anything you can see with your eyes,” he told *New Scientist*.

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

- (All levels) Read this Math Salamanders lesson about regular and irregular shapes. Look at the “Regular and Irregular Shape Sheet”. What ways can you move the shapes so that they look the same after you moved them? That might be rotating them by 90 degrees, or flipping them upside down. These are the symmetries of the shapes.
- What happens if you combine two of the symmetries of one shape — for instance, rotate it, and then flip it upside down. Is that a symmetry?
- Can you draw an irregular polygon that has symmetry? Can you draw one that has rotational symmetry (i.e. you rotate it by a given amount, and it still looks the same)?

- (High level) The symmetries of a shape, or other object, form an important mathematical structure called a
*group*. Read pages 96-106 of these lecture notes by Emanuel Lazar for an introduction to groups and symmetry. What insight does the previous exercise give you into Example 18? - (High level) Symmetry does not only pertain to shapes. Sometimes, objects in a set are interchangeable (this happens in quantum mechanics). The symmetries then form a
*permutation group*: The set of all possible ways to rearrange elements in the set. For instance, if you have three identical items in a row, you might swap items 1 and 3.- Figure out what permutations are possible on the set {1, 2, 3}. Now suppose these numbers represent the 3 sides of an equilateral triangle. What permutations correspond to symmetries? Does the symmetry group of the triangle capture all the permutations of the sides?

*—Leila Sloman*

*UC San Diego Today*, May 1, 2023.

What does math sound like? In your classes, it might sound like the clacking of a keyboard, the scribbling of a pen, or the squeak of an eraser. But to some people, math can sound like music. That’s because music *is* math. The spacing between frets on a guitar. The relationships between musical keys. And the feeling is mutual: “I find (my) mathematics very musical. There is harmony (and sometimes dissonance) in the way different ideas and structures come together in a mathematical proof,” said Todd Kemp, Professor of Mathematics at the University of California, San Diego, in an interview with *UC San Diego Today.* In this article, Michelle Franklin interviews three mathematicians about how they see math’s relationship to music.

*Note: The article attributes the quote “There is geometry in the humming of the strings, there is music in the spacing of the spheres,” to Pythagoras, but not enough evidence exists to prove that conclusively.*

**Classroom Activities:** *music, representation system*

- (All levels) Research famous mathematicians from history who were also musicians, and answer the following questions:
- Which person’s accomplishments do you find most impressive, and why?
- Which person’s accomplishments were most surprising, and why?

- (All levels) Watch this video on math and music.
- The video describes a magic number: the ratio between sequential steps within an octave. That ratio is approximately 1.0595. What would it be if there were 15 steps between octaves? 10? (More info here.)
- The function $y=\sin(x)$ is a wave. Find two functions that are harmonics of $y = \sin(x)$.

- (Mid level) One mathematician told the magazine “Both fields have their own representation system. Math has symbols and notation, and music has staffs and clefs.” Answer the following:
- What can music communicate that math cannot?
- What can math communicate that music cannot?
- What can both representation systems communicate?

*—Max Levy*

*Mint*, May 18, 2023.

Lots of mathematical concepts are named “Bernoulli”: The sequence of rational Bernoulli numbers; Bernoulli distributions, which describes a random outcome that can be equal to either 0 or 1. I was vaguely aware that Bernoulli pops up in physics, too. But it never occurred to me that these Bernoullis might all be different people. Writer Dilip D’Souza had the same misconception, but in this article for *Mint*, he reveals that there were many Bernoullis — all related to one another, with complicated familial relationships and wide-ranging scientific interests. Several of the Bernoullis had a mathematical bent, especially toward probability theory. In these activities, we’ll explore some of the probabilistic concepts the Bernoullis discovered.

**Classroom Activities:** *probability, Bernoulli distribution, Law of Large Numbers*

- (Probability) Watch this video explaining what a Bernoulli distribution is.
- (Mid level) Calculate the following Bernoulli distributions:
- The probability that a student randomly selected from your class has brown hair.
- The probability that a randomly selected United Nations member state begins with the letter A.
- The probability that two coin tosses in a row will come up heads.

- (Mid level) What are the mean and variance of a Bernoulli distribution with probability of success
*p*? (Answers here.) - Jacob Bernoulli proved the Law of Large Numbers. In his online probability textbook, Hossein Pishro-Nik describes this law as follows:
*“If you repeat an experiment independently a large number of times and average the result, what you obtain should be close to the expected value.”*- (All levels) As a class, brainstorm how students would test this law in class.
- (High level) Prove this law holds for an experiment that follows a Bernoulli distribution.
- Try this lesson from
*Statistics Teacher*, which uses Jimmy Fallon’s “Egg Roulette” game to explore the Law of Large Numbers.

*—Leila Sloman*

- What if nobody is bad at maths?

*The Guardian*, May 29, 2023. - Can you solve it? Rotation, rotation, rotation

*The Guardian*, May 29, 2023. - Ian Hacking, Eminent Philosopher of Science and Much Else, Dies at 87

*The New York Times*, May 28, 2023. - Tom Osler, celebrated Rowan math professor, national champion distance runner, author, and mentor, has died at 82

*The Philadelphia Inquirer*, May 26, 2023. - These Are the Most Bizarre Numbers in the Universe

*Scientific American*, May 23, 2023. - MIT students give legendary linear algebra professor standing ovation in last lecture

*USA Today*, May 23, 2023. - Math Patterns That Go On Forever but Never Repeat

*Quanta Magazine*, May 23, 2023. - The potency of pattern: Mind the gap

*The Irish Times*, May 18, 2023. - New Proof Finds the ‘Ultimate Instability’ in a Solar System Model

*Quanta Magazine*, May 16, 2023. - 10 Fantasy Books Where Math is Magic

*Book Riot*, May 10, 2023. - Unevenly packed coffee to blame for weak espresso, say mathematicians

*New Scientist*, May 9, 2023. - How Pythagoras turned math into a tool for understanding reality

*Science News*, May 9, 2023. - George Boole: Lincoln’s self-taught mathematical genius who changed the world

*The Lincolnite*, May 7, 2023. - Why the ‘Sleeping Beauty Problem’ Is Keeping Mathematicians Awake

*Scientific American*, May 4, 2023. - Mathematicians Discovered Something Mind-Blowing About the Number 15

*Popular Mechanics*, May 4, 2023. - The Most Important Machine That Was Never Built

*Quanta Magazine*, May 3, 2023.

A *Wall Street Journal* editorial “San Francisco Can’t Do the Math” (April 28, 2023) with subtitle “A battle over equity, achievement—and eighth-grade algebra” reports on a public petition, proposed by the *SF Guardians* (a group made up of “parents, teachers and concerned citizens”) asking the San Francisco Unified School District board to restore Algebra I to the eight-grade curriculum. This local piece of news earns mention on the editorial page of a national newspaper because Algebra I in 8th grade has become a symbol of a problem that transcends San Francisco and California: how to balance the need for well-prepared and well-supported students to receive appropriately challenging mathematics instruction with the need to close achievement gaps. In today’s United States this question has become yet another bitterly divisive issue. As the title suggests, the *WSJ* editorial board supports the petition.

How did this particular course become so important? Algebra I, Geometry, Algebra II, Pre-Calculus, Calculus is a 5-year sequence, so moving Algebra I to ninth grade jeopardizes the current ideal of high-school mathematics education, which culminates in a calculus course. (This is a fairly recent development. A kind of achievement inflation has made it seem necessary for a good student to take calculus in high school. In fact, building a solid foundation in algebra, geometry and trigonometry might well be a much better use of that student’s time.)

The *WSJ *‘s mention of “equity” alludes to the subtext that has made this question especially thorny. The SFUSD’s decision was in part motivated by statistics like the one graphed here which showed a large general failure rate for 8th-graders in Algebra I, and an even worse picture for Black and Latinx students. On the other hand, any change that diminished access to calculus would especially impact the 30 percent of SFUSD students identified as Asian, since nationwide 46 percent of that group take calculus in high school.

What actually happened? The SFUSD’s decision to move Algebra I from middle school to high school was made in 2014. One March 2023 preprint dives into the records of 23,309 district students and tabulates the consequences. As the authors (Elizabeth Huffaker, Sarah Novicoff, and Thomas S. Dee, all of Stanford University) explain, math classes the year after the policy was implemented were significantly more diverse, with almost all ninth graders now enrolled in Algebra I and the vast majority of tenth graders in Geometry. Enrollment in AP Calculus initially fell from 30% to 24%, with losses mostly coming from the group of Asian/Pacific-Islander students. These reductions, they report, eased off as students learned to take advantage of “acceleration options.” And the policy did not grant Black and Hispanic students more access to more AP classes, leaving large racial gaps in place.

Siobhan Roberts has an article in the *Times* for May 21, 2023, about the 50th anniversary of OEIS, the On-Line Encyclopedia of Integer Sequences. The OEIS was the brainchild of Neil J. A. Sloane (for many years at Bell Labs); he described its genesis in My Favorite Integer Sequences (2002). Here’s how it can be useful. I was once counting the number of mazes of a certain type that had $n$ path segments stacked atop one another ($n$ levels). With one, two or three levels there is only one; there are 2 with four levels, 3 with five levels, 8 with six: I was generating an *integer sequence.* It starts $(1, 1, 1, 2, 3, 8)$ and continues $(14, 42, 81, 262, \dots )$. The numbers grew very rapidly, as did the length of my counting process: the calculation for $n$ levels took on the order of $n^2$ as many steps as the calculation for $n-1$. I soon got stuck. But an article in the *New York Times* about Neil Sloane and his Encyclopedia caught my eye. Part of Sloane’s collection, reportedly, was the sequence of ways of folding a strip of $n$ stamps. For my maze-obsessed mind, counting folded strips of stamps was clearly related to counting my type of mazes. But the stamp-folder (John E. Koehler) had found a shortcut that made the number of steps grow at a much lower rate (although still exponentially). This insight (details here) allowed me to continue my calculations up to $n=22$, where there are $73,424,650$ mazes. The maze sequence became a new entry in the encyclopedia (it is now sequence A005316 and has been extended up to level 44, a 19-digit number).

