New Yorker staff writer Alec Wilkinson takes us along with him on his quest to “penetrate the mysteries of mathematics” (July 8, 2022). As he explains, he was bad in math as a child, and at 65, decided to revisit the subject and to document his experience in a book: A Divine Language: Learning Algebra, Geometry, and Calculus at the Edge of Old Age, published this year by Farrar, Straus and Giroux. In the New Yorker, Wilkinson tells us that he had been turned off by math’s “smugness” and “self-satisfaction” in high school. But he found himself enjoying his exploration anyway. Here are some insights he brought back from the trip. These insights and questions are shared by most mathematicians, and they refute the notion that math is smug and self-satisfied. On the other hand, they are certainly not obvious to someone experiencing mathematics at the introductory level. It is welcome to hear them from a relative outsider, and to read them in a wide-circulation, general-audience medium like the New Yorker.
Wilkinson’s book was reviewed in the New York Times, July 17: “Math Defeated Him in School. In His 60s, He Went Back for More.”
Rachel Crowell has an article on the Scientific American website (posted June 28, 2022) with the title “Mathematicians Are Trying to ‘Hear’ Shapes—And Reach Higher Dimensions.” This topic was the subject of one of our earliest “Feature Columns,” by Steve Weintraub back in June, 1997: “You Can’t Always Hear the Shape of a Drum.” Crowell’s article covers recent progress on the problem.
We are all familiar with a big bell sounding a lower note than a small one; similarly, many of us know that long strings vibrate more slowly than short ones. When the frequencies are in the audible range, we hear lower notes from the longer strings. So, all other properties being equal, one can tell by listening if two strings are the same length or not: in that sense, one can hear the length of a string.
Some more about strings. A string fixed at both its ends is limited in the ways it can vibrate. A few of the possibilities are shown in this animation:
Note from the animation that modes with shorter wavelength beat faster: in fact, for a string, the vibration speed of a mode is inversely proportional to its wavelength. As this image suggests, for a string of length $L$ the possible wavelengths are ${2L, L, \frac{2}{3}L, \frac{1}{2}L, \dots}$. So if $\nu$ is the frequency of the first mode, then the set of possible frequencies has to be ${\nu, 2\nu, 3\nu, 4\nu, \dots}$. This ordered list of frequencies is the spectrum of the string.
The head of a drum is the 2-dimensional generalization of a string fixed at both ends. Just as for a string the vibrations of such a membrane are sums of modes, each with its own frequency. For a circular drum, modes and frequencies can be identified explicitly.
The analysis shows that if mode $(0,1)$ has frequency $\nu$, the others have frequencies $1.593\nu$, $2.135\nu$ and $2.295\nu$, respectively (writing just the first three decimal places of each coefficient), and that in mode $(0,2)$ the inner circle’s radius is $0.436$ of the outer’s. Our numbers come from Russell’s Vibrational Modes of a Circular Membrane page, which has animations of these and several other modes.
Crowell’s survey reaches back to a 1966 work by Mark Kac: “Can one hear the shape of a drum?” Kac considers drums that are all cut from the same physical cloth, in that the combination of density, stiffness and tension that determines their mathematical behavior is the same for all of them. He proves for such an ideal drum that its area and perimeter can be “heard” (as above); moreover, he shows that any drum with a spectrum (its set of frequencies of vibration) of the same type as the disc, i.e. ${\nu, 1.593\nu, 2.135\nu, 2.295\nu, \dots }$ has to be circular.
So ideally one can tell by listening if a drum is circular or not. The question remained in general, if two ideal drums are isospectral (same spectrum, so they sound the same), beyond having the same area and perimeter, are they the same shape and size?
As Crowell tells us, it took more than twenty years for this question to be settled. “One Cannot Hear the Shape of a Drum” was published in 1992 by Carolyn Gordon, David Webb and Scott Wolpert. Here is one of the isospectral pairs the team exhibited in their paper.
Recent progress: as Crowell reports, it is now known for triangles and for quadrilaterals with two parallel faces (including parallelograms, rectangles and trapezoids) that sounding the same means having the same shape and size. Crowell goes on to examine isospectral problems in higher dimensions, but leaves us with two unanswered questions in the plane. Are there any convex examples of non-congruent, isospectral pairs? (Note that the Gordon-Webb-Wolpert isospectral pairs are very non-convex). And, quoting from a recent article, “Specifically, it is not yet certain whether the counterexamples [i.e., non-congruent, isospectral pairs] are the rule or the exception. So far, everything points towards the latter.” In other words, differently shaped drums will sound different in general. But this has to be proved.
Daniel Goldman (Georgia Tech) led a team of four publishing Coordinating tiny limbs and long bodies: Geometric mechanics of lizard terrestrial swimming in PNAS, June 27, 2022. They study the transition from four-footed ambulation in long-legged lizards to pure forward slithering in snakes in a sampling of reptiles with shorter and shorter legs. This work is part of a research direction that started some 45 years ago, and has roots in mathematics (topology and differential geometry) and in theoretical physics.
To recapitulate the history very briefly, it was discovered in the 1970s that two very different fields, topology and theoretical physics, had independently come up with the same concept. Topologists called this concept “connections in principal bundles”, and physicists called it “gauge fields”. The discovery led to important progress in both fields. It is also remarkable that not so long afterwards the same set of ideas turned out to be useful in robotics, in the study of how a mechanism could move forward over terrain or in water by periodically changing its shape (the basic Geometry of self-propulsion at low Reynolds number, by Alfred Shapere and Frank Wilczek, appeared in 1989). The next step was to apply these concepts, known as geometric mechanics, to natural history, in the investigation of how animals crawl and swim, which brings us to lizard terrestial swimming.
In this analysis, the standard use of the term gait to denote the different ways a horse (or other animal) moves forward (e.g., walk, trot, canter, gallop) is generalized to refer to any periodically repeating sequence of postures (in an animal) or configurations (in a robot) that results in locomotion. Mathematically a gait can be studied as a loop in the shape space of an animal or a mechanism.
How does geometric mechanics work? The basic process is easiest to visualize for a simple process like a snake’s slithering across a sandy surface, where the relevant part of shape space is 2-dimensional. It is encapsulated in the next image, taken with permission from an arXiv posting by Jennifer Rieser (also at Georgia Tech) and ten collaborators.
