Math Digests February 2024

Anyone can play Tetris, but architects, engineers and animators alike use the math concepts underlying the game

The Conversation, February 28, 2024.

In this article, education professor Leah McCoy argues that the game of Tetris is intrinsically linked to an area of mathematics known as transformational geometry. Introduced in middle school, transformational geometry involves linking two distinct mathematical objects through transformations like translation or reflection. What’s more, transformational geometry is used by a wide variety of professions, including animation and architecture. “There’s far more to Tetris than the elusive promise of winning,” McCoy writes.

Classroom Activities: geometry, optimization

  • (All levels) Have each student play five minutes of Tetris on their own. Then, teach the definitions of the four basic geometric transformations: Translation, reflection, rotation and dilation. Ask each student to answer the following questions:
    • Which transformations show up in Tetris?
    • How and when do they show up as you play? Be specific. For example, in what direction, by how much, and in what context are the objects transformed?
    • Which types of transformations do not show up in Tetris?
  • (All levels) Have students come up with their own variation on Tetris by changing at least one of the types of transformations used in the game. (Students cannot just add a new transformation to the game—they must “delete” at least one existing one. Other features of the game, such as the way new shapes appear, or the shapes that are included, can be changed.) Students will create prototypes of their game using posterboard and construction paper shapes, and present to the class.
  • (All levels) Play a game of “Tetris musical chairs.” Have students play Tetris in class. At random time intervals, alert them that it’s time to pause the game. Based on their paused screen, students will write down the transformations needed to get their piece where they want it to go.

—Leila Sloman 

Mathematicians have finally proved that Bach was a great composer

New Scientist, February 2, 2024.

To physicist Suman Kulkarni, music is more than just sound, rhythm, and notes. She sees the greatest works of Western classical music as complex networks of information. Kulkarni recently analyzed the works of revered composer Johann Sebastian Bach with tools from the field of information theory, which describes data based on its individual parts and the connections between them. It applies to cryptology just as well as it applies to linguistics. Kulkarni wanted to identify why Bach’s music is so revered. Bach “produced an enormous number of pieces with many different structures, including religious hymns called chorales and fast-paced, virtuosic toccatas,” writes Karmela Padavic-Callaghan for New Scientist. In this article, Padavic-Callaghan describes Kulkarni’s experiment and analysis.

Classroom Activities: information theory, graph theory

  • (Mid level) The article describes Kulkarni parsing Bach’s compositions into a graph of “nodes” connected by “edges.” Watch this short TED-Ed video about the origins of graph theory for more context and to answer the following questions.
    • Based on the article and video, describe what the “nodes” and “edges” are in the Bach graphs. What would it mean when two nodes are connected by an edge?
    • Can one node be directly connected to more than two other nodes? If so, what would this mean in the Bach piece?
    • How would you expect the graphs of a Bach Toccata to differ from a Bach chorale, based on the information in the article?
  • (High level) Learn more about graph theory methods, such as depth-first search (DFS), with this free TeachEngineering lesson. Complete the Making the Connection lesson worksheet, which asks you to define terms and analyze a graph. (An alternative to the “friends in class” criteria could be “students with more than one class together” or “students who share an extracurricular activity.”)

—Max Levy

Oakland Filmmaker’s Documentary Details Achievements Of Black Mathematicians

SFGate, February 5, 2024.

In 1876, physicist Edward Bouchet became the first African American to earn a doctoral degree at a university in the United States. That long-awaited feat was delayed not by a lack of willing students, but by policies barring those students from equal rights. A new documentary titled “Journeys of Black Mathematicians: Forging Resilience” tells the story of similar hurdles in even more recent memory. “African Americans are not only underrepresented in the field, their achievements have been overlooked because the public perceives the road to excellence for African Americans is limited to sports and the arts,” writes Francine Brevetti for SFGate. In this article, Brevetti speaks about the stories he uncovered by speaking with Black mathematicians.