Roberts gives some examples of sequences in the collection. The odd numbers $1, 3, 5, 7, \dots$, the evens $2, 4, 6, 8, \dots$, the prime numbers $2, 3, 5, 7, 11, \dots$ (no divisors except themselves and 1) and the Fibonacci sequence $1, 1, 2, 3, 5, 8, 13, \dots$ (each number is the sum of the two preceding ones). These sequences have purely mathematical definitions. The OEIS has other sequences, like the “eban” numbers $2, 4, 6, 30, 32, 34, 36, 40, \dots$ (numbers whose English names do not use the letter “e”), where the defining criterion may depend on the language we’re speaking or on the shape of the numerals we choose to represent numbers; these sequences are as much riddles or jokes as mathematics.

One sequence mentioned in the article looks like it might be of the latter type. It’s John Conway’s “Look and Say” sequence, which starts $1, 11, 21, 1211, 111221, 312211, \dots$. Here for example $312211$ is a verbal description of the preceding number $111221$, which reads as “three 1s, two 2s, one 1.” But this is not a joke: “Look and Say” turns out to have a mathematical life of its own (for example, no numeral bigger than $3$ ever appears). For more details, I recommend the YouTube video where the master himself talks about the sequence.

Coccolithophores are tiny algae that play a large role in regulating the concentration of carbon dioxide (CO$_2$) in the oceans. *Emiliana huxleyi*, pictured above, is one of the most important members of the family. As plants, they participate in *photosynthesis*, using energy from sunlight to synthesize organic matter from water and CO$_2$. This has the result of taking dissolved CO$_2$ out of the ocean. On the other hand, the production of their calcium carbonate shells has the side effect of *releasing* CO$_2$ into the ocean. The notation used for the relative strength of these two effects is PIC:POC; it represents the ratio of the *particulate inorganic carbon* (present in the shell) and the *particulate organic carbon* (absorbed during photosynthesis) produced by these organisms. If PIC:POC $>1$, they are releasing more CO$_2$ than they absorb, and coccolithophore growth is a net CO$_2$ source; if PIC:POC $<1$, it is a net CO$_2$ sink.

“Estimating Coccolithophore PIC:POC Based on Coccosphere and Coccolith Geometry” was published in *JGR Biogeosciences*, April 18, 2023. The authors, Xiaobo Jin and Chuanlian Liu (Tongji University, Shanghai) collected samples of *E. huxleyi* and another coccolithophore at locations in the South China Sea. It had been suggested, based on work with laboratory cultures of these algae, that the PIC:POC ratio was correlated with a geometric property of the individual coccoliths. Namely, the ratio of its average thickness to its length, called the *lateral coccolith aspect ratio* $A_L$. To verify this suggestion for a large sample of wild-caught specimens, the authors developed a method for measuring both PIC:POC and $A_L$ from photographs taken with a conventional (light) microscope.

Jin and Liu exploit the light microscope’s focusing sensitivity: only the material in or very close to the plane of focus gets recorded. Since calcium carbonate is translucent, focus on the equatorial plane of a coccolithosphere (below, left) gives an image allowing the measurement of the inside and outside diameters of the shell. The inside diameter of the shell is the diameter of the included organic cell. From this number one can calculate the volume of the cell, and estimate the POC. The difference between the outside and inside radii is the thickness of the carbonate shell, so the volume of the inorganic part can be calculated, leading to an estimate of the PIC. On the other hand, focus on the top or the bottom (below, right) will give useful images of individual coccoliths.

For the measurement of $A_L$, the ratio of a coccolith’s average thickness to its length, the authors used another property of light. The amount of light that gets through calcium carbonate depends on its thickness. Calibrating a grey scale with a precisely micro-machined calcium-carbonate wedge allowed them to assess the thickness of a coccolith at each pixel in the image. From there they could calculate the average thickness. Since the length of the coccolith can also be measured from the image, this procedure yielded $A_L$.

The correlation between PIC:POC and coccolith $A_L$ can be useful in the study of the long-term history of our climate, since in the fossil record intact coccolithospheres are rare (they do turn up, even in classroom chalk), whereas individual coccoliths are abundant. Margaret Xenopoulos surveyed this research in Eos, the American Geophysical Union’s news magazine, with the subtitle: “Math can be fun when reconstructing the ocean’s past and forecasting the future with algal geometry.”

]]>- AI and mathematics
- Mouse bone-building cells prefer negative curvature
- Persistent homology and the early universe

Davide Castelvecchi, a staff writer for *Nature*, has covered developments in the interaction between artificial intelligence and mathematics over the last few years. As he explains it in his most recent posting (“How will AI change mathematics? Rise of chatbots highlights discussion”, in the issue dated March 2, 2023), this interaction has two very different modalities. Traditionally, humans relied on computers’ speed, accuracy and large memory for tedious computational tasks. An early example was the computer-assisted proof of the four-color theorem in 1976. In this line came *proof assistants* like Coq, first released in 1989. In fact, one of Coq’s first big jobs was checking that the four-color theorem proof actually worked. Proof assistants were also in the news more recently, in connection with Peter Scholze’s “Grand unification theory,” as reported by Castelvecchi in June 2021.

The other way AI has become involved in mathematics is through *machine learning.* Machine learning enabled computers to beat humans at chess and Go. Here, instead of being a super-secretary, the computer gets to learn on its own, through trial and error. This newer manifestation of artificial intelligence has recently become a topic of great public interest because of its packaging as “chatbots,” programs that have learned to imitate human speech patterns (specifically by training on sentence completion) to the point that they give the appearance of having learned to think. This has also been applied to mathematics. As Castelvecchi tells us, a group at Google has come up with a chatbot called Minerva, which has trained on a special corpus: mathematics arXiv postings. We learn that Minerva can accurately factor integers into primes in a certain range, but that once the numbers are too large it makes mistakes, “showing that it has not `understood’ the general procedure.” This is always going to be the problem with a program based on language.

Will computers ever be able to “do” math? Castelvecchi spoke with Melanie Mitchell (Santa Fe Institute), who says that our jobs are safe until AI can learn “to extract abstract concepts from concrete information.” That is, until computer programs can understand mathematical facts, and not just process mathematical words.

A *Nature Communications* article published March 3, 2023 considers what happens when you grow cells on surfaces with varying *principal curvatures*.

**What are principal curvatures?** Consider a point P on a smooth surface in 3-dimensional space. Draw a vector ${\bf n}$ pointing outward, perpendicular to the surface at $P$. Now cut through the surface with a plane containing ${\bf n}$. The plane and the surface intersect along a curve; see some examples below.

For each of these curves, measure the curvature at $P$ by calculating 1 over the radius of the circle that best approximates the curve at $P$. Count it as positive if it curves *away from* ${\bf n}$. Generally at $P$ there will be a maximum $\kappa_1$ and a minimum $\kappa_2$ of these numbers. These are the principal curvatures at $P$.

The authors, a team of biomechanical engineers based mostly at the Delft University of Technology and led by Sebastien J. P. Callens and Amir Zadpoor, propose “a geometric perspective on cell-environment interactions that could be harnessed in tissue engineering and regenerative medicine applications.” In their experiments they cultured cells on a collection of microfabricated surfaces designed to manifest a variety of geometrical environments.

Those environments had a variety of curvatures. They included unduloids, surfaces where the *mean curvature *$\frac{1}{2}(\kappa_1 + \kappa_2)$ is constant. (Unduloids were first published by Charles-Eugène Delaunay in 1841). When the mean curvature is zero, unduloids form a cylinder. As the mean curvature increases, they morph into a line of spheres. The researchers also used strings of pseudospheres (surfaces where the Gaussian curvature, the product of the principal curvatures $\kappa_1\kappa_2$, is constant and negative) and shapes called catenoids. Finally, they threw in a cylinder bent into a sinusoidal curve, which unlike its straight, flat cousin has regions of positive and negative Gaussian curvature.

Techacute.com picked up this research on March 29 (Gwendoline Guy, “Can Geometry Help Your Organs Regenerate?”). Guy interviewed Zadpoor, who reminds us that geometry belongs to everybody. For use in the lab, it’s “easy, safe and inexpensive.”

Cosmology has been called “the physics of the universe”, but it is not an experimental science. Cosmologists develop theories to explain the origin and evolution of the universe and use them to generate models to be compared with observation. The currently most generally accepted model, the “standard $\Lambda$ Cold Dark Matter” model, depends on a small number of parameters, like the age of the universe and the Hubble constant governing its expansion. That includes a parameter labeled $\sigma_8$, which measures the initial fluctuations of mass density in a certain volume of space (the “8” comes from the size of that volume in appropriate units). It turns out that there is general agreement on the values of the main cosmological parameters except for $\sigma_8$. This issue is addressed in “Wasserstein distance as a new tool for discriminating cosmologies through the topology of large-scale structure” published in the *Monthly Notices of the Royal Astronomical Society* April 18, 2023. The authors, the Ukrainian-Italian team of Maksym Tsizh, Vitalii Tymchyshyn and Franco Vazza, ran a series of simulations of the evolution of one small sector of the universe (from our point of view) for various values of $\sigma_8$, while keeping the other parameters fixed.

For each value of $\sigma_8$, the simulations generate a picture like one of these for each of several different ages of the universe: each point represents a concentration of dark matter. For a given age, the authors want to measure the difference between the pictures corresponding to different values of $\sigma_8$. To do this, they use the *Wasserstein distance;* this is a way of measuring how different one cloud of points is from another. It involves “persistent homology”, a recently developed mathematical tool for taking a collection of data points and identifying clusters (and other, more complex structures) among them.