Roughly speaking, if the creature has taken a particular shape $s$, and starts to wiggle its body in a certain way, that will make it start to move in a certain direction. This change in position can be drawn as a vector in the copy of position space associated to the initial shape $s$. Now the creature has a new shape $s’$; it wiggles again, and produces a new change in position: a vector in the copy of position space associated to $s’$. As the creature continues to execute its gait, it will eventually return to its original position $s$. When these infinitesimal vectors are summed up around the closed loop (blue) corresponding to the gait, the integral gives the displacement resulting from one cycle.
Goldman and his team start from the observation that, just as Rieser et al. had observed for pure slithering, the periodic changes in reptilian body shape during locomotion can be approximated by a cycle of combinations of two modes, like the first two of the string analyzed above: one where the body oscillates between ( and ), to use a graphical shorthand, and one where it oscillates between S and Z. Their shape space is consequently very much like Rieser’s, but with fore and hind limbs also taken into account.
They report: “… we find that body undulation in lizards with short limbs is a linear combination of a standing wave and a traveling wave and that the ratio of the amplitudes of these two components is inversely related to the degree of limb reduction and body elongation.” This surprisingly mathematical statement implies that for a long-legged lizard, the loop in shape space corresponding to its gait collapses to a straight-line path going back and forth from ( to ); whereas for a limbless snake, the gait mixes the ( ) mode with the S Z mode in such a way as to produce a traveling wave moving down the creature’s body and making it progress forward.
For more background I recommend Geometric Phase and Dimensionality Reduction in Living (and Non-living) Locomoting Systems, a lecture given by Daniel Goldman in Trieste in 2019. It gives details about how the connection form—the mathematical name for the process illustrated in Rieser et al.‘s diagram—can be derived experimentally in each case. Towards the end of the lecture Goldman explains how a “height function” defined on shape space can be calculated for such a connection. This is the same computation that yields the Gaussian curvature in Riemannian geometry, and it allows optimal gaits to be designed for robots by choosing paths that enclose the largest amount of what would be positive curvature.
The link between connections and gauge theory has an elementary presentation in Fiber Bundles and Quantum Theory, an article Herb Bernstein and I wrote for Scientific American in July, 1981. Some historical details can be gleaned, among other topics, from a 2008 video interview of C. N. Yang and Jim Simons.
]]>New York Times, July 5, 2022
Math helps us make the most out of space. Sometimes that involves familiar dimensions: Geometric rules can dictate how to stack oranges in a three-dimensional box or arrange tiles on a two-dimensional plane so that as much space as possible is used up. Other times, mathematicians answer these questions in higher dimensions. This summer, mathematician Maryna Viazovska of the Swiss Federal Institute in Zurich was awarded the Fields Medal—considered by many to be math’s highest honor—for proofs of sphere packing in space of dimensions 8 and 24. The work represents an extreme twist on a 400-year-old conjecture that stacking bowling balls in a pyramid fills nearly 75% of the available space. Viazovska is only the second woman to win a Fields Medal. “I feel sad that I’m only the second woman,” Viazovska told the Times. “I hope it will change in the future.” In this article for the New York Times, Kenneth Chang describes Viazovska’s work and gives more context on her momentous award.
Classroom activities: stacking, geometry, higher dimensions
—Max Levy
Irish Examiner, August 24, 2022.
As the buzz surrounding Maryna Viazovska’s Fields Medal win demonstrated this summer, the granting of one of math’s most prestigious prizes to a woman remains, unfortunately, big news. But over a century ago, women like Sophie Bryant were already breaking barriers in mathematics: When Bryant earned her Ph.D. in 1884, no other women in the United Kingdom or Ireland had done so. In this article, Clodagh Finn sketches the accomplishments of Bryant, who tragically died in a hiking accident 100 years ago this August.
Classroom Activities: geometry, trigonometry, calculus
—Leila Sloman
Science News, August 22, 2022.
Packing problems seem to be a theme in math news this month. A new paper in Journal of the Royal Society Interface shows that sea urchin skeletons resemble a mathematical packing pattern called a Voronoi pattern. These patterns look like a web or mesh whose holes, or “cells”, appear somewhat irregular. But the cell shapes actually obey strict rules. Within each cell is a “seed” that governs the cell’s shape, writes Rachel Crowell for Science News. Cells must hug their seeds tightly—if you sit down anywhere on the Voronoi pattern, the closest seed should be in the same cell as you. In this article, Crowell explains the rules of the Voronoi pattern, and some of the engineering benefits this structure may offer both the sea urchin and, potentially, human technology.
Classroom Activities: geometry, Voronoi patterns
—Leila Sloman
The Independent, August 15, 2022
Here’s a question you might not know is contentious: how many holes are there in an ordinary drinking straw? You could say that a straw has two holes—one on each end—but you could also say that the empty space inside the straw is just one long hole. In an article for The Independent, mathematician Kit Yates examines the straw question through the lens of the mathematical field of topology. Yates compares topological shapes to objects made of dough, which can be stretched, pulled, or squished without being fundamentally changed. To topologists, a drinking glass is the same as a plate, a mug is the same as a donut, and a pair of binoculars is the same as a pair of glasses. And a straw can be compressed down into a ring—which has just one hole.
Classroom activities: topology, geometry
—Tamar Lichter Blanks
The Guardian, July 25, 2022
In a game show from the 1980s called Blockbusters, participants try to connect two sides of a map of tiles. Each tile on the map is an identical hexagon. Tiles on two opposing edges are blue (one team’s color), and the other opposing edges are white (for another team). When participants answer questions correctly, a tile between the edges becomes white or blue—a point for their team. The goal: connect your team’s two edges with a path of hexagons of the same color. In this article, puzzle expert Alex Bellos describes the Blockbusters game and imagines a diamond shaped map (shown below). Bellos poses a mathematical riddle. How many configurations of the map will contain a path connecting the two blue edges? The solution requires a little math, but it mostly uses a clever trick of logic.
Classroom activities: probability, logic puzzles, percolation
—Max Levy
National Geographic, June 6, 2022
If a giant wave hits a ship, and nobody survives to tell the tale, did the wave even exist? What if someone does survive, but the wave is simply too big to believe? In 1826, a French naval officer and scientist survived a 100-foot wave during a storm in the middle of the ocean. “His story, backed by three witnesses, seemed so outlandish that it was dismissed as fantasy,” writes Ally Hirschlag. But similar accounts of so-called “rogue waves” appeared many times through the years, and scientists now acknowledge their existence. Still, rogue waves—waves more than twice as tall as those around them—remain a mystery to physicists and mathematicians. Unlike tsunamis, the giant swells appear out of nowhere, often within just 10 to 15 seconds. In this article, Hirschlag describes recent research aiming to understand how rogue waves form, how common they are, and how to predict them in real time.
