Math Digests August 2024

Stand-Up Mathematician Matt Parker on Why Triangles are the Best Shape

Forum, August 20, 2024.

In a new book, mathematician-slash-comedian Matt Parker expounds on the many uses of triangles. They create the foundation of buildings, are the basis of much origami, and can be used to measure distances large and small. In this episode of Forum on the radio station KQED, Mina Kim interviews Parker about triangles, and takes questions from listeners.

Classroom Activities: trigonometry, geometry, algebra

  • (Mid level) In the podcast, Parker describes how he found a way to split a rectangular sandwich into three “triangular-ish” pieces that all have the same area and the same amount of crust. Consider the following diagram of a rectangle, whose red edges are made of crust, cut into three distinct “triangular-ish” regions.

    A rectangle with red edges. The vertical edges are 3 inches and horizontal edges are 6 inches. The rectangle is split into three regions. The first region is a triangle whose vertices are the two top corners of the rectangle and a point X in the interior of the rectangle. This triangle has height h. The other two regions are split by a vertical line that connects X to the bottom edge of the rectangle. The distance from this vertical line to the rightmost edge of the rectangle is marked b.
    Rectangle with width $w = 3$ inches and length $\ell = 6$ inches, split in three “triangular-ish” sections. Image created in tikZ by Leila Sloman.
    • What should $h$ and $b$ be so that the three pieces all have the same area and the same amount of crust? How might the diagram change if $w$ is 4 inches instead of 3? Show your work.
    • Is it possible to cut a rectangular sandwich into three equally sized triangles? If so, how would you do it? If not, why not?
    • Brainstorm in small groups why you might want to use triangles in this situation. Some questions to get you thinking: Is it possible to solve this problem with three rectangles or three pentagons? What if the sandwich were a different shape, like a circle, pentagon, or hexagon?
  • (Mid level, Trigonometry) Read Chapter 1 of Parker’s book. Draw a diagram showing how Parker used triangles to calculate the height of the Tokyo Skytree. Explain the calculation in your own words. Make sure to include enough detail that a reader can follow each step of the calculation.
    • As homework, find a building nearby where you live, and then use triangles to measure its height. Tools you might want to use: A ruler or yardstick, a protractor, a camera, and/or a to-scale map. Calculate the height, showing all your work. If data is available, check your results against the true height.

—Leila Sloman

Perplexing the Web, One Probability Puzzle at a Time

Quanta Magazine, August 29, 2024.

Daniel Litt loves puzzles. His online brainteasers went viral this year, stumping even people who felt confident in their answers. “Litt’s online project doesn’t just highlight the enduring allure of brainteasers. It also demonstrates the limits of our mathematical intuition, and the counterintuitive nature of probabilistic reasoning,” wrote Erica Klarreich. Litt, a mathematician at the University of Toronto, spoke with Quanta Magazine about his love for deceptively tricky puzzles and what can be gained by struggling through them.

Classroom Activities: probability, puzzles

  • (All levels) Attempt Litt’s puzzle about 100 balls in an urn (from the article’s introduction).
    • Write your answer and explain your logic. (Do not check the correct answer yet.)
    • If your answer differs from that of a classmate(s), compare your approaches.
    • Now, look at the solution. Were either you or your classmate(s) correct? Read the explanation described in Litt’s interview. (The explanation starts where Litt says, “My favorite way to think about this is due to George Lowther.”) How does this explanation compare with yours?
    • If your answer was incorrect, what idea or concept did you incorrectly include (or leave out) of your logic? Discuss what you’ve learned with your classmates.
  • (Mid level) Attempt the three puzzles shown in the graphic called “Test the limits of your probabilistic reasoning” (answers are revealed at the bottom of the article).
    • For each puzzle that you missed, research the answer, and discuss with a partner where you made a mistake.

—Max Levy

Stone Age builders had engineering savvy, finds study of 6,000-year-old monument

Nature, August 23, 2024.