How persistent homology works. To get an intuitive idea of the procedure, imagine surrounding each point with a solid ball of radius $r$ centered at that point, and tracking how the union of all those solid balls changes shape as $r$ increases. The mathematical way of characterizing a shape starts by listing its *Betti numbers*. The zero-th Betti number $\beta_0$ counts the number of disjoint pieces. The first, $\beta_1$, counts the number of distinct holes in the shape. (Similarly, $\beta_k$ counts the number of its $k$-dimensional “holes.” A 2-dimensional hole, for instance, would be the hollow interior of a sphere.) Typically, as $r$ increases, disjoint pieces will agglomerate into larger units; at the same time, holes may form (“be born”) and may “die” when their centers get filled in, with analogous phenomena in higher dimensions.

One characteristic of the persistent homology of a space is the set of its *barcodes*. There is one for each dimension; it is a display of the [birth, death] intervals for each of the holes (see for example Figure 4 in this article). Finally, the Wasserstein distance between two distributions of points is computed, very roughly speaking, in terms of the differences between the ends of corresponding intervals in the barcodes of their persistent homology.

The goal of this work is to see if topological criteria like the Wasserstein distance could help pin down the value of the cosmological constant $\sigma_8$. By testing them on models generated with different values of $\sigma_8$, the authors can determine which criteria are the most sensitive, and could be most usefully applied to actual observational data. In their conclusions, the authors report that the Wasserstein distance gives the best $\sigma_8$ discrimination when used with images of the universe at earlier epochs, and that the distance calculated using dimension-0 barcodes shows more sensitivity to changes in $\sigma_8$ than calculations with higher-dimensional ones.

On a more terrestrial note, Franco Vazza started this collaboration with Maksim Tsizh just after the start of the current war in Ukraine. He remains active in encouraging Ukrainian astrophysicists to work with him and his department at the University of Bologna.

]]>*Scientific American*, April 10, 2023.

The newest version of Unicode will allow you to type using an Inuit number system called “Kaktovik”. Kaktovik was developed in the 1990s as part of a class project, and draws on the counting practices of Iñupiaq, the language of the Alaskan Inuit. Kaktovik is “strikingly visual”, writes Amory Tillinghast-Raby for *Scientific American*. Performance on standardized math tests soared after it was implemented in middle schools in 1997. But in many schools, Kaktovik became a casualty of the No Child Left Behind Act. Its inclusion in Unicode may help bring it back to life.

**Classroom Activities: ***arithmetic, counting*

- (All levels) Study the “Kaktovik numerals” chart from the article in class. Write out the following numbers using Kaktovik: 5, 11, 25, 33, 99.
- (Mid level) Now study the chart “Solving equations with the Kaktovik System.” Using only Kaktovik numerals, calculate the following:
- $25 – 5$
- $99 \div 33$
- $25 \div 5$
- $99 + 11$

- (Mid level) Now try the same calculations, this time using the Hindu-Arabic number system. Reflect: Which calculations did they find easier? What did they notice while doing the calculations?
- (High level) Another type of number system involves using Hindu-Arabic numbers, but with a different base. The binary system is a base-2 number system. Read this online lesson and do the questions under “Your Turn”.
- Kaktovik uses a base 20 system. Convert the Hindu-Arabic numbers from the previous exercises to base 20, and repeat the calculations in base 20.

*—Leila Sloman*

*Quanta Magazine*, April 4, 2023.

Sometimes being a good mathematician is all about having a knack for finding patterns, and sometimes it’s about showing there are no patterns at all*.* That was the case in a recent breakthrough in tiling, the study of how to arrange shapes so that they completely cover a surface without any overlaps or gaps. Tiling usually creates repeating patterns, like the honeycomb pattern that hexagons make. But for decades, mathematicians and hobbyists searched for a more elusive phenomenon: one shape that can tile a plane without creating a pattern. (A German geometer dubbed the hypothetical tile an “einstein” — a pun on the German phrase “ein stein,” which means “one piece.”) Recently, puzzle enthusiast David Smith found an einstein. “We were all pretty blown away,” one computer scientist told Erica Klarreich, who tells the story of how Smith finally cracked the code with a shape that looks like a hat.

**Classroom Activities: ***tiling, periodicity*

- (All levels): Print and cut out 20 identical equilateral triangles, 20 identical pentagons, and 20 identical hexagons. Then answer the questions below.
- Using your cut-outs, try to tile a plane with each of the three shapes.
- Which shapes tile the plane?
- If a shape does not tile the plane, do you think that changing the length of its sides will help? Why or why not?

- (Mid level): Watch this Veritasium video about Penrose tiles. Print out 20 copies of kites and 20 copies of darts. Are you able to arrange the tiles in a repetitive pattern? Based on the video and your experimenting, how many different patterns and arrangements do you think are possible? (Infinity is a possible answer!)
- (High Level) Read Klarreich’s article about the hat tile. Klarreich writes that “the hat is one of infinitely many different tiles of this type.” Based on what you have read, why are there an infinite number of “hat tiles” possible?

*—Max Levy*

*Wired*, April 7, 2023.

In this article for *Wired*, Rhett Allain answers a simple question: Why can’t you point a camera at the moon, zoom in, and walk away with a photo that captures the moon’s intricate topography? The answer has to do with how light behaves as it squeezes through the small opening of a camera lens. Light waves far away from the lens are “diffracted,” or spread out. As a result, light waves coming from different points on the moon overlap with one another, making them impossible to distinguish. The upshot? You’ll never find a smartphone camera good enough to get a clear, detailed photo of the moon — at least, not unless you make use of some fancy image correction software. “It’s not a limit on the build quality of the optical device; it’s a limit imposed by physics,” says Allain.

**Classroom Activities: ***optics, trigonometry*

- (Mid level) Allain writes that whether you can distinguish between two points on the moon depends on their
*angular separation*. That is, draw a line from your eye to Point A, and another from your eye to Point B. The angle between these lines determines whether Points A and B are distinguishable. Try the following exercises.- Imagine you are looking directly at two objects, A and B, which are 10 meters away from you. A and B are about 10 centimeters apart from each other. According to the formula found in the article, what is the angular separation between A and B?
- What if A and B are one meter apart? Five meters?

- (High level) In his formula $\theta = h/r$, Allain is taking advantage of a common approximation: $\sin(\theta) \approx \theta$. Calculate the angular separations from the previous exercise precisely using the inverse sine function. How do the answers compare?
- (High level) Allain’s smartphone camera has a lens about 0.5 cm in diameter. Suppose Allain wanted to take a photo of two spots of light that were 1 meter apart, and the wavelength of the light was 500 nm. How far away could those objects be before they would be impossible to distinguish?

*—Leila Sloman*

*Wired*, April 3, 2023.

If you like to buy lottery tickets, mathematical thinking can help you increase your potential winnings. In a video from *Wired*, mathematician Skip Garibaldi explains lottery strategies that can theoretically help you win more money. You can try choosing unpopular numbers to increase the odds that you’ll be the only person who wins the jackpot, or carefully select state scratch-off games with a lot of unclaimed prizes. There have even been rare cases of lotteries where people were able to consistently profit, but as Garibaldi explains, it’s usually a losing bet: “It’s really hard and unusual to be in a situation where you could reasonably expect to make money on the lottery.”

**Classroom Activities:** *probability, combinatorics*

- (Probability; All levels) Imagine a small lottery with just one prize, worth \$70. There are 100 tickets and each ticket costs \$1. At the drawing, one of the 100 tickets will be selected randomly to win the prize.
- If you buy one ticket, what is the probability that you will win the prize?
- If you buy all 100 tickets, what is the probability that you will win the prize? In this situation, how much money would you spend, and how much would you receive?
- The
*expected value*of a lottery ticket is the amount you would expect to win, on average, if you played the same lottery many times in a row. (See this summary of the concept of expected value.) To compute the expected value, take the amount of money you would receive if your ticket won the prize, and multiply that by the probability that that ticket will win. What is the expected value of a ticket in this lottery?

- (Probability; High level) In the video, Garibaldi talks about a type of lottery that you win by correctly guessing a 4-digit number. He discusses a particular kind of entry in this lottery called a “six-way box.”
- The winning “ticket” in this lottery is a 4-digit number. How many different 4-digit numbers are there, including numbers that start with 0?
- A six-way box is a type of entry in this kind of lottery, where you choose two numbers—say, 1 and 2—and bet on all of the 4-digit numbers that have two of each of those digits, such as 1122 and 2121. How many combinations of two 1’s and two 2’s are there? Justify your answer.
- If you bet a six-way box, what is the probability that you will win this kind of lottery? (Garibaldi answers this question in the video: start watching at time 5:26.)

*—Tamar Lichter Blanks*

*The New York Times*, April 7, 2023.

Mathematics is the language of the universe. It lets us communicate about intangible forces in physics, and double the amount of flour in a cake recipe. And it also shows up in literature, as mathematician Sarah Hart notes in an article for the *New York Times*. “Leo Tolstoy writes about calculus, James Joyce about geometry. Fractal structure underlies Michael Crichton’s ‘Jurassic Park’ and algebraic principles govern various forms of poetry,” Hart writes. “We mathematicians even appear in work by authors as disparate as Arthur Conan Doyle and Chimamanda Ngozi Adichie.” In this essay, Hart admires the symbiosis between math and writing.