Classroom activities: waves, linear and nonlinear effects
Related Mathematical Moments: Knowing Rogues.
—Max Levy
/Film, June 8, 2022
Math makes a cameo in the animated television show Futurama via an ordinary-looking string of digits. In the show, a robot named Bender has the serial number $1729$, which has a special property: It is the smallest number that can be written as the sum of two positive cubes in two different ways. In other words, two different choices of whole numbers $a$ and $b$ are solutions to the equation $1729 = a^3 + b^3$. This fact was discovered by Srinivasa Ramanujan, the prolific mathematician who was the subject of the 2015 film The Man Who Knew Infinity. Futurama writer Ken Keeler, who holds a PhD in applied math, suggested using $1729$ in the show because of its mathematical significance. In an article for /Film, Witney Seibold discusses $1729$ in Futurama and some riffs on the sums of cubes property.
Classroom activities: number theory, exponents, nested radicals
—Tamar Lichter Blanks
Scientific American, July 2022 issue
A flock of birds, a swarm of locusts, a crowd of people—large groups of animals often seem to move as a cohesive unit rather than as unconnected individuals. This phenomenon is called collective motion, and there’s a lot of math to model it. Now, researchers studying Antarctic krill say they have figured out how these tiny crustaceans form and maintain underwater swarms. Krill seem to pay the most attention to the neighbors swimming above and below them instead of those to the left and right—a behavior not seen in fish and birds that exhibit collective motion. In this article, Andrew Chapman explains how the scientists conducted the study and what questions remain unanswered.
Classroom activities: emergence, geometry, vectors
time (s) | $x_9$ (mm) | $y_9$ (mm) | $z_9$ (mm) | $x_{10}$ (mm) | $y_{10}$ (mm) | $z_{10}$ (mm) |
0.000 | 98 | -34 | 520 | 117 | -18 | 503 |
0.100 | 97 | -38 | 518 | 116 | -21 | 500 |
Related Mathematical Moments: Getting It Together.
—Scott Hershberger
The Conversation, June 16, 2022
Random sequences of letters and numbers make great passwords because they are hard to guess. Unfortunately, they are also hard to recall. Consider the sequence “UKCIQCKDEWIAOMF.” It might take you a minute to memorize. How about “QUICK.WICKED.FOAM,” three random words made from the same characters? Much easier to remember—and still useful as a unique identification. A web app called what3words divides the entire surface of Earth into 3-meter-by-3-meter squares and uses three random English words to represent every square. The order of the words matters, and they don’t appear in alphabetic order based on longitude or latitude. So while quick.wicked.foam takes you to the US president’s Oval Office, wicked.quick.foam sits near a bus stop in Louisville, Kentucky. In this article, professor Mary Lynn Reed explains how what3words works: “The secret behind this power is just a little bit of math.”
Classroom activities: combinations, permutations, surface area, encryption, algorithms
—Max Levy
There will be no July Math Digests. We will return with the August Math Digests in early September. Until then, explore coverage of the recipients of the 2022 Fields Medals in Nature, Quanta Magazine, and The New York Times.
]]>That is part of what Jordan Ellenberg has to say about geometry in an interview with Anne Strainchamps of Wisconsin Public Radio, broadcast on May 28, 2022. More completely, “Geometry’s power lies in how it endangers received wisdom.” At the center of the interview Ellenberg tells a “parable” (you can hear the excerpt on YouTube) that illustrates his point. It’s the story, from Edwin A. Abbott’s 1884 novella Flatland, about the Square’s encounter with the Sphere. (Flatland is a 2-dimensional universe inhabited by intelligent polygons; Abbott spends several chapters explaining in detail how their society works). Our narrator is the Square, a kind of flat Everyman, who has his life drastically changed by an encounter with a Sphere.
The Sphere explains that his world has another dimension. Initially the Square cannot conceive of an “up” that does not mean “North.” But the Sphere argues by analogy: a point (1 terminal point) moving North produces a segment (2 terminal points) and a segment displaced sideways produces a square (4 terminal points). He remarks that 1, 2, 4 are in geometric progression and asks the Square for the next number. “Eight.” “Exactly.” A square, when displaced “up,” will sweep out what “we call” a Cube, with 8 terminal points. The Square gets involved in the calculation and figures out that since a point has no sides, a segment has 2 “sides” (both points), and a square has 4 sides (all segments), a cube will have to have 6 sides, all squares. Through all of this exposition the Sphere speaks condescendingly to the Square. E.g. “Distress not yourself if you cannot at first comprehend the deeper mysteries of Spaceland.”
Later in the story, after the Sphere has dragged the Square out into Spaceland and shown him how he can see inside all of his fellow polygons, the Square understands, and asks to be taken into into the Fourth Dimension to get the same perspective on the Sphere and the other solids. Now it’s the Sphere’s turn to be obtuse: “There is no such land. The very idea of it is utterly inconceivable.” And here is Ellenberg’s point. As he puts it, “The Square is figuring something out. And the Sphere, who was moments ago in this position of authority is suddenly like, ‘Wait, wait, I don’t like this!'” Geometry is endangering received wisdom.
“Seifert surfaces in the 4-ball” by Kyle Hayden, Seungwon Kim, Maggie Miller, JungHwan Park and Isaac Sundberg was posted on ArXiV on May 30, 2022 and picked up by Josh Harnett for Quanta Magazine on June 16, with the title “Surfaces So Different Even a Fourth Dimension Can’t Make Them the Same.” The team had solved a problem open since 1982 regarding what happens to two topologically equivalent Seifert surfaces of the same knot when their interiors are allowed to move in four dimensions.
A Seifert Surface for a knot is a surface with that knot as boundary.
A knot can have more than one Seifert surface. This happens, for example, when the knot can be drawn without self-intersections on a closed surface so as to cut the surface into two parts. Each of these is a Seifert surface.
The surfaces $\Sigma_0$ and $\Sigma_1$ look quite different, but the difference comes from the way they are embedded in 3-space. Their intrinsic topologies are actually identical. We can verify this by adding vertices and segments so as to cut them up into simple pieces (topologically, polygons) and showing that the resulting cell complexes have the same Euler characteristic.
