Math Digests September 2021

How Math Solved the Case of the Volcanic Bombs That Didn’t Explode

The New York Times, September 11, 2021

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

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

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

—Scott Hershberger


Algebra: the maths working to solve the UK’s supply chain crisis

The Guardian, September 12, 2021

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

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

  • (Algebra, geometry) In discussing the history of algebra, Brooks mentions a medieval text entitled “Problems to Sharpen the Young” that contains several era-appropriate word problems.
    • After solving some of the problems, ask students to write their own modern-day versions of these word problems. Have pairs of students exchange their new problems and try to solve them.
  • (Pre-calculus) Imagine a truck departing Albany, NY with packages to be delivered to all 48 of the US mainland’s state capitals before returning to Albany. Have students try to guess the shortest route. Afterwards, compare their result to the answer found by data scientist Randy Olson. How close were they?
    • When teaching permutations and combinations, ask students to calculate the number of possible routes using a tool like WolframAlpha. If a supercomputer can test 200 quadrillion ($2\times 10^{17}$) routes per second, how long would it take to test all of the possibilities by brute force? Compare this to the age of the universe.

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

—Leila Sloman


Mathematicians discover music really can be infectious – like a virus

The Guardian, September 22, 2021

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

Classroom Activities: exponential growth, modeling in Excel

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

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

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

—Max Levy


The Simple Math Behind the Mighty Roots of Unity

Quanta Magazine, September 23, 2021

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

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

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

—Leila Sloman


Why is an egg shaped like an egg? Turns out, there’s some serious math behind it

ZME Science, September 10, 2021

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

Classroom Activities: symmetry, geometry, functions

  • (All levels) Birds lay all sorts of eggs. Ural owls lay almost spherical eggs, emus lay ellipsoid-shaped eggs, ospreys lay ovoid eggs, and king penguins lay pyriform eggs. Figure 7 in an earlier paper cited by the researchers shows the corresponding shapes in 2D: circles, ellipses, ovals, and pyriforms.
    • Where are the axes of symmetry for each?
    • Is one of these shapes more symmetrical than the rest?
    • How do the other shapes compare in terms of their symmetry?
    • (High school) For the two shapes that have the same type of symmetry (oval and pyriform), discuss why you think one is more complicated to describe mathematically than the other.
  • (High school) The mathematical formula described in this new study depends on four variables: the egg’s length $L$, its maximum width, the location of the line of maximum width, and the width at a distance $L/4$ from the pointed end. Put students in pairs and have them do the following. Using graph paper (hidden from your partner), sketch out some shape that is reasonably simple, yet more complicated than a circle or regular polygon. Now, try to come up with words and numbers to describe your shape. Put it to the test by having your partner draw your shape using just your description. Compare and discuss what conditions make this task easy or hard.

—Max Levy


Some more of this month’s math headlines:

 

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