**Classroom Activities: ***cycloids, the universal language*

- (All levels) Based on the essay, sketch what a cycloid should look like for a wheel that is 2 inches in diameter and traveling at 1 inch per second. Would this shape change if the speed was 4 inches per second? If so, how?
- (Mid level) Have each student bring in their favorite book. Then form small groups, and discuss the following questions.
- Can you remember (or find) a reference to mathematics in your book? Why does the author include this reference (e.g. what do they find interesting enough to write about)?
- Math also influences literature in a more subtle way than explicit references. Choose a paragraph or two from the book without any quotes, and read it aloud to the group. Then count the number of words in each sentence. Are they all about the same? How do they vary? Do you notice anything else? Compare the results for each group member’s book.
- Based on what you read, discuss why the number of words in sentences might matter. (Hint: Think about style, emotion, and readability.)

- (High level) Hart writes, “It’s worth pointing out as well that the links between mathematics and literature do not run in just one direction.” In what ways is mathematics similar to a language, like English?

*—Max Levy*

- Why Mathematicians Re-Prove What They Already Know

*Quanta Magazine*, April 26, 2023. - Maths in a minute: Peano arithmetic

*Plus Magazine*, April 20, 2023. - A New Kind of Symmetry Shakes Up Physics

*Quanta Magazine*, April 18, 2023. - Interview: Why AI Needs to Be Calibrated for Bias

*Undark*, April 14, 2023. - Venn: the man behind the famous diagrams – and why his work still matters today

*The Conversation*, April 13, 2023. - More than maths: understanding infectious diseases in care homes

*Plus Magazine*, April 12, 2023. - Gloria Gilmer Makes History as the First Black Woman Mathematician to Have Work in the Library of Congress

*Because of Them We Can*, April 11, 2023. - Renowned Mathematician Sujatha Ramadorai Honored With Padma Shri

*Femina*, April 8, 2023. - How Mathematics Can Predict—and Help Prevent—the Next Pandemic

*Scientific American*, April 6, 2023.

]]>

- Aperiodic tilings in
*The New York Times* *The New York Times*covers the Abel Prize- The geometry of color space

Siobhan Roberts contributed “Elusive ‘Einstein’ Solves a Longstanding Math Problem” to the *Times* on March 28, 2023. The longstanding problem is the existence of a 2-dimensional shape that can tile the plane but only *aperiodically:* the pattern never repeats no matter how far out you go. Roberts starts by summarizing its history, which goes back some sixty years and has intriguing connections with “undecidability” problems in logic. There was an explosion of interest in the problem around the time of the discovery of the aperiodic Penrose tilings (they made the cover of Scientific American in January, 1977). But the simplest Penrose tiling uses two different tiles, and the question still remained if the phenomenon could be achieved with just one.

The breakthrough was announced in the ArXiV posting An aperiodic monotile (the title says it all) on March 20. Among the authors, David Smith, Joseph Samuel Myers, Craig S. Kaplan and Chaim Goodman-Strauss, only Kaplan (University of Waterloo) and Goodman-Strauss (University of Arkansas, Fayetteville) are academics. The tile — known as “the hat” for its fedora-ish shape — was discovered by Smith. But the proof that it, and its mirror image, could tile the plane, and could only do it aperiodically, is extremely intricate and took the whole team.

Roberts goes much deeper than usual into the nitty-gritty details of the work. For example, the *Times* displays this illustration of the intricate recursive nature of the tilings, the key to generating larger and larger assemblies of the tiles and to proving that they can cover the whole plane.

These images can be compared with the corresponding ones for a Penrose tiling. The gnarly geometry of the hat tile makes the procedure here considerably more complicated.

More from the Gray Lady. On March 22, this year’s Abel Prize was awarded to Luis Caffarelli of the University of Texas, Austin. Kenneth Chang covered the award for the *Times*, calling it “like a Nobel Prize for mathematics”. The print-edition headline was “Math Whiz Awarded Abel Prize for Applying Equations to Real-Life Problems.”

The Abel Prize site cites Caffarelli’s “seminal contributions to regularity theory for nonlinear partial differential equations.” Real-life problems often involve partial differential equations. A simple and important example is the 1-dimensional *heat equation* $$\frac{\partial u}{\partial t} = k\frac{\partial^2 u}{\partial t^2}$$ governing the flow of heat in an insulated bar. In the heat equation, $u$ is temperature; at any point, the time-derivative of $u$ is proportional ($k$ is a positive constant depending on what the bar is made of) to its second space-derivative. In general, a partial differential equation describes how the change of a measured quantity (in this case, temperature) with respect to one variable (often *time*) is related to the way it changes with respect to another variable (in this case, *space*).

The equations Caffarelli studies are more complicated (note the word *non-linear* in the citation). Chang gives three examples of where such equations occur in real-life phenomena.

*Melting ice: *How does the shape of an ice-cube, for example, change as it melts? If the ice-cube is not a perfect sphere, some parts will melt more quickly than others, so the problem itself changes with time.

*The obstacle problem:* In Chang’s example, a balloon is pressed against a wall. What shape does it take? If the wall is flat, we can imagine what the solution would look like, but what if the wall has a protrusion? The shape of the balloon changes as you move both side-to-side and up-and-down along the wall. These two directions correspond to two variables in the equation.

*The Navier-Stokes equation:* This is the target of one of the Millenium Prizes. It describes the flow of a fluid like water or air. Obviously very important, but we still cannot prove that solutions to the equation are reasonably behaved. For example, we don’t know whether the speed will always stay finite if it starts out with certain properties that you would expect from a real-life fluid. Chang tells us that Caffarelli, along with the late Louis Nirenberg and Robert Kohn (both then at the Courant Institute, NYU), was able to show that “regions of infinite speed, if they existed, would have to be extremely small.”

Our perception of color is mediated by the cone cells in our retina. These come in three types, each one “tuned” (responding maximally) to a certain wavelength: red, green or blue. So our *color space* is naturally 3-dimensional. But what about its geometry? Can we quantify our feeling that, say, blue is closer to purple than to yellow? In 1855 Hermann von Helmholz published his experiments showing that the Euclidean distance in the (red, green, blue) coordinate system does not correspond to the way we perceive differences between colors (see a recent paper by Giulio Peruzzi and Valentina Roberti).

These ideas developed over the next fifty years. Researchers viewed color space as an example of a *Riemannian manifold*, a space where the distance between two points can be defined as the length of the *geodesic* (the shortest curve) between them. If we use the Euclidean distance, a geodesic is a straight line, but this is not true in every Riemannian space. (Riemann himself had given the space of colors as an example of a manifold that everyone should be familiar with).

It turns out that this concept only goes part way in modeling perceptual reality. “Scientists disprove 100-year-old understanding of color perception” was posted on *Newswise* March 16, 2023. The item is a press release from Los Alamos National Laboratory calling attention to a paper titled “The non-Riemannian nature of perceptual color space”, published in *PNAS* on April 29, 2022. The authors are five members of their research staff, a team led by Roxana Bujack and Terece Turton. They show that the way distances work in color space is incompatible with the geodesic definition of differences. This is because of the perceptual phenomenon known as *diminishing returns. *A 2018 conference paper written by Bujack and others describes diminishing returns this way: “When presented with two colors, $A$, $C$, and their perceived middle (average/mixture) $B$, an observer usually judges the sum of the perceived differences of each half greater than the difference of the two outer colors $\Delta(A, B) + \Delta(B, C) > \Delta(A, C)$.” In a Riemannian manifold, the average $B$ would be halfway along the geodesic from $A$ to $C$, and this equation would read $\Delta(A, B) + \Delta(B, C) = \Delta(A, C)$.

The authors refer to the Weber–Fechner law: *the intensity of a sensation is proportional to the logarithm of the intensity of the stimulus causing it* (as reported in Merriam-Webster Medical, where it is characterized as “an approximately accurate generalization in psychology.”) This law is often taken for granted. For example, we measure loudness in decibels, which are logarithmic units: a sound 10 decibels louder than another has 10 *times* the physical intensity. The authors suggest that the non-Riemannian phenomena they are detecting may correspond to a “second-order Weber–Fechner law describing perceived differences.”

*Wired*, March 14, 2023.

March 14 is “Pi Day,” a celebration of the mathematical constant $\pi$ (which rounds off to 3.14). While $\pi$ famously appears in equations that have to do with circles, its applications extend beyond the ones you might see in a geometry class. In an article for *Wired, *physics professor and science communicator Rhett Allain takes a deep dive into some ways that $\pi$ shows up in physics. The article explains why $\pi$ appears in equations that model the physical world, including equations for magnetic fields, oscillations, and even the uncertainty principle.

**Classroom Activities:** *geometry, physics, astronomy*

- (Geometry) Allain describes how $\pi$ appears in a formula for measuring the intensity of sunlight. Work through the following steps to see how this formula works.
- Write down the formula for the surface area of a sphere in terms of its radius.
- According to a NASA fact sheet, the radius of the sun is about $695,700$ kilometers. Use this number to calculate the approximate surface area of the sun.
- Allain writes that the sun emits almost $4 \times 10^{26}$ joules of energy every second. Based on the surface area that you found in the previous step, how much energy is being sent out of each square kilometer of the sun every second?
- (Advanced) Read the section of the article titled “Pi and Symmetry” up through the paragraph that ends “…equidistant from the center of the sphere.” How would you calculate the intensity of sunlight as it reaches Earth? What information do you need?
- In your own words, what does $\pi$ have to do with the intensity of sunlight?

- (Trigonometry, Physics) Read the explanation of simple harmonic motion in the OpenStax physics textbook, up until the Khan Academy video.
- Give a real-world example of a deformation, and give a real-world example of an oscillation. You can use examples from the book, or get creative with your own examples.
- Based on the text, what does $\pi$ have to do with simple harmonic motion?
- Watch the Khan Academy video embedded in the textbook up until time 7:12, then answer the question under the video about displacement, amplitude, and period. Using a calculator or other software, try graphing the function $A\cos(\omega t)$ for different values of $A$ and $\omega$.
- (Calculus) Finish watching the Khan Academy video, then watch the other two videos in Khan Academy’s lesson about understanding harmonic motion. Discuss: where does the $2\pi$ come from in the formula for the period in simple harmonic motion?