As the Sphere explained to the Square in our first item, some problems of a topological nature can be resolved by moving to a higher dimension. For example, a point can escape from a circular prison in Flatland by slipping into and out of the third dimension.
Similarly, any knot in 3-space is isotopic to a circle if we are allowed to move some of its points into and out of the fourth dimension. For the problem at hand, many cases were known of a knot with two topologically equivalent Seifert surfaces that are not isotopic in 3-space, but which become isotopic when their interiors are allowed to move in the fourth dimension. The open question was whether or not is this the general situation.
In Harnett’s words, “The new work identifies the first pair of Seifert surfaces that are as provably distinct from each other in four dimensions as they are in three.” $\Sigma_0$ and $\Sigma_1$ are topologically identical Seifert surfaces for the same knot, but even when their interiors are pushed into 4-space, there is no isotopy taking one to the other. So the answer to the question is “no.” How do they show this? As they describe it, their argument is “extremely elementary,” but this only means that it doesn’t use any topological machinery invented since 1970. After all, the question they answer has been on the table since 1982. Elementary can be quite different from easy.
]]>“The Grothendieck Mystery” is the title of Rivka Galchen’s article in the issue for May 16, 2022; the subhead is “Alexander Grothendieck revolutionized mathematics—then he disappeared.” Galchen takes us from Grothendieck’s birth (Berlin, 1928; Grothendieck is his mother’s last name) and childhood as a refugee in France, through a brilliant mathematical career that essentially ended in 1970, and through the last half of his life—which he spent in self-imposed exile, farther and farther from Paris, finally as a kind of hermit in a village in foothills of the Pyrenees. He died in 2014.
Galchen was not going to tax the New Yorker readership with mathematical details, but manages to convey, through metaphors Grothendieck used, some understanding of his approach to mathematics. The first is thinking of a problem as a hard nut. One way to get to the meat is to go at it with “sharp tools and a hammer.” But a better way would be “to put the nut in liquid, let it soak, even to walk away from it, until eventually it opened.” You build a theory, step by simple step, and by the time you get around to your problem, the solution is obvious. The second metaphor is “the rising sea.” You have to get your boat across a rocky reef. Instead of attacking the reef, you can “wait for the sea to rise, providing a smooth surface to cross effortlessly.”
The metaphors pay off when Galchen speculates on why Grothendieck dropped out from what seemed the top of the mathematical world. One of his main motivations had been the solution of the Weil Conjectures. These had been formulated during the war by André Weil, another towering figure of 20th-century mathematics. What makes the conjectures striking is that they relate numerical properties of the set of solutions to a polynomial equation over a finite field—a totally discrete gadget—to topological invariants of a geometric object: the complex variety corresponding to that same equation. There were four conjectures. Bernard Dwork had proved the first in 1960; Grothendieck and collaborators proved two of the others by 1965; but the last one just would not succumb to his methods. Instead, after he quit, his student Pierre Deligne found a different approach and scored the goal. Galchen quotes Ravi Vakil: “It was as if, in order to get from one peak to another, Deligne shot an arrow across the valley and made a high wire and then crossed on it.” Whereas, Galchen tells us, “Grothendieck wanted the problem to be solved by filling in the entire valley with stones.”
The digital magazine Popular Science ran “What tangled headphones can teach us about DNA” on April 18, 2022. The story, by Ryan F. Mandelbaum, is about the mathematical biologist Mariel Vazquez, a professor of mathematics and of microbiology and molecular genetics at UC Davis, and her research into topological and geometric features of DNA molecues. Here’s how it starts: “It’s a truth universally acknowledged that if you shove wired headphones into your pocket, they’ll eventually emerge in a jumble of knots.” Then he reminds us that each of our cells [average diameter $30 \mu\text{m}$] contains about 6 linear feet of DNA. A lot of stuffing, a lot of tangles. And tangles in DNA are Vazquez’s field of expertise.
Mandelbaum gives a brief definition of topology, with a useful link to Quanta magazine. A sphere may be equivalent to a cube, but “Doughnuts are a different beast […] : Turning an orb into a ring requires slicing a hole in it or sticking its ends together, making them two fundamentally different shapes.” We learn how Vazquez, as a math major at the National Autonomous University in Mexico, discovered that topology was just what she needed to link her interest in math with her curiosity about the natural world: It was a key to begin understanding how living creatures handle the hugely complicated task of copying and interpreting the genetic information encoded in their DNA.
The image below is not in Mandelbaum’s report, but it gives an elementary example of how topological structures occur in Vazquez’s earlier work on DNA. It comes from “DNA knots reveal a chiral organization of DNA in phage capsids” by Vazquez and five co-authors in Proceedings of the National Academy of Sciences, June 28, 2005. They were studying DNA from a virus, the bacteriophage P4 (the phage). The DNA extracted from the head (capsid) of this virus presents a large proportion of highly knotted DNA circles. One conclusion of their research is that those knots capture information about the original packing of the DNA.
Ellen Phidian centributed “Molecular Möbius strip: chemists make a geometric anomaly from atoms” to the web science magazine Cosmos (May 20, 2022). As Phidian explains it, “sometimes, chemists want to make molecules that are simply geometrically interesting.” These adventurous experiments can have important real-life consequences—Phidian gives the example of carbon nanotubes. In fact the Nagoya University team (Yasumoto Segawa, Kosuke Itawa and collaborators) that synthesized the Möbius strip carbon nanobelt (MCNB) had made an untwisted carbon nanobelt (CNB), which is an ultra-short nanotube, back in 2017.
Both CNBs and MCNBs are assembled from hexagonal benzene rings (schematically represented in the figure above); what makes both syntheses hard is wrestling a strip of rings into the proper shape. As Phidian describes it: “Unlike a paper Möbius strip, the molecule took more than some scissors and tape to create. In fact, because of its unusual shape, placing strain on the carbon atoms in its belt, making the molecule was deeply complicated.” Segawa et al. report the synthesis of the MCNB to have taken 14 steps. Interestingly, they remark “a CNB can be generated when the number of repeat units is even, whereas an MCNB can be obtained when the number is odd.”
Once synthesized, the MCNB is a very lively molecule: Segawa and collaborators report that, as predicted in 2009, the twist travels rapidly around the belt. The period is on the order of several picoseconds ($10^{-12}$ seconds) at room temperature.
