Math Digests April 2024

Language doesn’t perfectly describe consciousness. Can math?

Vox, April 10, 2024.

When people say that “a picture is worth a thousand words,” they usually mean that one image can relay a lot of information. However, you could also conclude that one word doesn’t say much. In this article, Vox’s Oshan Jarow writes about how mathematics might be better suited than words for describing our conscious experience. “Words could offer you a poem about the feeling of standing on a sidewalk when a car driving by splashes a puddle from last night’s rain directly into your face,” Jarow writes. “A mathematical structure, on the other hand, could create an interactive 3D model of that experience, showing how all the different sensations — the smell of wet concrete, the maddening sound of the car fleeing the scene of its crime, the viscous drip of dirty water down your face — relate to one another.”

Classroom Activities: math in language, data analysis

  • (All levels) The limitation of language, according to the article, is that an adjective like delicious mainly exists in one dimension. A chocolate mousse can be delicious and so can a burrito. The word tells you something about food, but not enough to distinguish between them. Answer the following questions.
    • Describe the last thing that you ate with one adjective.
    • Now describe the last thing you ate with four adjectives.
    • Find a partner and describe your food with the four adjectives you chose without saying what the food is. Try to guess each other’s food. (Avoid words like “cheesy” which reveal the ingredients. A word like melty might be better, for example.)
  • (Mid level) List five pairs of adjectives (e.g., “salty/unsalted,” “dry/moist,” and “fragrant/odorless”) that you can use to describe a variety of different foods by assigning a score of 1 to 10 for each pair. (For example, a saltine cracker might be a [8,1,3] meaning very salty, very dry, and fairly odorless).
    • Assign scores to the following foods: apple, chocolate bar, raw broccoli, parmesan cheese, mozzarella cheese.
    • Tabulate the numbers for each of the foods.
    • Write five observations based on the data, such as “XX is similar to YY in ZZ way but different in VV way.”
    • In what ways is this system useful for comparing and contrasting the experiences of eating different foods? In what ways is it insufficient?
  • (Mid level) Explain how the above exercises relate to the mathematics of consciousness described in the article.
    • Explain what “ineffability” refers to in your own words.
    • Upload this image of Earth to this online tool for pixelating images. Pixelate the image with a block size of 1, 50, and then 100.
    • Describe how you would apply the idea of ineffability to this example. What happens when ineffability is low versus high?
    • Is there a block size limit where the image no longer looks like anything?

—Max Levy

First known fractal molecule is a natural mathematical marvel

New Atlas, April 15, 2024.

When you look closely at a fractal, patterns that are obvious in the zoomed-out view repeat infinitely on smaller and smaller scales. Fractals are present all over nature, from snowflakes and trees to mountains and coastlines. However, they’ve never been observed in proteins, the building blocks of life, until now. “We stumbled on this structure completely by accident,” Franziska Sendker, a microbiologist who recently discovered a fractal molecule, said in an interview with New Atlas. Sendker was studying a protein isolated from cyanobacteria when she noticed its strange shape. Its atoms were arranged in countless triangles with triangular holes in their center. The pattern repeated at different sizes, “totally unlike any protein assembly we’ve ever seen before,” according to Sendker. This article describes the research and the specific type of fractal observed.

Classroom Activities: fractals, Sierpiński triangle

  • (All levels) Read the description of the Sierpiński triangle observed in Sendker’s research and the instructions on this worksheet from the National Oceanic and Atmospheric Administration.
    • Complete the worksheet for 6 iterations of the Sierpiński triangle.
    • How many white triangles do you count at steps 3, 5, and 6? Describe the pattern.
  • (Mid level) Follow this activity from the Fractal Foundation, “Fractal Trees,” where you calculate the ratios of branch length.
  • (All levels) For more, refer to previous Math in Media digests about fractals from 2021 and 2022.

—Max Levy

Computer scientist wins Turing Award for seminal work on randomness

Ars Technica, April 10, 2024.

On April 10, the Association for Computing Machinery announced the winner of the 2023 Turing award, one of the most prestigious awards in computer science. It went to Avi Wigderson of the Institute for Advanced Study in Princeton, New Jersey. In this article for Ars Technica, Jennifer Ouellette describes some of Wigderson’s accomplishments, including work showing that the advantages of random algorithms could be reproduced without randomness and on zero-knowledge proofs.

Classroom Activities: randomized algorithms, algebra

  • (High level) In Richard Karp’s 1991 survey, he gives several examples of randomized algorithms. One is for testing whether a polynomial equation is valid or not.
    • Read Section 4.1 of the survey. Describe in your own words the randomized algorithm for testing whether a polynomial $f$ is zero or not.
    • Graph the polynomial $f(x,y) = x^3-5x^2y + 7xy^2 -3y^3$ using an online 3D plotter like WolframAlpha. Is $f(x,y) = 0$?
    • Evaluate $f(x,y)$ at the points $(0,0)$, $(3,1)$, and $(2,2)$. Then use a random number generator to choose 3 coordinate pairs from the set ${-3, -2, -1, 0, 1, 2, 3}^2$ and evaluate $f(x,y)$ again. Interpret the results.
    • Based on Theorem 4.1, what are the chances that you got $0$ for all 3 of your choices in the last exercise? Why do you think it’s important that the variables $a_1,\dots,a_n$ are chosen randomly?
    • Come up with your own secret polynomial, and have a classmate try to guess whether it’s zero using the randomized algorithm.
  • (High level) Read the Ars Technica article and state, in your own words, what Wigderson showed about randomized algorithms like the one you just used.
    • Come up with a non-random algorithm for figuring out whether a polynomial of the form $ax^2 + bx + c$ is zero, by testing it on various values of $x$.
  • Check out some activities on zero-knowledge proofs from our October 2022 digests.