*—Tamar Lichter Blanks*

*Fast Company*, March 3, 2023.

Have you ever been to a crowded event, like Comic-Con or a local concert, and watched the crowd as everyone exited at once? Despite the chaos, people manage to navigate without knocking each other over. How can this be? Math has the answer. The mathematical scientist Tim Rogers and his team recently discovered hidden mathematical patterns in human movement. They conducted an experiment that asked volunteers to walk within a maze of entrance and exit gates, finding that pedestrians subconsciously fell into orderly “lanes.” “At a glance, a crowd of pedestrians attempting to pass through two gates might seem disorderly,” according to Rogers. “But when you look more closely, you see the hidden structure. Depending on the layout of the space, you may observe either the classic straight lanes, or more complex curved patterns, such as ellipses, parabolas, and hyperbolas.” In this *Fast Company *article, Connie Lin describes Rogers’ work and inspiration.

**Classroom Activities: ***Brownian** motion, chaos*

- (Mid level) Ask the class to walk through the classroom doorway as quickly as possible without running or shoving each other. (Optional: Use a stopwatch to get the fastest time.) Take a video from behind. (If the class is small, consider inviting another class to join the activity.) Then read the article and play the video.
- What did you notice? (e.g. roughly how many people were around, and how many people could pass through an entrance/exit at once)
- How many “lanes” did you notice? What shape were the lanes?
- Did the lanes persist throughout the video?
- Discuss anything else you may have observed.

- (High level) The researchers drew inspiration from Albert Einstein’s theory of Brownian motion. Watch this video for some ideas of how to simulate Brownian motion in class, or have them try the following recipe:
- Mix one tablespoon of cornstarch with one cup of water to make a thin, milky solution. Add a few drops of green food coloring to the solution and stir. Pour the mixture into a clear container. Shine a flashlight on the container to illuminate the solution. Observe the movement of the particles in the solution under a microscope. Discuss how you expect the particles to be distributed if you let the experiment run for an hour, and why. (Hint: Should Brownian motion move the particles randomly or in a particular direction?) Compare what you observe to the video of the class moving through the doorway.

*—Max Levy*

*The Washington Post*, March 3, 2023.

After a snowstorm hits and leaves a thick blanket in its wake, cities are left with a difficult math problem. They need to ensure roads get plowed as quickly as possible using a limited number of snowplows. What’s more, routes need to be designed with many details in mind, such as prioritizing high-traffic streets and roads to hospitals. In this article, Kasha Patel explains how mathematics and computer software are helping some cities solve this problem more efficiently.

**Classroom Activities: ***optimization, graph theory*

- (Mid level) Patel relates route optimization to the famed “Königsberg bridge problem”, solved by Leonhard Euler. In class, read this
*Encyclopedia Britannica*article about the Königsberg bridge problem. Ask students to solve the following questions:- Suppose the town of Königsberg wanted to plow its seven bridges without having any plow cross a bridge more than once. How many plows would they need to use? Justify your answer, and describe the routes of each plow using the bridge numbers in this map.
- Suppose it costs the city \$500 to buy a snowplow, and \$100 to drive a snowplow across one bridge. (Assume it is free to travel between bridges.) Using the route you came up with in the previous problem, find the most cost-effective solution for the city. How much would a snowplow need to cost for your answer to change?
- Now suppose it takes a snowplow one hour to cross a bridge, and each hour that the bridges are covered in snow costs the city $150. What is the best solution? At what cost per hour would your answer change?

- (Mid level) Patel notes that several cities have declined to use computer software in their snowplow routes. When Shrewsbury, Massachusetts tried it, she writes, “the number of routes didn’t change, still requiring 33 snowplows. The computer-generated pathways also didn’t prioritize main roads as well as the status quo.” Ask students: Does this surprise you? Imagine that bridge 3 in the Königsberg problem was especially crucial. How would you try to prioritize it in a mathematical model?

*—Leila Sloman*

*Popular Mechanics*, March 31, 2023.

At the AMS Spring Southeastern Sectional Meeting last month, one talk was given by an unlikely pair. Two high school students from New Orleans presented a proof of the Pythagorean theorem. Though the Pythagorean theorem has been proven many times, the two students, Calcea Johnson and Ne’Kiya Jackson, came up with a new argument using trigonometry. Darren Orf covers their talk for *Popular Mechanics*. (Although Orf writes that “almost none of [the proofs of the Pythagorean theorem]—if not none at all—have independently proved it using trigonometry,” there are other trigonometric proofs out there. Orf links to one from cut-the-knot.org; the same site lists a few others here.)

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

- (All levels) The Pythagorean theorem relates the three side lengths in a right triangle. Learn about it with this online lesson from Leeds Beckett University. Do the practice problems and activities at the bottom of the lesson.
- Have students draw their own right triangles using rulers and graph paper, without measuring. Then ask them to measure the lengths of the sides and verify that they satisfy the Pythagorean theorem.
- Now, ask students to draw several copies of their triangles, cut them out, and use them to verify Proofs 3 and 4 at this link.

- (Trigonometry) In their proof, Jackson and Johnson used a formula called the Law of Sines. The Law of Sines says that the length of one side of a triangle divided by the sine of its opposing angle is the same for all three sides of the triangle. Learn the Law of Sines here, and do the practice questions at the bottom.

*—Leila Sloman*

*Scientific American*, March 3, 2023.

We often think that mathematicians prefer objectivity to subjectivity. Square numbers are not “special” or “exciting,” they’re products of an integer and itself. But it turns out that mathematicians do gravitate toward certain numbers more than others. For example, the numbers $e \approx 2.72$ and $\pi \approx 3.14$ stand out. In an article for *Scientific American*, Manon Bischoff discusses how mathematicians analyzing a database of more than 360,000 integer sequences revealed a clear split, as certain numbers appeared in the database more frequently than others. “To mathematicians’ great surprise, research in 2009 suggested that natural numbers (positive integers) divide into two sharply defined camps: exciting and boring values,” writes Bischoff.

**Classroom Activities: ***boring numbers, integers*

- (Mid level) Watch this Numberphile video about why 63 and -7/4 are special. In the video, 63 appears in the list of “Mersenne numbers.” Research Mersenne numbers and write out the first 10 of these numbers. Are most Mersenne numbers prime? Find examples of where Mersenne numbers are used in mathematics.
- (High level) Bischoff writes that although the mathematician Godfrey Harold Hardy thought 1,729 was a boring number, Srinivasa Ramanujan knew that it was in fact special, remarking
*“It is a very interesting number; it is the smallest number expressible as a sum of two cubes in two different ways.”*Have students do research on why the following formulas create “special” numbers.- $2^n – 1$
- $2^n + 1$
- $2^{2^n}+1$
- $n!+1$

*—Max Levy*

- Mathematicians have finally discovered an elusive ‘einstein’ tile
*Science News*, March 24, 2023.- Mathematician wins Abel prize for solving equations with geometry

*New Scientist*, March 22, 2023. - The World’s Simplest Theorem Shows That 8,000 People Globally Have the Same Number of Hairs on Their Head

*Scientific American*, March 20, 2023. - Jack and Rose could have both fit on the door in Titanic and Ross could have pivoted the sofa in Friends

*Daily Mail*, March 20, 2023. - March Madness brackets: A mathematician’s formula for the best outcome

*Scripps New*, March 17, 2023. - Euler’s Number Is Seriously Everywhere. Here’s What Makes It So Special.

*Popular Mechanics*, March 16, 2023. - This Pi Day, how to BAKE pi(e) — and have mathematical fun

*NPR Short Wave*, March 14, 2023. - Meet an Ohio State researcher introducing students to Black mathematicians through history

*The Columbus Dispatch*, March 10, 2023. - Simpler Math Predicts How Close Ecosystems Are to Collapse

*Quanta Magazine*, March 6, 2023.

*Nature* ran an editorial on January 31, 2023 with the title “Why we have nothing to fear from the decolonization of mathematics.” What might be feared? Through the *international decolonizing movement*, “university faculty members and students are exploring the contributions that people from many cultures have made to the story of different research fields,” they explain. The editors reference a backlash from people who insist that mathematics exists in a domain beyond culture. To give the example mentioned in the editorial, the roots of a quadratic equation do not depend on the particular identity of the person doing the extraction.

The editors remind us that universal truths may have very local origins. Their subhead here, “Maths made the modern world — and everyone stands to gain from the acknowledgment that the world made maths,” is actually illustrated by the history of the quadratic equation. That equation may exist in the world of platonic ideals, but our methods for solving it were developed here on Earth, starting in Mesopotamia about 4000 years ago. Similarly, zero could hardly be more disembodied, but as the editors mention, our notation for it dates from a certain time (“as early as the third or fourth century”) and a certain place on the Indian subcontinent.

Mathematicians know that whether math is discovered or invented, it takes people to do the discovering or the invention. Reminding students of this fact — and that those people lived in many different times, places, and cultures — may possibly dispel some of the forbidding aura that often cloaks the discipline. The *Nature* editors suggest that it can be “particularly empowering for people from historically marginalized groups.”

Some ongoing examples of more radical decolonization were spotlit by Rachel Crowell in a “Work” rubric article, also in *Nature:* “Maths Plots a Course to Cultural Equality.” We learn to distinguish between *indigenous mathematics* (teaching the mathematics native to a given culture) and *indigenizing mathematics* (teaching standard mathematics with reference to a given culture). Crowell gives a nice example of the latter: Kamuela Yong (University of Hawai’i-West O’ahu) indigenizes his pre-calculus course by working with Polynesian ocean-navigation techniques. Yong observed to Crowell that “although students might be especially interested in maths examples derived from their own cultural backgrounds, they also benefit from engaging examples that are rooted in other communities.” (Unfortunately this information seems only to be anecdotal: there is no reference to any systematic evidence showing that students learn more useful mathematics when their instruction has been indigenized.)

When the National Assessment of Educational Progress released “the Nation’s Report Card” on Grades 4 and 8 mathematics and reading last fall, the picture they gave was initially reported as catastrophic. “The Pandemic Erased Two Decades of Progress in Math and Reading” was a typical headline (*New York Times*, September 1, 2022). On February 8, 2023 *The 74* (“America’s Education News Source”) posted Kevin Mahnken’s interview getting the perspective of Sal Khan, the founder of Khan Academy. As Mahnken remarks, Khan Academy is “an internationally known learning tool reaching tens of millions students in over 100 countries”; it has become the go-to destination for online mathematics tutoring, particularly in the United States. So Khan is uniquely well-positioned to see where our students are. His own report: “It’s not as if scores went from decent to bad; they went from horrible to even-more-horrible.” (He gives the example of Detroit public schools, which went from 5% of eighth-graders being proficient in math to 4%.)

The interview goes beyond the pandemic and its effects. Khan emphasizes the *cumulative* aspect of math education. If you’re a little shaky on addition then you’ll never be comfortable with multiplication, and when you get to exponents and word problems, you will founder and sink. This explains why Algebra I is such a stumbling block for students in high school and college: their weakness in pre-algebraic and even arithmetic skills catches up with them. Khan does not believe in “direct instruction” — teachers should not lecture, but ask questions and make students think. When asked about memorization versus more conceptual learning, Khan says, “I absolutely think it’s got to be both.”

… that doesn’t quite rhyme. Petals of Venus: The Fascinating Geometry Behind Planetary Motions was posted January 23, 2023 on the website *culturacolectiva.com.* The Earth and Venus both have near-circular orbits around the Sun. “The Petals of Venus” refers to the orbit of the planet Venus when the Sun-Venus-Earth system is tracked in an Earth-centered coordinate system. (The mental image we share now of a Solar System with planets orbiting the Sun is relatively modern, and less intuitive than a point of view setting us at the center).

In Earth-centered coordinates, the two quasi-circular motions (Sun around Earth and Venus around Sun) combine to approximate a kind of curve called a (generalized) trochioid.

Note in this image that the path described by Venus almost closes up after 8 years: the planet has circled the Sun just a bit more than 13 times. (An Earth year is 365.256 days while a Venus “year” is 224.772 Earth days. The ratio is 1.6250067, slightly over 13/8 = 1.625). The *Culturacolectiva* posting ignores the slight discrepancy and displays an image like the one below, where the two ends of the 8-year curve match up exactly.

(They compound the error in Venus Pentagram, their animation of the orbit, which shows Venus tracing the same lovely, symmetrical curve over and over for more than 50 years). As they put it, “In a nutshell, if you plot the Venusian trajectory for eight years with the Earth as the center, you get a geometric poem that reminds us that everything in the cosmos is perfect.” Well, almost perfect.

]]>*The New York Times*, February 7, 2023.

In December, mathematical physicist John Carlos Baez posted a question about rectangles on the social network Mastodon. The problem was as follows. Take a square and break it into four rectangles, where the rectangles can be of any size and orientation, but must be “similar.” In other words, the rectangles should all have the same proportions, that is, the same length-to-width ratio. Baez asked: how many different solutions are there? Since the original post, mathematicians and math enthusiasts around the world have shared their progress, chipping in with a combination of computer code and mathematical theory. The unofficial team found 11 possible ways the rectangles can be proportioned in a solution, as Siobhan Roberts reports for the* New York Times*.

**Classroom Activities:** *geometry, ratios, irrational numbers*

- (All levels) Read the original Mastodon post about the problem, where Baez shows three possible ways to break a square into three similarly proportioned rectangles.
- Draw a square on a piece of paper. Try to divide it into four similar rectangles. Then discuss your ideas with another student. If you both found a solution, did you find the same one, or two different ones?

- (High level) Read the first two paragraphs in the beginning of Baez’s blog post on the problem, then take a look at the image with the three squares. (For this activity, it may help to have printouts of this image of the three squares.)
- Assume that the square has dimensions 1 unit by 1 unit. For the first two pictures, label the lengths of the sides of each rectangle. For a hint, note that Baez describes these rectangles’ proportions in the two bullets under the image.
- Now, read the third bullet under the image, along with the paragraph that begins “What’s $x$?” Label the lengths of the sides of the rectangles in the third square, using “$x$” the way that Baez does in his description.
- For a challenge, read through the algebraic explanation of $x$, up through the line $\rho^3 = \rho + 1$. Discuss: Does it seem surprising to you that the height-to-width ratio of the rectangle is an irrational number? Why or why not?

*—Tamar Lichter Blanks*

*Swimming World Magazine*, January 31, 2023.

In 2021, when members of the University of Virginia swim team attempted to qualify for the Beijing Olympics, they had a secret weapon: mathematician Ken Ono. They needed more than just muscles to swim faster than they ever had before. By combining sensors like accelerometers with mathematical analysis, Ono carefully studied the swimmers’ form and suggested tweaks to improve their speed. One swimmer named Paige had ranked seventh in a 200-meter contest and eleventh in the 400-meter. Only the top six swimmers could advance, so she considered only focusing on the 200-meter. But Ono’s data revealed a different story: “I had been telling [UVA’s head coach] and Paige, you really want to be focusing on the 400,” Ono told *Swimming World *writer Mathew De George. “Our speculation was that Paige would have almost a near lock to make the Olympic team.” That prediction came true. This article describes Ono’s analysis and how he found this niche.

**Classroom Activities:** *movement analysis,*

- (Mid level) Install an accelerometer app on your cell phone (individually or in groups) with a “record” function.
- The forces graphed on this app should have three components: $x$, $y$, and $z$. Which dimensions/directions of movement does each variable correspond to? (up and down, side to side, or forward and back)
- Start a recording on the accelerometer. Place your phone in your hand and move it from side to side as if waving hello. The accelerometer will produce a chart showing force versus time. Based on this chart, how long does one side-to-side movement take?
- Predict how the accelerometer graphs for $x$, $y$, and $z$ will look for the following movements and sketch them. Then perform the movement and compare the graph to your prediction.
- A full circle at constant speed (as if wiping a window)
- A full circle at constant speed (as if wiping a table)
- A zig zag motion facing you

*—Max Levy*

*The New York Times*, February 15, 2023.

On February 7, the basketball player LeBron James scored the 38,388th point of his career. The achievement made headlines, as it meant James had broken the record for most career points in the NBA, previously held by Kareem Abdul-Jabbar. The* Times’ *analysis of the event included a graph showing how the 250 top scorers in the NBA accumulated points throughout their career, highlighting the line representing James. Two days later, the Learning Network posted the graph as part of their “What’s Going On in This Graph?” series, which asks students to analyze and interpret a graph and post their thoughts for others to consider.

**Classroom Activities: ***data analysis, point-slope form*

- (All levels) Answer the four questions asked by the Learning Network in activity 1. When you’re done, find a partner and share your answers. What did your partner pick up on that you didn’t notice? What did your answers have in common?
- (Algebra) Read this online lesson on point-slope form. Now, use the graph to apply it to real data. By eyeballing two data points from James’ scoring graph, estimate the formula of the line. Repeat this with 3 different choices of points. Do the same for Abdul-Jabbar.
- Print out the graph, and draw the lines whose formulas you estimated. How well do they match the data? Which choices of points created a line that best matched the data?
- James is 38 years old, four years younger than Abdul-Jabbar was when he retired. Using your formula, estimate how many points James will have if he plays until he’s 42. How accurate do you think your guess is?

- (Algebra) Create your own graph with data you collect from your daily life. For example, you might use your phone to track how many steps you take each day, and graph your total steps for the week. After one week, repeat the activity above, finding a line and predicting what the total would be if you continued for three more days. After three more days, check how good your estimates were.

*—Leila Sloman*

*Cleveland19 News*, February 2, 2023.

Every year, a groundhog in Pennsylvania predicts the weather. Or at least, he tries to. If Phil sees his shadow on Groundhog Day, it is believed that there will be six more weeks of winter. The tradition dates back to the 1800s, and in 2023, Phil predicted that winter would stick around for another six weeks. “To verify whether or not his predictions have been accurate, we look at temperatures across the country during the months of February and March,” writes Erika Paige, in an article for *Cleveland 19 News*. Paige questions the accuracy of Phil’s predictions and describes her process for conducting the analysis. Since records have been kept, Phil has failed to see his shadow only 20 times. Over the last 30 years, Phil’s success rate is only around 37%.

**Classroom Activities:** *data analysis, spreadsheets *

- (Mid level) Transfer the table of data from the article into a spreadsheet, such as Microsoft Excel. Use spreadsheet functions to do or answer the following.
- In the last 20 years, how many times did Phil see his shadow? (Use equation functions, do not count manually)
- What percent of Phil’s predictions of winter were correct in the last 20 years?
- What percent of Phil’s “spring” predictions were correct, in the last 20 years?
- Use the software’s Conditional Formatting options to make all “verified” predictions green.
- What was the longest streak of verified predictions in the last 30 years?

- (High level) The article determines a “true” winter or spring based on whether the actual temperatures are warmer or colder than average for February and March. As a class, talk about whether you believe this is a fair classification or not. If not, then how would you change it? Do you expect that this change should increase Phil’s calculated accuracy or decrease it?

*—Max Levy*

*HowStuffWorks*, February 14, 2023.

Pick a number, any number. If it’s odd, multiply it by 3 and add 1. If it’s even, divide it by 2. Apply this recipe again and again until you get 1. The Collatz conjecture states that this is guaranteed to happen, no matter what number you pick. But as Jesslyn Shields writes for *HowStuffWorks*, no one has actually been able to prove this — they’ve only been able to check all the numbers that have 19 or fewer digits. There are still infinitely many numbers for which the conjecture is still unknown.