“How to Slice a Pie in Four Dimensions, According to Math” ran in Popular Mechanics on May 17, 2022. The author, Juandre, tells us: “If you’ve ever drooled over a pie, cut into eight beautifully equal slices, you’re already a little bit familiar with the concept of equiangular lines—those that intersect at a single point, with each pair of lines forming the same angle.” This is not quite right: with eight equal angles, some lines intersect at $45^{\circ}$ and some at $90^{\circ}$. (We take the angle between two lines to be the smaller of the two angles they form). But changing the eight to six gets us out of trouble.
Notice that slicing a pie in four equal slices, or taking the three coordinate axes in 3-space, also produces a set of equiangular lines. The examples in the figure above represent the largest possible collections of equiangular lines in 2 and 3 dimensions. What happens in higher dimensions? Juandre seems to imply that researchers recently solved all the higher dimensional cases of equiangular lines, but in fact the maximum number $N(d)$ of equiangular lines in $d$-dimensional space is not known exactly in general. Some recent contributions to the problem are in Equiangular Lines in Low Dimensional Euclidean Spaces published last year (ArXiv version here), including $N(14)=28$ and $N(16)=40$.
The work that motivates Juandre’s posting addresses a modification of the question. That is, given a fixed angle, what is the maximum number of lines in $d$-space which are pairwise separated by that angle? This is the problem that was recently solved (for values of $d$ sufficiently large) by Zilin Jiang, Jonathan Tidor, Yuan Yao, Shengtong Zhang, and Yufei Zhao, all of them at MIT at the time. Yao and Zhang (both 2018 Putnam Fellows) were undergraduates, Tidor (MIT ’17) was a graduate student, Jiang was a postdoc, and Zhao (MIT ’10) was an assistant professor. Their work was published in the prestigious Annals of Mathematics on November 2, 2021.
]]>The New York Times, May 1, 2022
Have you ever wondered how instruments make music? Pluck a guitar string, and watch it vibrate. Vibrations reverberate in the instrument’s wooden body to sustain a note, and the geometry of the body and string influences the sound. The same mathematical music exists in other objects that may not seem musical—even construction tools. In an article for The New York Times, Nicholas Bakalar writes about the math of musical saws. Musicians have played the saw for around 200 years. They must bend the tool into an S shape and draw a bow across a precise sweet spot to sustain a note. Recently, researchers analyzed the math behind that bend. One scientist told Bakalar that the saw “sings,” like a violin: “Musicians have of course known this experientially for a long time, and scientists are only now beginning to understand why the saw can sing.” (See also this Harvard article.)
Classroom activities music, frequency and wavelength, geometry, inflection points
—Max Levy
The Atlantic, May 13, 2022
You’ve been invited to a party! You’ve figured out what to wear and what to bring—but when should you arrive? Too early, and you might enter an awkwardly empty room. Too late, and you might miss out on the best conversations. Maybe math can help. Mathematician Daniel Biss devised a (somewhat silly) formula to calculate your personally optimal arrival time. It includes seven factors, such as how punctual your friends are and how excited you are about the party. This article by Joe Pinsker makes Biss’s formula interactive and goes on to explore the tension between “clock time” and “event time.”
Classroom activities: mathematical modeling, algebra
—Scott Hershberger
NPR, May 11, 2022
This year, the entire US has dealt with rising economic inflation at levels not seen in decades. Basically, the price of stuff is going up shockingly fast. A loaf of bread. Gasoline. A pair of jeans. Stuff that’s essential for people to subsist and hold down a job that allows them to subsist on their own. Over time, the prices naturally inch up—many economists consider around 2% per year to be ideal. But compared to this time last year, consumer prices are up an average of 8.3%, according to the most recent data. In this article, Scott Horsley discusses a possible fix with roots in economic math—raising federal interest rates—and why some communities are disproportionately burdened. “When inflation is high, everyone pays the price, but research suggests that lower-income families suffer the most.”
Classroom activities: economics, inflation, supply and demand, data analysis
—Max Levy
Science News, April 27, 2022
On a winter walk in the woods, you might notice that the bare trees around you have intricate shapes. The branches of deciduous trees fork into smaller and smaller offshoots until they end in narrow twigs. To describe how branches divide, Leonardo da Vinci proposed a mathematical rule: When a tree limb splits into smaller branches, the area of a cross-section of the big branch is the same as the sum of the cross-sectional areas of the smaller branches. In an article for Science News, James R. Riordon reports on a new paper that tweaks Leonardo’s rule. Instead of cross-sectional area, the researchers model tree branching using surface area, which uses information about the length of a branch as well as its diameter. The paper also incorporates more advanced techniques such as numerical Fourier analysis, but the researchers still consider their rule to be “Leonardo-like.”
Classroom activities: geometry, area, circles
—Tamar Lichter Blanks
Frank Wilczek’s column in the the Wall Street Journal (April 14, 2022) had the title “A Quantum Leap, With Strings Attached; The Inca system of quipu—tying a series of knots to record information—is providing a surprising model to modern physics and quantum computing.”
This textile document is a typical quipu in that its data is numerical and recorded in a decimal system. (For another example and more details, see Nicole Rode’s YouTube video from the British Museum). While they were used by earlier pre-Columbian Andean cultures, most of the surviving specimens date from the period of Inca domination, c. 1400–1532 CE. (We can only guess what the numbers recorded on quipus were actually counting —these civilizations left no written records).
Wilczek compares quipus with the information storage and transmission systems we encounter today: “written human language, the binary code of computers and the DNA and RNA sequences of genetics,” and remarks that the Andean system involves “something unique: topology, the science of stable shapes and structures.” In fact the difference between one knot and another, which is contrastive in quipus, to borrow a term from linguistics, is one of the most basic examples of a purely topological concept.
The connection between quipus and “modern physics and computing theory” comes precisely through topology. The equivalents of Andean cords are the world-lines of particles. Wilczek asks us to suppose our particles are only moving in two dimensions, and that we add a third dimension to represent time. Then as time progresses the successive positions of a particle trace out a curve: this is its world-line. And if several particles are observed at once, their world-lines can tangle (“these are not our ancestor’s strings”) and form what mathematicians call braids.
