# Math Digests July 2023

June’s record-smashing temperatures — in data

Nature, July 5, 2023.

This June, extreme heat scorched across the northern hemisphere. “Records for individual climate phenomena have been broken in previous years, but this June felt different,” writes Katharine Sanderson for Nature. Sanderson is referring to records of climate data kept since 1850. In all that time, neither air temperatures nor sea surface temperatures have been so high in June. Even in the southern hemisphere, where June is winter, it’s been unseasonably warm: Antarctic sea ice reached a record low compared to previous Junes. Sanderson’s article breaks down important metrics that help us measure just how hot Earth is becoming due to climate change.

Classroom Activities: climate data, scientific analysis

• (All levels) Some data in the article come from Climate Reanalyzer. Use the website to answer the following:
• List three metrics, other than air temperature, that climate scientists can use to compare climate from year to year.
• What does “2m air temperature” mean? Why is the “2m” important?
• Describe what the “Daily 2m Air Temperature” plot is revealing. What can you conclude about this year so far?
• (Mid level) Create a table showing the extent of northern hemisphere sea ice (in million km$^2$) on September 1 of the first 5 years recorded and the newest 5 years recorded.
• What is the average over the first 5 years? Over the most recent 5 years?
• What is the standard deviation for each set?
• Make a conclusion based on your analysis.
• (Mid level) Follow this Climate Data Activity from UCAR (University Corporation for Atmospheric Research).
• (High level) Learn about probabilities and uncertainty related to climate change with this activity from NOAA (National Oceanic and Atmospheric Administration).

—Max Levy

Evelyn Boyd Granville, barrier-breaking mathematician, dies at 99

The Washington Post, July 14, 2023.

Mathematician Evelyn Boyd Granville, who made important contributions to the American space program, died at age 99 on June 27, 2023. Granville, who received her doctorate in mathematics from Yale University in 1949, was one of the first Black women to obtain a mathematics PhD in the United States. In an article for The Washington Post, Brian Murphy describes some of Granville’s experiences, her teaching, and her role in the space program. As a mathematician at IBM and at North American Aviation, Granville worked on NASA satellites, on the Mercury program to launch the first American into space, and on moon landing calculations for the Apollo missions. Granville also taught mathematics in multiple universities and wrote a college mathematics textbook that was used in more than 50 schools.

Classroom Activities: space, pi, physics, astronomy

• (Geometry) Evelyn Boyd Granville worked on mathematical problems for NASA. To see one way that math is used in the space program, take a look at this NASA poster about using $\pi$ to measure properties of planets and other objects in space. Discuss: did any of the uses of $\pi$ described in the poster surprise you? Was there anything you found especially interesting?
• The poster says that using $\pi$ and an asteroid’s radius and mass, scientists can calculate the asteroid’s density and find out what it is made of—for example, ice, rock, or iron. If a sphere-shaped asteroid has radius $r$ and mass $m$, what is its density? (Hint: the density of an object is its mass divided by its volume.) Once you know the density of an asteroid, how would you figure out what it is made of?
• (All levels) Evelyn Boyd Granville’s work included calculations for orbit trajectories. Recently, astronaut Chris Hadfield and mathematician Matt Parker discussed the mathematics of orbits in a video on YouTube.
• Watch the video from time 6:57 to time 10:28. Note that in this video, $v$ refers to the speed of the spacecraft in orbit, $G$ is a number called the gravitational constant, $M$ is the mass of the earth, and $r$ is the radius—that is, the distance between the spacecraft and the center of the Earth.
• Hadfield said that when he was in orbit at about 420 kilometers above the Earth’s surface, he was moving at a speed of about 8 kilometers per second. Based on the video, if another astronaut had also been in orbit further away from the Earth, would that second astronaut have been moving at a faster, slower, or identical speed to Hadfield? How does your answer relate to the equation $v^2 = GM/r$?
• If you are interested in more mathematical details, you can watch the video from the beginning or check out the accompanying notes. For more about Hadfield’s experiences with orbits as an astronaut, watch the second part of the video, from 10:28 until the end.
• How did this video affect the way you think about orbits? What else would you like to understand about orbits, or about the math of space travel?

Tamar Lichter Blanks

Mathematicians find 27 tickets that guarantee UK National Lottery win

New Scientist, July 28, 2023.

Participants in the United Kingdom’s “Lotto” game try to predict the 6 winning numbers between 1 and 59. Players who get two numbers right win a free, randomly-generated ticket (a “Lucky Dip”) for the next game. If a player gets three or more numbers right, they win cash — potentially millions of pounds if they choose all six numbers correctly. In a new preprint, David Cushing and David Stewart of the University of Manchester show that you can buy just 27 tickets and guarantee yourself a free Lucky Dip. Matthew Sparkes reports on their finding for New Scientist.