**Classroom Activities: ***sequences, proofs, programming*

- (All levels) As Shields writes, the sequence of numbers you get by applying the Collatz recipe step-by-step is called the “Hailstone sequence”. Find the Hailstone sequences that start with 1, 3, 20, and 13.
- (Mid level) Imagine a simplified version of the recipe: At each step, multiply by 3 and add 1, whether the number is odd or even. What will happen if you apply this recipe over and over?
- What if you divide by 2 at each step? What will happen if you apply that recipe over and over?

- (High level) Change the Collatz recipe so that if you get an odd number, you simply add 1. Re-calculate the sequences from the last activity with this change. Prove that with this recipe, you’ll always end up with 1.
- Now tweak the recipe so that if you get an odd number, you multiply by 2 and then add 1. Re-calculate the sequences from the last activity with this change. What’s different about this sequence? What do you think will happen if you follow the sequence forever?

- (Programming) Write a computer program that applies the Collatz recipe to every integer between 1 and 100, and outputs the number of steps it took to reach 1. Describe the results. Do they make you more or less convinced that the conjecture is true?
- Shields writes: “[Terence Tao]’s results point to a new method for approaching the problem and note how rare it would be for a number to diverge from the Collatz rule. Rare, but not necessarily nonexistent.” Is this consistent with the data from your program?

*—Leila Sloman*

- The Surprisingly Scientific Roots of Monkey Bars

*Smithsonian Magazine*, March 2023. - Nalanda: The university that changed the world

*BBC Travel*, February 23, 2023. - How will AI change mathematics? Rise of chatbots highlights discussion

*Nature*, February 17, 2023. - Cancer evolution is mathematical – how random processes and epigenetics can explain why tumor cells shape-shift, metastasize and resist treatments

*The Conversation*, February 10, 2023. - No one can predict exactly where birds go, but this mathematical model gets close

*Popular Science*, February 1, 2023. - ChatGPT might be taking over the internet, but a computer scientist explains why some problems are still too hard to solve—even for AI

*Fortune*, January 30, 2023.

Paul T. von Hippel posted ChatGPT Is Not Ready to Teach Geometry (Yet) (last update January 4, 2023) on the Math Technology blog of *Education Next*. Von Hippel’s point is that even though ChatGPT gives fluent, coherent answers to mathematics questions, the program can give wrong information as glibly as the truth. He put it to the test with questions about the Pythagorean theorem: it did OK for a while, but then veered into nonsense. “[I]t had revealed something important: it didn’t have a coherent internal representation of the triangle that we were talking about.”

As I understand it, ChatGPT is based on the premise that knowing a huge amount of things that people have written about the world can substitute for knowing about the world itself. (In that way, it is the digital epitome of “book learning”). I encourage readers to follow von Hippel’s example and experiment with ChatGPT and math. It’s fun if you know the stuff, but it might be dangerous for a student.

Here’s an example. I asked: “Can you chart a great-circle route between Chicago and Beijing?” ChatGPT answered: “Yes, a great-circle route is the shortest path between two points on the surface of a sphere. To find the great-circle route between Chicago and Beijing, you can use a map projection that displays the surface of the Earth on a two-dimensional plane. One such projection is the Mercator projection, which is commonly used for navigation purposes. On a Mercator projection, a great-circle route between two points appears as a straight line. To chart a great-circle route between Chicago and Beijing on a Mercator projection, you would simply draw a straight line between the two cities on the map. … ” The last two sentences are completely wrong. (See for example this Feature Column for a detailed explanation).

How did this happen? One can imagine sentences in Chat’s corpus like “A great-circle route is the shortest path between two points on the surface of a sphere.” “A straight line is the shortest path between two points.” “The Mercator projection displays the surface of the Earth on a two-dimensional plane.” And one can imagine Chat stitching pieces together to get “On a Mercator projection, a great-circle route between two points appears as a straight line.” It’s engineered to sound plausible but it’s just not true, confirming von Hippel’s observation: ChatGPT has literally no idea what it is talking about.

*Nature* ran Davide Castelvecchi’s review of Alma Steingart’s *Axiomatics: Mathematical Thought and High Modernism*, on January 16, 2023. Steingart is a historian of science; part of her thesis in *Axiomatics*, as Castelvecchi explains it, is that the axiomatizing movement in mathematics was not special to math but was a central part of *Modernism,* a movement that “dominated the mid-twentieth century in the arts and social sciences, particularly in the United States.” Whether or not axiomatization (more than relativity theory, for example) had an impact on art, Steingart gives evidence, in Castelvecchi’s words, of scientists “who saw their liberation from merely explaining the natural world as analogous to how abstract expressionism freed painting from the shackles of reality.”

Castelvecchi is more interested in the impact of the axiomatic method on mathematics itself. As he explains, the mathematical tradition exemplified by Euclid had traction because its concepts were “rooted in physical reality.”

Starting around the turn of the 20th century, mathematicians turned to *structures* as a way of encapsulating known phenomena and engendering new problems. For example, even though every finite group is the permutation group of its own elements, freeing the concept of *group* from these and other concrete examples was an essential step in the development of modern mathematics.

In Castelvecchi’s presentation of Steingart’s narrative, this new love of abstraction caught on first in the United States. But those mathematicians, far from pursuing abstraction for its own sake, understood that the ability to recognize abstract patterns, to “reveal a hidden skeleton of conceptual relationships” was what mathematicians could contribute to the study of the real world. Unfortunately, the story continues, most academic mathematicians, even in the U.S., were seduced by the thrill of ever purer abstraction and left the important connections with the concrete and perceivable to applied mathematics departments. The tide only turned in the last part of the century: William Thurston is mentioned as “an enormously influential topologist who delighted in making his complex geometric constructions feel physically real.”

Castelvecchi notes that Steingart’s story omits the interaction between very abstract mathematics and theoretical physics, which has become a central area of activity in both fields. In fact, what seems to be missing in this discussion is the realization that mathematics is part of the structure of the universe, and that the distinction between pure and applied is essentially an illusion. This is not to say that in *teaching* mathematics one should jump into abstraction prematurely. The subject is difficult, and as Byrne understood, we need all the help that our senses and terrestrial experience can give us.

The Hopf fibration is a historically and conceptually central ingredient of modern topology. Heinz Hopf discovered around 1930 that the 3-dimensional sphere $S^3$ (we can think of it as regular 3-dimensional space plus a point at infinity) can be completely filled up with circles: any two are linked, and each one corresponds to a point on the 2-dimensional (ordinary) sphere $S^2$ in such a way that nearby circles match with nearby points. The term *Hopf fibration* represents the whole picture, packaged in the function $h: S^3\rightarrow S^2$ that sends each circle to the corresponding point.

In an article in *Advanced Photonics* (January 10, 2023) a U.K.-China team led by Yijie Shen (Southampton), Zhihan Zhu (Harbin) and Anatoly Zayata (King’s College, London) explain how they have used lasers to generate stable physical instantiations of the Hopf fibration (they call them *hopfions*) and how they can even get them to propagate through space.

Where is the physics? Splitting a laser beam and recombining the two halves after manipulation, the team created an optical field where the polarization varies from point to point. Polarization is a physical phenomenon associated with waves, electromagnetic waves for example, that oscillate transversely to their direction of propagation.

Each of (a), (b) and (c) above shows three rays, represented by their electric fields, one drawn in blue, one in red and one in black (they all have the same wavelength, so if they were in the visible range they would all have the same actual color). In each case, the black field is the sum of the other two. The blue field oscillates back and forth along the $x$-axis: it is *linearly polarized*; likewise for the red field. When (a) blue and red fields are exactly in phase, their sum (black) is also linearly polarized. When (b) the red field has phase $\frac{1}{4}$-wavelength ahead of the blue, the sum rotates clockwise (looking down the $z$-axis). This is *circular polarization*. When (c) the red field is $\frac{1}{4}$-wavelength *behind*, the rotation is counter-clockwise. (The angle of the linear polarization in (a) can be controlled by varying the proportions of red and blue in the mix. Similarly for (b) and (c), when red and blue are not equal in magnitude the sum will manifest *elliptical polarization*; this also happens for phase differences different from $\frac{1}{4}$-wavelength.)

The polarization of a light beam can be represented by a point on the sphere: the south pole corresponds to clockwise circular polarization, the north pole to clockwise. Points along the equator represent linear polarization at the angle of their longitude. Intermediate points represent the gradation from a circle through more and more eccentric ellipses to a line, eccentricity varying with the latitude from 0 (circular, north pole) to infinite (linear, equator) and back to 0 (circular, south pole), with color corresponding to the orientation of their major axis.

When the points in the beam field are colored by polarization, using the coloring of the sphere described above, the lines of constant polarization (*isospin lines*) form the circles of the Hopf fibration. This is the simplest hopfion.

The team reports that they were able to make their hopfion move through space, by varying a phase angle in the preparation of the beam. In the newsletter of the International Society for Optics and Photonics, the report of this achievement had the headline “Light shaped as a smoke ring behaves like a particle.”

]]>*The New York Times*, January 1, 2023.

The British-American mathematician and artist Henry Segerman can’t visualize shapes or scenes with his eyes closed. Yet this “aphantasia” seems to motivate him to bring math out of imagination and into his hands: as puzzles and tools. Last year, Segerman debuted a new puzzle called Continental Drift. It’s a sort of spherical blend between a Rubik’s Cube and a sliding puzzle. The puzzle represents Earth, and its surface is split into 12 pentagons and 20 hexagons like a soccer ball. To beat the game, players must slide the hexagonal tiles until they have recreated the correct map of the globe. It’s a complex game — Segerman has calculated that there are $7 \times 10^{31}$ possible arrangements of the tiles. (A Rubik’s Cube has one-trillionth as many.) This article by Siobhan Roberts describes Continental Drift as well as Segerman’s other mathematical creations from 2022.