“There are certain particles, called anyons, whose quantum behavior keeps track of the braid that their world-lines form [and therefore could be used to store information]. The anyon world-lines form a quantum quipu.” Wilczek’s terminology has to be taken with a grain of salt. Knots and braids are very different mathematical objects, although fundamentally related (see the drawing above and Alexander’s Theorem on Wikipedia); quipus use one and not the other. Nevertheless it is striking that the topology of curves in space turns up both in an antique recording system and in the latest quantum science.
The science really is very new. Anyons were only experimentally detected two years ago (Wilczek himself had conjectured their existence, and named them, some 40 years back). He tells us that “the simple quantum quipus that were produced in those pioneering experiments can’t store much information” but that only last month “Microsoft researchers announced that they have engineered much more capable anyons.” This is presumably the research described in the Microsoft Research Blog on March 14.
Siobhan Roberts contributed Is Geometry a Language That Only Humans Know? to the March 22, 2022 New York Times. The subtitle is more specific: “Neuroscientists are exploring whether shapes like squares and rectangles — and our ability to recognize them — are part of what makes our species special.” The neuroscientists in question are Stanislas Dehaene (Université Paris-Saclay and Collège de France) and his collaborators.
The first part of Roberts’s article concerns the research that Dehaene and his team published last year in PNAS: “Sensitivity to geometric shape regularity in humans and baboons: A putative signature of human singularity.” In a typical experiment they report, subjects were presented with a display of polygons. Five of the six were similar, differing only in size and orientation; the sixth was like the others except that its shape had been changed by moving one vertex. Subjects were asked to pick out the oddball.
The “normal” polygons were chosen from a family of eleven quadrilaterals that can be ranked, starting with a square, by how unsymmetrical they are.
The first experiment, involving 605 French adults, showed that the number of errors they made “varied massively” with the lack of symmetry/orthogonality/parallelism of the “normal” exemplar.
The team repeated the experiment with French kindergarteners and with Himba adults (“a pastoral people of northern Namibia whose language contains no words for geometric shapes, who receive little or no formal education, and who, unlike French subjects, do not live in a carpentered world.”) The results correlated strongly with those of French adults. “Both findings converge with previous work to suggest that the geometric regularity effect reflects a universal intuition of geometry that is present in all humans and is largely independent of formal knowledge, language, schooling, and environment.”
The experimenters had access to a colony of baboons (Papio papio) in the south of France; they managed to train the baboons to where they had a “clear understanding of the task”—they could recognize the oddball apple in a group of watermelon slices, and even a regular hexagon in a group of non-convex polygons, but “although error rates differed across the 11 shapes, with a consistent ordering across baboons, […] they correlated weakly and nonsignificantly with the geometric regularity effect found in human populations.”
After speaking with Moira Dillon (a psychologist at New York University) Roberts puts this research in a historical context: “Plato believed that humans were uniquely attuned to geometry; the linguist Noam Chomsky has argued that language is a biologically rooted human capacity. Dr. Dehaene aims to do for geometry what Dr. Chomsky did for language.” But Frans de Waal (a primatologist at Emory University) cautioned her: “Whether this difference in perception amounts to human ‘singularity’ would have to await research on our closest primate relatives, the apes.”
Roberts reviews connections between this research and work in artificial intelligence, and then moves on to Dehaene et al.‘s latest project, essentially figuring out what in the human mind makes geometric regularity so significant. Here’s a clue, quoting from Dehaene: “We postulate that when you look at a geometric shape, you immediately have a mental program for it. You understand it, inasmuch as you have a program to reproduce it.” The team explored an algorithm, DreamCoder (the authors overlap with Dehaene’s collaborators) that “finds, or learns, the shortest possible program for [drawing] any given shape or pattern.” Then they tested human subjects on the same shapes. “The researchers found that the more complex a shape and the longer the program, the more difficulty a subject had remembering it or discriminating it from others.”
]]>Discover Magazine, April 7, 2022
If you’re a sports fan, you’ve likely witnessed the “home-field advantage”: Teams tend to win more often when they’re playing at their home venue. Statistical studies have shown that this phenomenon is real. As Cari Shane explains, the primary cause of the home-field advantage is that hometown fans exert a subconscious sway over referees, leading to a few more calls in favor of the home team. Travel-related factors contribute, too. But in recent years, the magnitude of the advantage has been decreasing, statistical analyses show. Rule changes such as the addition of instant replays and coaches’ challenges are leveling the playing field. Plus, travel isn’t as hard on athlete’s bodies as it used to be, thanks to increased budgets and advances in sports medicine.
Classroom activities: statistics, sports
Related Mathematical Moments: Holding the Lead.
—Scott Hershberger
Wired, April 1, 2022
How fast should you drive to work? You can find a precise answer to that question, at least if your goal is to save money. In an article for Wired, physicist and science popularizer Rhett Allain explains how to calculate the most cost-effective driving speed given your car’s fuel efficiency, your hourly pay rate, and the price of gas. Allain says that the cheapest speed is not too slow (if you’re idling, you’re getting 0 mpg) and not too fast (because of factors like air resistance and friction), but at a sweet spot in the middle. As a bonus, Allain shares a snippet of code that does the math, which you can edit to calculate your own optimal speed.
Classroom activities: fractions, ratios, physics, optimization, coding, programming
basempg
, G
, R
, and dx
, then clicking the play/run icon. Try some of the following:
R
will increase or decrease the cheapest speed. Then test your prediction using the code.dx
have any effect on the optimal speed? (The answer to this question is in the article.)—Tamar Lichter Blanks
Popular Mechanics, April 25, 2022
In an April 20 tweet that attracted over 75,000 likes, attorney Paul Sherman claimed that “It is physically impossible to exceed the 70-pound domestic weight limit for a small flat rate box” shipped by the US Postal Service. A small flat rate box filled with pure osmium—the densest naturally occurring element—would weigh in at around 61.5 pounds, he wrote. In this article, writer Caroline Delbert compares osmium to other dense materials. While osmium may top the scales on Earth, trying to mail the material that neutron stars are made of would leave you on the hook for a hefty overweight charge.
Classroom activities: density, mass, volume, media literacy
—Scott Hershberger
Ars Technica, April 6, 2022
To Ben Orlin, a teacher and author, playing is essential to learning. That’s especially true in math. Orlin recently wrote a book called Math Games with Bad Drawings. It’s an illustrated collection of multiplayer games in which intriguing puzzles emerge from simple mathematical rules. You just need household items like pencil and paper, your hands, the internet, and Goldfish crackers, writes Jennifer Ouellette in Ars Technica. Ouellette interviewed Orlin about his new book, which includes more than 50 “math-y” games. According to Orlin, the book doesn’t try to create educational puzzles for math-lovers. He instead sees each game as a thought experiment enjoyable for anyone. “Games are constantly generating new puzzles,” he says. “With a puzzle, you solve it, you’re done. A game is like a fountain of puzzles that’s constantly pouring out new puzzles for you.”