—Leila Sloman

Can you solve it? Tiler swift

The Guardian, April 29, 2024.

Every two weeks, Alex Bellos publishes a mathematical puzzle for readers of The Guardian. On April 29, Bellos had readers come up with a grid whose tiles were colored in black and white according to some constraints. The problem is reminiscent of mathematical concepts like map coloring.

Classroom Activities: graph theory, coloring

  • (All levels) Solve Bellos’ puzzle. Write a short paragraph describing how you came up with your solution.
  • (Mid level) Bellos’ puzzle, along with the map coloring problem, can be represented on mathematical objects called graphs. Complete this lesson on graph theory from the Park School in Baltimore, up through Exercise 17.
    • (Mid level) Rewrite Bellos’ puzzle as a problem about graphs. Do you find it easier or harder to solve now? Why?
  • (High level) Read about the map coloring problem in the first section titled “Map Colorings” here.
    • Come up with a graph that cannot be colored with only 4 colors. Describe why your graph is not a planar graph.
    • Read the proof of the simpler six-color theorem given in the notes. Does the proof help you understand the examples you’ve worked with?

—Leila Sloman

Mathematicians Explain Why Some Lengths Can’t Be Measured

Scientific American, April 18, 2024.

In mathematics, the concept of measure is used to quantify the length of sets of real numbers. For example, the measure of the interval $[0,1]$ is 1; the measure of the interval $[0,0.01]$ is 0.01. More complicated sets can be measured, too. The measure of the natural numbers is 0, while if you take each natural number $n$ and surround it by an interval of length $10^{-n}$, the total measure of all those intervals is $\frac{1}{9}$. But in 1905, Giuseppe Vitali showed that there are sets of real numbers that can’t be measured. In this article for Scientific American, Manon Bischoff explains the concept of measure and Vitali’s proof. (Note: The proof given that Vitali’s set is non-measurable contains some minor errors. The reason $\mu(V^*) = \sum_p \mu(V_p)$ is because the sets $V_p$ are disjoint, and there are uncountable sets with measure zero.)

Classroom Activities: measure

  • (Mid level) Read the first section of the article. Based on the information provided, guess the measure of the following sets, with a brief justification for your answer:
    • The interval $[0,4]$
    • The union $[0,1] \cup [2,3]$
    • The union $[0,1] \cup [1/2,1]$
    • The set of all real numbers bigger than 1
    • The set containing only 0
  • (All levels) In dimension 1, the measure described by Bischoff represents “length.” The concept of measure can be generalized to two-dimensional sets, three-dimensional sets, and higher. Brainstorm as a class how this generalization should work before moving on to the next activity.
  • (Mid level) Usually, the simplest measures in two-dimensional space represent area and the simplest measures in three-dimensional space represent volume. Guess the measure of the following sets, with a brief justification for your answer:
    • The inside of a $1 \times 1$ square in the $xy$-plane
    • The inside of a circle of radius 1 in the $xy$-plane
    • The line $y = 2x$ in the $xy$-plane
    • A sphere of radius 2 in 3D space
    • The surface of the sphere of radius 2 in 3D space

—Leila Sloman 

Here’s the math keeping housing inventory so low

Yahoo Finance, April 13, 2024.

Usually, sellers are happy when the price of what they’re selling is abnormally high. Homes are different. Most homeowners who sell would need to purchase another home — either upgrading to a more expensive house or moving “laterally” to a similar one — and right now, the math is not in their favor. They’d wind up paying far more on their monthly loan than they currently do. “That gap is convincing many American homeowners to stay put, stunting the number of homes for sale and buoying prices on the paltry supply that exists,” writes Janna Herron in Yahoo Finance. In this article, Herron explains the math behind the mounting pressure on buyers and sellers in the housing market.

Classroom Activities: mortgage math, data analysis

  • (All levels) Most people purchase homes with a long-term loan called a mortgage, which requires monthly payments based on the total loan amount, plus interest. Read this interactive guide about mortgage math, and test the mortgage calculator with different numbers to answer the following questions:
    • What would be the monthly payment on a \$500,000 loan with 7% interest paid over 30 years? What total amount will be paid by the end of the 30 year period? How much of that is interest?
    • How about a \$500,000 loan with 7% interest paid over 20 years?
    • A \$600,000 loan with 5.5% interest paid over 30 years?
  • (Mid level) Using a spreadsheet such as Google Sheets or Excel, create your own mortgage payment calculator.
    • What would be the initial monthly payment on a \$700,000 loan with interest of 4.5% paid over 30 years? What total amount will be paid by the end of the 30 year period?
  • (Mid level) Read the example given in the article comparing the “payment shock” of two homeowners A and B.
    • Suppose you and your neighbor each own homes worth \$400,000. You have \$100,000 in home equity, and your neighbor has \$300,000 of equity. If each of you decides to sell and use your equities for a down payment on \$500,000 homes, what will each of your monthly payments be? (Assume 5% interest and a 30-year loan)
    • Define the payment shock as the percent change in the mortgage balance after selling your home and buying a new one. Calculate the payment shock for you and your neighbor.

—Max Levy

Some more of this month’s math headlines