**Classroom Activities: ***holonomy, math puzzles*

- (All levels) Watch the videos of Segerman’s various puzzles and inventions. Which is the most interesting to you and why?
- (Mid level) As Roberts writes, the mathematical trick that complicates this game is called holonomy. If you slide the unique hexagonal tiles (and the vacant space) to loop around a pentagon between them, each tile will arrive slightly rotated. You must keep looping 5 more times to recreate the starting arrangement.
- Suppose you are moving 7 unique
*octagonal tiles*and an empty space clockwise around the perimeter of an octagonal tile. How many moves would it take to return to the start configuration? (Hint: watch Segerman’s video that explains this holonomy math.)

- Suppose you are moving 7 unique
- (High level) Segerman studies topology, the study of geometry that ignores lengths or angles. Watch this video explaining a classic topology joke: “A topologist is somebody who can’t tell the difference between a coffee mug and a donut.” That’s because a coffee mug has one hole (the hole created by its handle) and so does a donut.
- If a donut is “the same” as a coffee mug, which of the following pairs of shapes would also be considered the same:
- Needle and Macaroni
- Pretzel and College-ruled Paper
- Gold brick and Bowtie tied in a knot

- If a donut is “the same” as a coffee mug, which of the following pairs of shapes would also be considered the same:

*—Max Levy*

*Science News*, January 5, 2022.

Though jumping beans are really nothing but seed pods, they house moth larvae who prefer a cool, shady spot. When it gets hot, the larva inside starts to move, and the jumping bean jumps along with it. A new study analyzes the motion of jumping beans, concluding that — unable to see where it’s going — the jumping bean follows what’s called a random walk. As James Riordon reports for *Science News*, this means that if it keeps jumping around indefinitely, the jumping bean is sure to find a spot of shade.

**Classroom Activities: ***probability, random walk*

- (Mid level) The jumping bean “walks” around on a two-dimensional surface, but you can also define a one-dimensional random walk: At each step, you choose randomly to go either one foot left or one foot right. Each option has probability ½.
- What is the probability of taking 2 steps left, then one step right?
- What is the probability of taking 2 steps left and one step right, in any order?
- What is the probability that after 10 steps, you’re less than 9 feet away from the starting point?
- How do your answers to the first three questions change if the probability of going left is ⅓, and the probability of going right is ⅔?

- (High level) As Riordon explains for the jumping bean, if you follow the one-dimensional random walk forever, you’ll explore the whole number line. For instance, you’ll eventually be 1000 feet to the right of the starting point. Does this surprise you? Why or why not?
- (All levels) For a more in-depth lesson on random walks, try Ralph Pantozzi’s award-winning lesson plan. In this 2-day lesson, students spend the first day collecting data on the random walk by participating in a “flip trip”. On the second day, students will examine the probability model in more detail.

*—Leila Sloman*

*The Daily Californian,* January 8, 2023.

Martin Davis, who made important contributions to mathematics and computer science, died on January 1, 2023. Davis became one of the earliest computer programmers after receiving a mathematics Ph.D. from Princeton University in 1950. As Ani Tutunjyan writes in an obituary in *The Daily Californian*, Davis is famous for his work on Hilbert’s tenth problem, one of 23 unsolved problems mathematician David Hilbert posed in 1900. Hilbert’s tenth problem is to find an algorithm that can take any polynomial equation with integer coefficients, such as $x^2 + 3y^3 – 4 = 0$, and determine whether or not that polynomial has any integer solutions. The algorithm should be able to give a yes or no answer for any polynomial after a finite number of steps. Thanks to the work of Davis and other mathematicians, it is now known that Hilbert’s tenth problem is unsolvable: it is impossible to find an algorithm that checks for the existence of integer solutions.

**Classroom Activities**: *integers, algebra, polynomials, algorithms*

- (Algebra) The integers are the positive and negative whole numbers, along with zero. They include $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$, but not fractions like $\frac{1}{2}$ or irrational numbers like $\pi$.
- Find an integer solution for $x^2 – 4 = 0$. In other words, find an integer $a$ with the property that $a^2 – 4 = 0$.
- Find integers $a$ and $b$ with the property that $a + 4b – 17 = 0$. Discuss your answer with another student. Did you find the same solution, or two different ones? Can you find another solution?
- (High level) Explain why $2x – 1 = 0$ does not have an integer solution.
- (High level) Explain why $x^4 + y^2 + 5 = 0$ does not have an integer solution. (Hint: If $a$ and $b$ are integers, can $a^4 + b^2$ be a negative number?)

- (All levels) An algorithm is a list of steps that starts with some kind of input and produces a result after a finite number of steps. The quadratic formula is a kind of algorithm: it is a process for taking any quadratic equation $ax^2 + bx + c = 0$ and finding its solutions.
- Read the examples of polynomials with integer coefficients listed on the Wikipedia page for Diophantine equations. Do you find it surprising that there is no algorithm that can always check whether or not a polynomial has integer solutions, as Davis and other mathematicians have shown? Why or why not?
- Watch the introduction to algorithms video on Khan Academy, then try the guessing game. Try to answer the question after the longer guessing game: Why should you never need more than 9 guesses?

*—Tamar Lichter Blanks*

*Conde Nast Traveler*, January 3, 2023.

Electric vehicles are popular among drivers who want to reduce their carbon footprint. But the thought of embarking on a long journey can be intimidating due to concerns about battery range. In an article for *Conde Nast Traveler,* an EV owner describes journeying through India in an electric car to test the feasibility of using an EV for long-distance travel. The trip covered approximately 8,849 kilometers over 70 days, and included 35 city stops. Sushil Reddy, the author who is also an engineer, describes what factors drivers must consider to extend their range. “Lower speeds will generally give a higher range due to the lesser air drag and rolling resistance,” Reddy writes. The feasibility of an EV is also a financial math problem. Within the article, Reddy describes his analysis of cost savings.

**Classroom Activities: ***algebra, efficiency*

- (Mid level) Suppose you are going on a 1,100 mile road trip in an electric car from Los Angeles to Seattle. Your car has a range of 300 miles, but must not let the range drop below 15% of the maximum.
- Assuming you leave Los Angeles with a full charge, what is the fewest number of chargers required to drive to Seattle and back?
- Where should these chargers be (in terms of the number of miles away from Los Angeles)?
- Write an algebraic expression for the remaining range,
**y**, in terms of miles driven,**x**.

- (High level) Suppose that heating the car makes the car 10%
*less efficient*. Write a new algebraic expression.- How much less efficient would the car have to be to require one extra charging station between Los Angeles and Seattle?

*—Max Levy*

*FiveThirtyEight*, January 20, 2023.

Every other week, Zach Wissner-Gross posts two mathematical puzzles as part of *FiveThirtyEight*’s series “The Riddler”. The second puzzle on January 20 (the “Riddler Classic”) combined geometry and randomness into a problem about how a delivery drone compares to a delivery scooter. In the puzzle, the drone can travel in a straight line to its destination — flying over buildings, cars, and trees — while the scooter must follow city streets. Wissner-Gross asks how much better the drone does at its deliveries than the scooter, requesting the ratio between the average number of deliveries each performs. (In this puzzle, you only have to worry about the distance from the restaurant to deliveries, not distances between two deliveries.) We’ll use this puzzle as an opportunity to build students’ intuition about probability and expected value.

**Classroom Activities: ***probability, geometry*

- (Mid level) Imagine you are at the center of a circle whose radius is 1 mile, and someone randomly chooses a point inside the circle. What is the probability that the point will be less than half a mile away from you? Why?
- Is it more likely that the point will be less than 1/10 of a mile away, or more than 9/10 of a mile away? Why?
- Imagine you draw a square anywhere inside the circle. The square’s area is $\pi/4$. What are the chances that the random point will fall inside the square? Why?
- How do your answers change if you replace the circle with a square whose sides have length 2?

- (Mid level) The puzzle is about two expected values: The expected distance the drone travels, and the expected distance the scooter travels. An expected value is an average over all possible (random) outcomes, weighted by their probabilities.
- Imagine playing darts on the diagram below. The smallest circle has radius 1, the middle circle has radius 3, and the large circle has radius 5. Your score is the number labeling the region where your dart lands. What is the expected value of your score? (Assume you’re not very good at darts, and your dart lands randomly anywhere on the diagram.)

- Draw a circle with three or four dots randomly placed inside. Measure the distances the drone and the scooter would have to travel to reach the dots from the circle center. Does
*FiveThirtyEight*’s puzzle answer ($4/\pi$) match your measurements? If you know integral calculus, read the full puzzle solution.

*—Leila Sloman*

- Mathematician Explains Infinity in 5 Levels of Difficulty

*Wired*, January 30, 2023. - Charting a course to make maths truly universal

*Nature Careers*, January 30, 2023. - Only a geometry master can stop irresistible Bukayo Saka

*The Times*, January 26, 2023. - Can you solve it? Prisoners and boxes

*The Guardian*, January 23, 2023. - Are We Living in a Computer Simulation, and Can We Hack It?

*The New York Times*, January 17, 2023. - How mathematics stopped being defined by reality — and started to invent new ones

*Nature*, January 16, 2023. - The Mystery And Power Of The “Non-Archimedean” World

*Science Blog*, January 15, 2023. - ‘Random is random’: A mathematician’s view on lotteries, betting and why it’s best to just have a little fun

*WSFL-TV*, January 11, 2023. - Curious Kids: what are gravitational waves?

*The Conversation*, January 11, 2023. - Why maths, our best tool to describe the universe, may be fallible

*New Scientist*, January 10, 2023. - A Crucial Particle Physics Computer Program Risks Obsolescence

*Wired,*January 1, 2023. - Exploring the mathematical universe – connections, contradictions, and kale

*The Conversation*, January 1, 2023.