Classroom Activities: math games
—Max Levy
CNET, April 1, 2022
Math is a language, and humans aren’t the only creatures who understand it. Nature is full of animals that seem to perform simple mathematical operations. Animals as big as lions tally their competitors; animals as small as bees distinguish small numbers from large ones. The list is always growing as scientists put more animals to the test. In an article for CNET, Monisha Ravisetti writes about new research involving fish. Researchers from Germany taught cichlids and stingrays to add and subtract 1 from numbers up to 5. Their experiment worked like this: Fish can’t read our numerals, so the team showed the fish some number of blue or yellow shapes. Blue meant “add 1,” and yellow meant “subtract 1.” The fish earned treats for choosing correct answers and eventually, they learned. “The fish passed with flying colors,” writes Ravisetti.
Classroom activities: counting, symbolic math, numerals, box plots, p-values
—Max Levy
On March 14, 2022, the Times ran “Pi Day: How One Irrational Number Made Us Modern” by Steven Strogatz. Strogatz focuses on the nature of $\pi$ as a number that can only be reached by an infinite process, so that while we can calculate more and more digits of the decimal expansion of $\pi$, the knowledge of exactly where $\pi$ sits on the number line will always be out of reach.
As Strogatz tells us, the mathematician who first made the process explicit was Archimedes (c. 287–c. 211 BCE). The Greeks had already been using the approximation of curves by polygonal lines—for example, this is how Euclid proved that the area of a circle is proportional to the square of its diameter. But determining the exact constant of proportionality does not seem to have been on his agenda. (Euclid did not explicitly consider the ratio of circumference to diameter.) It was left to Archimedes, who was an engineer as well as a mathematician, to try to nail it down.
Strogatz leads us through the process, imagining measuring the length of a circular track by walking around it and counting your steps, and multiplying that number by the length of your stride. As he explains it, each of your steps is a shortcut “in place of what really is a curved arc,” so the product you obtain will underestimate the actual length of the track. But by taking smaller and smaller steps you can get better and better estimates.
Archimedes started with an inscribed regular hexagon as a first approximation to a circle (say, of radius $r$).
Since a regular hexagon is made up of six equilateral triangles, the perimeter of a regular hexagon inscribed in a circle of radius $r$ is equal to $6r$. This gives a lower bound of $3$ for the ratio of circumference to diameter. As Strogath tells us, Archimedes went on to calculate the perimeters of inscribed regular polygons with 12, 24, 48 and 96 sides.
What allowed Archimedes to do these calculations is the (relatively) simple relation between the side of an inscribed regular $n$-gon and the side of the $(2n)$-gon obtained by putting a new vertex halfway between each pair of original adjacent vertices.
The red chord cuts the radius bisecting it into two segments of lengths $x$ and $y$, with $y=r-x$ . The calculation involves two applications of the Pythagorean Theorem. In triangle $*$, the theorem gives $x^2 = r^2 – (\frac{a}{2})^2$. Then in triangle $**$, it gives $b^2 = (\frac{a}{2})^2 + (r-x)^2$. The two steps, each involving extraction of a square root, give $b$ in terms of $a$ and $r$.
Archimedes repeated this calculation three more times, starting with the hexagon and $a=r$ and ending with the side of a $96$-gon. This allowed him to set $3 + \frac{10}{71}$ as a lower bound for $\pi$, while a similar calculation with circumscribed polygons yielded $3 + \frac{10}{70}$ as an upper bound. As Strogatz describes it, “The unknown value of pi is being trapped in a numerical vise, squeezed between two numbers that look almost identical, except the first has a denominator of 71 and the last has a denominator of 70.”
For Strogatz the accuracy of Archimedes’ estimates is less important than the model he gave of linear approximations to curves and the importance of iterative calculations giving sharper and sharper bounds on a number of interest. As he puts it, “Archimedes paved the way for the invention of calculus 2000 years later.”
The multimedia web portal Big Think ran a piece by Michael Brooks on February 2, 2022, with a nonstandard perspective on mathematical research: “More math, more money: How profit-seeking has sparked innovations in mathematics.” Brooks, the author of The Art of More: How Mathematics Created Civilization (Penguin Random House, 2022), tells us here that while we often think of math as somehow above everyday, sordid life, in fact “math and money are like Bonnie and Clyde.”
The article starts off with a reference to the “The Anatomy of a Large-Scale Hypertextual Web Search Engine” on the Stanford website, where Sergey Brin and Larry Page give among other things the mathematical definition of PageRank, an idea we know turned out to be worth a big chunk of change. But it didn’t start there. Brooks links to an article on Mesopotamian Bronze Age Mathematics by the eminent historian of science Jens Høyrup to show us how back in Ur, about 4000 years ago, King Shulgi used his knowledge of arithmetic to “implement a kingdom-wide, tamper-proof accounting system,” prevent fraud in the collection of taxes, and fill his coffers.
Moving rapidly through the centuries, Brooks takes us to When pirates studied Euclid, where Margaret Schotte (York University, Toronto) tracks in wonderful detail the merchant and privateers in 17th-century Europe as they attended special academies to learn how to use astronomy and spherical trigonometry “to deliver goods faster or, in the case of the pirates, perform better interceptions.”
Additional illuminating examples cover algebra, calculus, and statistics. The moral of the story: “No one should be aiming to become a singer or a sports star. Math is a much more reliable road to riches.”
“The geometry of ancient Greece has stood for more than two millennia, even after relativity and quantum mechanics” is the rest of the title of a short article by Frank Wilczek. The Nobel prize-winning physicist is a regular contributor to The Wall Street Journal, where the piece ran on February 4, 2022.
Euclid’s Elements was written around 300 BCE. It built geometry up from some more or less obvious definitions and from a few “axioms,” supposed to be self-evident assumptions about how our geometrical world works. As Wilczek reminds us, the Elements, beyond training generations of students “not only in the science of space and measurement but in the art of clear thinking and logical deduction,” provided the framework for Newton’s physics and later for Maxwell’s theory of electromagnetism.
