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
- (All levels) In high-dimensional spaces, like those that Viazovska worked in, it’s tough to visualize geometric objects. For more information about math and geometry in higher dimensions, watch “A Journey into the 4th Dimension”.
- In the video we learn that we can imagine higher dimensions by picturing their projection onto a lower dimension. Can you draw a 2D projection of the following 3D shapes? 1) Sphere, 2) rectangular prism, 3) triangular prism from the top, 4) triangular prism from the side.
- (Mid level) Demonstrate a type of sphere packing with a small box (such as a shoebox) and ping pong balls. (The box should have straight sides and the balls should all have the same diameter.) Cover the bottom layer of the box with as many ping pong balls as possible.
- How many can you fit in that one layer? What is the area of the floor of the box? Looking from the top down, what fraction of the area is empty space? (Hint: calculate the total area covered by ping pong balls.)
- (High level) Now add a second layer of ping pong balls on top of the first. Place them such that you are maximizing the number of ping pong balls that will fit in two layers.
- What is the total volume of ping pong balls in that space? What total volume spans the bottom of the first layer to the top of the second? (Measure the height of the second layer from the bottom of the box.) What fraction of the volume is empty space?
- Look at Figure 1 of this article about sphere packing. Do your arrangements from the last exercise resemble the photos? Do the packing densities (the fraction of volume taken up by ping pong balls) match the numbers in Table 1 for dimensions n = 1 and 2?
- Figure 2 shows the densities of the best packings mathematicians have found in each dimension. What do you notice about the chart? Are you surprised by its appearance?
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
- (All levels) For a more in-depth look at Sophie Bryant and her mathematical work, read this 2017 biography by Patricia Rothman for Chalkdust Magazine.
- Now, choose another mathematician who broke gender barriers to research and write a 500-word report about. You may find this list of women mathematicians from Agnes Scott College helpful for getting started.
- Bryant’s 1884 paper, the first by a woman for the London Mathematical Society, studied the shapes of cells within honeycombs. Read this article by Philip Ball for Nautilus on that same topic.
- (Trigonometry) Ball writes: “Hexagonal cells require the least total length of wall, compared with triangles or squares of the same area.” Suppose you have a regular hexagon, an equilateral triangle, and a square, each with total area 1. Calculate their perimeters. Hint: A hexagon is made up of 6 equilateral triangles. Why?
- (Trigonometry) Now suppose you have a regular polygon with n sides and total area 1. What’s the perimeter?
- Calculate the perimeter of the area 1 10-gon, 100-gon, and 1000-gon. What shape of area 1 do you think has the smallest perimeter? (Calculus) Prove it.
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
- (All levels) Try out computer scientist Alex Beutel’s interactive Voronoi diagram generator.
- (All levels) For more on Voronoi patterns, read this Scientific American article by Susan D’Agostino.
- Now, make your own Voronoi pattern: Draw three dots anywhere you like on a piece of paper, and treat these as the “seeds” for the cells. Try to figure out where the boundaries of the cells should be! Come up with a guess and write a brief justification.
- Once you have your guess, read this tutorial on making a Voronoi diagram from Plus Magazine. What did you learn from their solution? Does it match your guess?
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
- (All levels) Look at some objects in the room and try to determine how many holes they have, topologically. Try, for example, to determine the number of holes in a book, a rubber band, a water bottle, and a pair of pants.
- Pair up with another student and discuss your answers. If you disagree about the number of holes in a particular object, work together to figure out the right number.
- Scavenger hunt: Find an object that has one hole, an object that has two holes, and an object that has three or more holes.
- (All levels) Read the section on cylinders in this article about topological tic-tac-toe, then play a few games of cylindrical tic-tac-toe with another student.
- (Advanced) Try some of the tic-tac-toe puzzles in the article. Note: the puzzles get progressively harder!
- (Geometry, Precalculus) Watch this video about using topology to solve problems by 3Blue1Brown.
- Draw an ellipse on a piece of paper and find an inscribed rectangle inside the ellipse.
- (Advanced) Draw a few different closed loops and try to find an inscribed rectangle in each one. Discuss: Why is this hard to do, even after watching the video? Notice that the video proves that there is always a solution to the inscribed rectangle problem, but doesn’t show how to construct a solution for any particular loop. (A proof that demonstrates that a solution exists without giving a method for finding it is sometimes called an “existence proof.”)
—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
- (All levels) Assuming there are 100 non-edge tiles, solve the riddle. Compare your answer to the solution published here.
- (All levels) Play Blockbusters in class using this online version. (Find some default questions here.)
- (All levels) Demonstrate percolation as introduced in the article. What you’ll need: 1 strainer or sieve; 1 bowl or cup with an opening large enough to hold up the strainer; 3 materials of different size (coffee grounds, sand, small rocks, little rocks, marbles, or something similar). Fill the strainer with one material. Place on top of the cup/bowl. Pour a cup of water over the material in the strainer. Measure how long it takes for the entire fluid to percolate. Repeat for each material. What do you notice about the relationship between fluid flow and material size?
Some more of this month’s math headlines:
- Octonions: The strange maths that could unite the laws of nature
New Scientist, August 16, 2022
- Here’s the quickest way to grill burgers, according to math
Science News, July 26, 2022