Classroom Activities: equity, math history

  • (All levels) Watch the trailer (or rent the full film on Vimeo) and note the names of interviewees in the documentary.
    • Research at least two of these mathematicians, and list five facts about their work or life stories.
    • Describe the research or recent work of at least one of these mathematicians, in your words.
    • Choose one mathematician and present what you’ve learned about them in class.
  • (All levels) Read this profile of Christine Darden from Quanta Magazine: The NASA Engineer Who’s a Mathematician at Heart. Discuss what barriers Darden mentions existing in her career. How have things changed or not changed since Darden worked at NASA?

—Max Levy

The Strangely Serious Implications of Math’s ‘Ham Sandwich Theorem’

Scientific American, February 17, 2024.

In 1938, mathematicians proved a principle called the Ham Sandwich theorem. The theorem states that you can always “cut” $n$ objects in half simultaneously under certain conditions: If those objects are $n$-dimensional, and if your cut has $n-1$ dimensions. For example, if you have 2 circles in a 2-dimensional plane, there exists a 1-dimensional line that bisects both circles simultaneously. And they don’t have to be circles; they can be any shape, including discontinuous shapes like scatters of random points and blobs. “Contemplate the bizarre implications here,” writes Jack Murtagh for Scientific American. “You can draw a line across the U.S. so that exactly half of the nation’s skunks and half of its Twix bars lie above the line.” Murtagh’s article explains the Ham Sandwich theorem and how it complicates efforts to prevent political gerrymandering, a geographic trick often used to devalue the votes of minority groups or to favor one party.

Classroom Activities: coordinate systems, geometry

  • (All levels) For each example below, draw a coordinate grid spanning $x=[0,8]$ and $y=[0,10]$ using blank or graph paper. Draw the two shapes/sets and find the line that bisects both. (All squares are upright; that is, their sides are perfectly horizontal and vertical, rather than tilted.)
    • One circle with a radius of 8 centered at (1,9); one circle of $r = 2$ centered at (4,5)
    • One square with side lengths of 2 centered at (4,9); one oval with a width and height of 2 and 1, respectively, centered at (7,2)
    • One set of 4 squares, each with areas of 1, centered at (2,2), (3,3), (4,1), and (6,5); one circle of $r = 2$, centered at (2,7)
    • What is easiest and most challenging about these examples?
  • (Mid level) Murtagh writes “with sufficiently many voters, any percentage edge that one party has over another (say 50.01 percent purple vs. 49.99 percent yellow) can be exploited to win every district.” Demonstrate how this happens with the following exercise based on the images with purple and yellow dots in the article:
    • Describe the tally of purple and yellow votes in each district before and after the straight lines shown in the article. Which group is being advantaged and disadvantaged?
    • Starting from the second image, draw four more bisecting lines to create 8 “districts.”
    • Is it possible to use the “Ham Sandwich” lines to increase the power of the minority group?

—Max Levy

Geometry can shape our world in unexpected but useful ways

Science News Explores, February 29, 2024.

In this article, Lakshmi Chandrasekaran recounts several ways that geometry bleeds into research. One group that she covers, at George Mason University’s Experimental Geometry Lab, modeled four-dimensional shapes by 3D-printed projections of the shapes. Other examples were more practical. Laura Schaposnik at the University of Illinois Chicago is studying the possible geometries of viruses, which can help researchers get ahead of as-yet-undiscovered viruses. “Many people may find it hard to see the appeal or everyday uses of such math,” writes Chandrasekaran. “But modern geometry is full of problems both beautiful and useful.”

Classroom Activities: geometry, tiling

  • (All levels) Read the article. In class, brainstorm other possible applications of geometry. As homework, write a paragraph about one of these possible applications. Be creative—your paragraph can be hypothetical, it can incorporate research about what related geometric work has been done, or it can involve your own experiments, but it must be specific about how geometry shows up.
  • (All levels) The article describes projection in terms of shadows. Try to work out what the following shadows would look like, then check your work with physical objects:
    • A sphere
    • A cube oriented upright
    • A cube balancing on one corner, with the opposing corner directly above it
    • A rectangular prism balancing on one corner, with the opposing corner directly above it
    • (High level, Linear Algebra) Now work backwards by calculating the projections explicitly. Assume light is coming from straight above your objects.
  • (All levels) Explore tiling with this lesson plan from PBS. For more on tiling, check out the April digests.

—Leila Sloman

Some more of this month’s math headlines