But then something happened. Euclid’s Axiom 5, the “parallel postulate” (equivalent to the statement that the angles in a triangle add up to $180^{\circ}$), felt different from the others; in the 19th century, mathematicians realized that it could be modified in two different ways and still give a coherent, uniform geometry. In one direction the sum of the angles in a triangle is always more than $180^{\circ}$. This corresponds to the situation on the surface of the Earth, where navigators build triangles out of segments of great circles, the spherical equivalent of straight lines. In the other direction, the sum is always less than $180^{\circ}$. This second geometry was discovered by Gauss (who didn’t publish it) and independently by Lobachevsky and Bolyai. As opposed to the curvature of a spherical surface, which we call positive, these “hyperbolic” surfaces have negative curvature, like the surface of a saddle, curling up in one direction and down in the other.
Not many years later Bernhard Riemann came up with the concept of what we now call a Riemannian manifold, where the curvature can vary from point to point. The equivalent of a straight line on such a surface (a path giving the shortest distance between two of its points) could be quite far from straight. Wilczek gives an example from sports: “an Alpine skier racing down a bumpy mountain will keep doing her best to go straight down, but over the course she will trace a curve.” Riemann himself was interested in the geometry of the three-dimensional space we live in, and his manifolds can have 2, 3, 4, … any number of dimensions.
Wilczek leads us through the next development. It started with Einstein’s special theory of relativity, developed to explain the non-intuitive fact that your measurement of the speed of light does not depend on how fast you are moving. The mixture of space and time coordinates that Einstein required was given a geometric formulation by Minkowski who stated in 1908, as Wilczek quotes him, “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.” Einstein used this formalism for his next step, the general theory of relativity, where space-time has curvature, analogous to the Riemannian concept we just saw, and intimately related to gravity. (The measurement, made during the 1919 total solar eclipse, of how much light rays—by definition, straight lines in space-time—would curve when passing close to the sun was experimental proof of the theory.)
This sounds pretty far removed from similar triangles and the Pythagorean theorem, but Wilczek assures us that “Einstein’s framework is still recognizably Euclidean, extended and adapted to bring in time and large-scale curvature.” In fact, he considers this “the most striking example of what Eugene Wigner called ‘the unreasonable effectiveness of mathematics in the natural sciences.'”
Kate Golembiewski wrote “Life’s Preference for Symmetry Is Like ‘A New Law of Nature,'” including the above image, for The New York Times. It appeared in the Trilobites section online on March 24, 2022, and a shorter version appeared in print on the 29th. The article reports on “Symmetry and simplicity spontaneously emerge from the algorithmic nature of evolution” by Iain Johnston, Kamaludin Dingle, and collaborators, published in PNAS on March 15.
As we know, symmetry is everywhere in nature: Starfish have 5-fold symmetry, flowers have bilateral or rotational symmetry (of various orders), virus capsids have icosahedral symmetry, our left hands are almost exact mirror-images of our right. The question is why. “Biologists aren’t sure,” Golembiewski tells us, “—there’s no reason based in natural selection for symmetry’s prevalence in such varied forms of life and their building blocks.”
It turns out that the study of algorithms can point to an answer. As Golembiewski relates, the authors of the PNAS paper concluded that symmetry is evolutionarily favored because it makes the storage and replication of genetic data easier and more reliable. “Dr. Johnston […] likens it to telling someone how to tile a floor: It’s easier to give instructions to lay down repeating rows of identical square tiles than explain how to make a complex mosaic.” This is an explanation distinct from natural selection! Chico Camargo, another of the authors, told her: “It’s like we found a new law of nature. […] This is beautiful, because it changes how you see the world.”
]]>Dot Esports, February 28, 2022
It was only a matter of time until someone came up with an even nerdier version of Wordle, the viral word game. Enter: Nerdle. Data scientist Richard Mann created Nerdle for people who enjoy math games, according to writer Sage Datuin. In Nerdle, players guess equations instead of words. Each equation contains digits, operations ($+$, $-$, $\times$, and/or $\div$), and one equals sign. Players must guess equations until they have the right pieces to find the mystery equation. Much like Wordle, you only get 6 guesses, and each guess has to be mathematically correct—or else the game will tell you, “This guess does not compute!” In this article, Datuin highlights the differences between Nerdle and Wordle and shares some tips on how to win.
Classroom activities: Nerdle, combinatorics
—Max Levy
Quanta Magazine, March 24, 2022
Is it possible for every person at a party to shake hands with an odd number of people? Patrick Honner suggests you try it out at your next social gathering. In this article, he explains the solution using the tools of graph theory. The puzzle connects to the concept of subgraphs, which is an open area of research. The latest advance in describing odd subgraphs, Honner writes, came just last year.
Classroom activities: graph theory, even and odd numbers
—Scott Hershberger
Scienceline, March 4, 2022
Every computer you own is in a sense the same under the hood: Desktops, laptops, and phones all run on the same kind of math. Encryption is designed with this in mind. But quantum computers are different, exploiting properties of quantum physics to do their calculations. This allows them to solve certain problems that have been the basis of cryptography until now, such as factoring large numbers efficiently. Quantum computers are hard to build—they require extremely cold metals and other specialized engineering—but if they get big enough, they’ll be able to break much of the encryption that keeps digital communications private. The US National Institute of Standards and Technology is running a competition to find new methods of encryption that are safe against both regular and quantum computers. Most of the finalists rely on mathematical objects known as lattices, as Daniel Leonard writes in Scienceline.
Classroom activities: cryptography, quantum computing, number theory
Related Mathematical Moments: Securing Data in the Quantum Era.
—Tamar Lichter Blanks
Scientific American, March 14, 2022
March 14, or $\pi$ (Pi) Day, has turned $\pi$ into the most famous number. Writing in Scientific American, mathematician Alissa S. Crans takes issue with pi’s fame. “I’m tickled that honoring something mathematical has become a widespread phenomenon,” she writes. “But, at the same time, I’m disappointed that this numerical celebrity seems to be somewhat of an accident.” Crans wants people to understand that math is more than just weird numbers. Math is full of fascinating mysteries and clever solutions to deceptively hard problems. In that spirit, the International Mathematical Union has turned March 14 into the annual International Day of Mathematics. In this article, Crans shares examples of other fascinating facets of math worth celebrating every March.
Classroom activities: pi, cake-cutting, irrational numbers, infinity
—Max Levy