Classroom Activities: combinatorics, probability

• (Mid level) In the Lotto, numbers are chosen “without replacement” — once a number is selected, it cannot be selected again.
• There are 45,057,474 possible draws of six distinct numbers in the real Lotto. Suppose you’re playing a game where tickets are made up of 2 distinct numbers between 1 and 5. (Call it Lotto(2,5).) How many possible draws are there? Find the smallest set of tickets that, if played together, guarantees you get at least one of the numbers right.
• Now suppose you’re playing Lotto(3,5), where tickets are made up of 3 distinct numbers between 1 and 5. How many tickets would you need to play to get at least two of the numbers right?
• In the alternate versions of Lotto, how many draws would be possible if the same number could be drawn more than once?
• (All levels) Sharpe points out that in the Lotto, each ticket costs 2 pounds, while the prize guaranteed by Cushing and Stewart’s proposed set of 27 tickets — a free Lucky Dip — may not result in any actual cash winnings. Indeed, on July 1, Cushing and Stewart won 3 Lucky Dips with their 27 tickets, and won nothing. How much would the prize have to be worth to guarantee that a player using the 27 tickets ends up gaining money?
• (Mid level, Probability) According to Lottery.co.uk, the chance of winning a cash prize in the Lotto is about 1%. Before playing their 3 Lucky Dips, what were Cushing and Stewart’s chances of a cash prize?
• (High level) In the previous question, you might have used the fact that the 3 Lucky Dips were independently generated. This is not true with the 27 tickets that Cushing and Stewart came up with. Do you think that their 27 tickets are more or less likely to result in a cash prize than 27 independently generated tickets?
• Try to justify your prediction by randomly generating tickets using this random number generator, and comparing to the winning sets of tickets you came up with for Lotto(2,5) and Lotto(3,5).

—Leila Sloman

How M.C. Escher Created His Mathematical Artwork

Popular Mechanics, July 6, 2023.

Mathematicians love M.C. Escher for his art depicting complex geometry and the concept of infinity. Yet the artist failed classes and never finished high school. In his twenties, he visited a part of Spain known for Islamic tilings that fit together in unusual ways. Those patterns inspired him. He spent years perfecting his own infinite patterns, known as tessellations. He also created “hyperbolic tessellations,” based on a geometry that deals with curved surfaces where parallel lines diverge infinitely. “Escher became excited about this drawing because he saw this technique could be used to capture infinity in a circle,” writes Kat Friedrich in this Popular Mechanics article, which describes how Escher created hyperbolic tessellations and other mathematical masterpieces.

Classroom Activities: tessellations, Poincaré

• (All levels) Create tessellations using this interactive web app.
• Which different shapes can fit together? Can triangles fit together with hexagons? With pentagons?
• Which shapes can tessellate with themselves?
• Which shapes cannot tessellate with themselves?
• (High level) Escher consulted with a mathematician to figure out arcs for his hyperbolic tessellations, using the Poincaré disk model. Hyperbolic geometry distorts lines and distances. In Euclidean geometry, the shortest distance between two points is a straight line. But in hyperbolic geometry, the shortest distance between two points is actually a curved path. The disk represents an infinite space — one inch at the edge represents a greater distance than one inch in the center. The distortions allow us to mathematically confine that infinite space to a circle. Use this interactive model of the Poincaré disk to explore the math behind Escher’s art. Explore this model and then click the “Circle Limit III” button to see Escher’s artwork.
• What do each of the three intersection points have in common with the artwork that the “Circle Limit III” button reveals?
• What do the curves between intersection points trace on the artwork?
• What do the medians, altitudes, angle bisectors, and circumcircle trace?

—Max Levy

Explainer: What is chaos theory?

Science News Explores, July 10, 2023.

Those of us who struggle with bowling may find it validating to know that, according to the magazine Bowling This Month, bowling is a chaotic system—meaning even a minuscule change in your form and throw can dramatically affect what happens when the ball reaches the pins. Many processes that occur around the world are chaotic like this, perhaps most notably the weather. In this explainer for Science News Explores, Sarah Wells details what it means for a system to be chaotic, and how scientists can analyze them using special states called “strange attractors.”

Classroom Activities: chaos theory, mechanics

• (Mid level) For a quick overview of chaos theory, watch the embedded video featuring Maren Hunsberger.
• (All levels) Hunsberger opens with a classic example of a chaotic system: The double pendulum. Play with this simulation of a double pendulum by myPhysicsLab. How is the system affected if you play with the rod lengths and the masses?
• The simulation creator, Erik Neumann, points out that the system is only chaotic when the angles are large. When the masses are both 2, describe the motion you see if you start the pendulum (1) about a 15-degree angle from the bottom, (2) a 45-degree angle from the bottom, (3) a 90-degree angle from the bottom (so, horizontally) (4) from the very top position.
• (High level) At what angle does chaotic behavior begin? How would you quantify this? By using the graph to quantify the pendulum’s position, verify whether or not your assessment of chaotic behavior is correct.
• (All levels) In this article, researchers Jonathan Borwein and Michael Rose give billiards as another example of a chaotic system. As a class, brainstorm five more examples of systems that you think might be chaotic.

—Leila Sloman

Some more of this month’s math headlines