6,000 years ago, in a tectonically active region of southern Spain, humans constructed a giant stone chamber. This site, now called the Dolmen of Menga, still stands today. In a new study, archaeologists marvel at how it was built. Researchers conclude based on engineering principles and the geometry of the site that the ancient builders “possessed a good rudimentary grasp of physics, geometry, geology and architectural principles,” wrote Roff Smith in an article from Nature.

Classroom Activities: unit conversion, friction

  • (All levels) The sandstone slabs used in the Dolmen of Menga weighed up to 150 tonnes. Answer the following questions based on the text, your own calculations, and web searches:
    • Convert 150 tonnes into a) tons and b) pounds.
    • Find three other ways of approximating 150 tonnes. (e.g., 150 tonnes = approximately 300 adult cows)
  • (Mid level, Physics) The article mentions that “wooden tracks” may have been used to facilitate transport of heavy stones because they “minimized friction.” Explain in your words what it means to minimize friction.
  • (High level) In physics, a simplified expression for calculating the force required to move an object is Force = weight $\times \mu$, where $\mu$ is a “coefficient of friction.” Answer the following questions, referring to the equation and this resource from Engineering Toolbox.
    • On the table in the resource, what material combination has the highest kinetic friction coefficient? Which has the lowest? Is it easier to move an object on a material with a higher or lower friction coefficient?
    • Explain the difference between static and kinetic friction. Why is one value generally higher than the other?
    • How many pounds of force are required to move a 100-pound wood box on a concrete surface, if the box is currently at rest?
    • How many pounds of force are required to move a 100-pound concrete box on a wooden surface, if the box is currently at rest?

—Max Levy

Finnegans Wake: mathematicians find method in the madness

Cosmos, August 22, 2024.

“Finnegans Wake” by James Joyce is an unusual book. “It’s either nonsense or evocation,” wrote the New Yorker in its 1939 review. In this article for Cosmos, writer Ellen Phiddian describes a new study that found another way the book is unusual: The mathematics of its punctuation.

Classroom Activities: statistics, data analysis

  • (All levels) Choose your favorite book, article, or other work of prose. Look at the first page. Write down the length of each clause, defined as the number of words until you hit a new punctuation mark, including question marks, periods, commas, semicolons, exclamation points, colons, ellipses, dashes, or parentheses. So, in the sentence

    My mother, whose name is Linda, went to the park today.

    you would write the numbers 2, 4, 5. Create a histogram of your data. Describe the results.

  • (Mid level) Calculate
    • The mean and standard deviation of your data.
    • The probability that a clause has 10 words.
    • The probability that a clause has 5 words.
  • (High level) In the new paper, the authors compare the hazard functions of several books. These functions depend on a number $k$. The number $h(k)$ is the probability that, if you know that a clause has more than $k$ words, it has exactly $k + 1$ words.
    • Describe how you would calculate this for your data.
    • Calculate $h(k)$ for your data at $k =$ 5, 10, 15.
    • Based on the article, what is unique about the hazard function of “Finnegans Wake”?

—Leila Sloman 

The Lazy Way to Cut Pizza

Numberphile, August 12, 2024.

How would you go about cutting a pizza into as many slices as possible, with as few cuts as possible? This Numberphile video explains the math behind this problem and the interesting pattern that arises from its solution.

Classroom Activities: geometry, number theory

  • (Mid level) You can cut a pizza the same number of times and end up with a different number of slices based on how and where you cut the pizza. Sketch how the following combinations of cuts and desired pieces are possible:
    • 2 cuts to make 3 pieces
    • 3 cuts to make 4 pieces
    • 3 cuts to make 7 pieces
    • 4 cuts to make 10 pieces
  • (Mid level) Based on the above example, explain in your own words how you decide where to cut to increase or decrease the number of pieces.
  • (High level) Watch the video. Explain in your own words what the equation at 9:15 represents.
    • Create a word problem that requires this equation to solve. (The problem can be about anything, not just pizza cutting, so be creative!)

—Max Levy

Some more of this month’s math headlines: