Math Digests December 2021

Adele’s ‘30’: A mathematician explores number patterns in album titles

The Conversation, December 1, 2021

Pop superstar Adele has a habit of titling her albums in a peculiar way. Each album title is an integer that represents how old the singer was when she began writing the songs. David Patrick of the Art of Problem Solving searched the On-Line Encyclopedia of Integer Sequences to see if Adele’s albums match any known integer sequence. Nine sequences turned up—some of them easy to describe and others less so. For The Conversation, Anthony Bonato explains the rules underpinning one of the possible “Adele sequences.”

Classroom Activities: sequences, prime numbers

(Middle school / high school) Explore the OEIS by having students come up with the first four terms of an integer sequence. They can choose four integers based on something in their own life, or they can choose four integers less than 40 that could represent the ages at which a musician releases new albums.
- Have students search the OEIS to see if their sequence matches any known sequences in mathematics, then share their findings with the class.
- Discuss: what do the results suggest about the mathematical significance of the “Adele Sequence”?
(Number Theory, Problem-solving) The sequence A072666 that Bonato analyzes depends on the sequence of prime numbers. Ask students to prove that there are infinitely many primes using the following hints (full solution here):
- Assume that there are finitely many primes which can be denoted $p_1,p_2, \dots, p_n$.
- Consider the number $(p_1 \times p_2 \times \dots \times p_n) + 1$.

—Leila Sloman

From icebergs to smoke, forecasting where dangers will drift

Science News for Students, December 16, 2021

Turbulence in water affects where fish eggs end up. Massive icebergs drift on ocean currents. Wildfire smoke from the U.S. West can make its way to Europe. In all of these cases, mathematics helps researchers predict what will happen, offering potential solutions to environmental threats. As Rachell Crowell explains, the computer models used to simulate how objects drift depend on the math of differential equations. These equations relate quantities that change as time passes and vary at different locations in the environment. Crowell interviews several researchers about what they have learned about the science of drifting and why it is important.

Classroom activities: mathematical modeling, differential equations, heat equation

(Algebra I, Algebra II) The article mentions that satellites helped reveal drifting wildfire smoke. Satellites are also useful in studying the fire itself. Explore one real-life example with this assignment from NASA’s Jet Propulsion Laboratory.
(Differential Equations) The article also mentions that researchers use the heat equation to predict icebergs’ melt rate. When introducing partial differential equations, show students this 3Blue1Brown video that visually explains the heat equation.

—Scott Hershberger

Unraveling How an Extinct Mollusk Got Its Strange Shell

The New York Times, December 10, 2021

Math shapes the world around us—and it shapes the shapes around us too. A trio of researchers from France and the United Kingdom are known for investigating the math and physics of how seashells form. Their newest advance deals with ammonites, an extinct group of mollusks. A type of ammonites called Nipponites had weird shells that twisted, turned, coiled, and bulged in unexpected directions. “The first time you look at it, it’s just this tangled mess,” mathematician Derek Moulton told reporter Sabrina Imbler. “And then you start to look closely and say, oh, actually there is a regularity there.” In this article, Imbler shares how Moulton’s team came up with a mathematical model that explains the origins of these ammonites’ weird shapes.

Classroom activities: growth rates, symmetry, golden ratio

(All levels) Nipponites appear so strange because they are very asymmetrical, whereas nature is normally full of symmetry. Ask students why symmetry tends to be beneficial in the living world. (As an example, think of symmetry in birds and fish as they move.)
(All levels) Nipponites provide one example of an initial symmetry giving way to an asymmetric result. Have students explore another example using a thin string. Hold the string above a table so it hangs freely. Now lower it gently so that one end touches the table, and continue to lower it until it has folded over itself a few times. Inspect the bundle and take a picture. Repeat this a few times and share your observations. Does the bundle look the same every time or does it appear unique? Discuss how this compares to what happens inside an ammonite shell.
(Algebra II, Pre-calculus) To learn more about how a simple mathematical rule can give shells and flowers fascinating shapes, watch this Numberphile video on the Golden Ratio.

Related Mathematical Moments: Going Into a Shell.

—Max Levy

Who Was Lotfi Zadeh? Google Doodle Honors the Azerbaijani American Scientist

Newsweek, November 30, 2021

When Lotfi Zadeh first came up with the idea for fuzzy logic, a way of enshrining in mathematics the uncertainty and imprecision of life, it was not received well by everyone. But almost sixty years later, Zadeh’s impact on the world is indisputable. On November 30, a Google Doodle celebrated this impact, marking the 57th anniversary of Zadeh’s seminal paper, “Fuzzy Sets.” For Newsweek, Soo Kim articulates what was so important about that work: “Considered an early approach to artificial intelligence, Zadeh’s fuzzy logic structure formed the basis of various modern everyday technologies including facial recognition, air conditioning, washing machines, car transmissions, weather forecasting, stock trading and rice cookers.”

Classroom Activities: set theory, logic, fuzzy logic

(High school) Teach an introduction to set theory with this online lesson, assigning the “Try It Now” boxes as problems. For extra practice, assign the following problems:
- Let $S = \{ 1, 10, 19 \}$. Write down all the subsets of $S$.
- Suppose the universal set is all the integers. Let $A = \{ \text{even integers} \}$ and $B = \{ \text{multiples of 3} \}$.
  - Find 3 numbers in $A^c \cap B$.
  - Find 3 numbers in $A \cap B^c$.
- (Advanced) Prove that for general sets $A$ and $B$, $(A \cap B)^c = A^c \cup B^c$.
(High school) In classical logic, an item either belongs to a set or it doesn’t—there is no in-between. By contrast, fuzzy logic allows items to be partly in sets. Have students read this Britannica Kids explainer on fuzzy logic. Ask them to think of three situations in which fuzzy logic might be more useful than classical logic and justify their claims.
- (Advanced) Have students read the first four pages of Zadeh’s original paper and then come up with an example of two membership functions $f_A(x)$ and $f_B(x)$ in the real numbers such that $A$ is a fuzzy subset of $B$.

—Leila Sloman

DeepMind’s AI helps untangle the mathematics of knots

Nature, December 1, 2021

Mathematics might sometimes seem very dry. You follow fixed rules to find right answers and check your work carefully to avoid wrong answers. That process doesn’t seem creative, but when it comes to discovering new proofs and theorems, creativity is arguably the most important trait. In an article for Nature, Davide Castelvecchi explains how math researchers have recruited a new ally to track down creative solutions to math’s mysteries—artificial intelligence. Researchers used machine learning, a type of AI, to comb through huge datasets and find patterns related to the study of knots and symmetry. It takes creativity and intuition to find these hidden patterns. “As mathematical researchers, we live in a world that is rich with intuition and imaginations,” one researcher said. “Computers so far have served the dry side. The reason I love this work so much is that they are helping with the other side.”

Classroom activities machine learning, knots

(All levels) The machine-learning algorithm helped researchers solve a question about knots that eluded researchers for decades. Read more about this study’s “knot” result and/or watch this Numberphile video about knot theory.
(Middle level) A central question that researchers ask is whether a set of knots that appear different are actually equivalent. With a shoelace, string, or yarn, try making some of the distinct knots found here. Discuss what properties make them different. (For example, how many times the thread crosses over itself.)
(High level) Discuss why machine learning can speed up innovation, based on this article and others. (Here’s a primer on machine learning.) One researcher told the writer: “Without this tool, the mathematician might waste weeks or months trying to prove a formula or theorem that would ultimately turn out to be false.” If artificial intelligence can help find patterns quickly, what other areas of science or society could benefit from these quick solutions? In what situations would machine learning be more dangerous or unethical?

Related Mathematical Moments: Being Knotty.