Math Digests July-August 2026

July & August Digests:

A Mathematical Tribute to the Soccer Ball

New York Times, July 17, 2026

What shape is a classic soccer ball? It is round, so one might be tempted to simply call it a sphere. But the seams along its suggest another, more fascinating answer: The ball is a truncated icosahedron. The icosahedron is a Platonic solid, a polyhedron where all corners, edges, and faces look identical. The other four Platonic solids are the tetrahedron, cube, octahedron, and the dodecahedron. These five shapes “are the foundation of mathematical symmetry,” writes Siobhan Roberts in this New York Times story about the many symmetries available in three-dimensional objects.

Classroom Activities: platonic solids, symmetry

  • (High level) Read the article. Explain why the classic soccer ball is an icosahedron, despite having more than 20 sides.
    • Sketch a square, and truncate it by chopping off its corners. How many edges does the truncated square have?
    • Sketch how the basic units of a regular icosahedron (interconnected triangles) can transform into the units of the classic soccer ball.
    • Sketch a regular octahedron, which has 8 triangular faces. (Hint: It can be broken into two pyramids.)
    • Sketch how the basic units of the regular octahedron can transform into a truncated octahedron, which has hexagonal and square faces.
    • How many hexagons and squares does the truncated octahedron have?
  • (Mid-level) Count the vertices and edges for all five platonic solids.
    • For each, calculate Vertex + Face – Edge. What do you notice about this value?
    • Watch this video from 3Blue1Brown for an explanation.
    • Based on the video, what would the “points” and “edges” represent in the following “graphs”?
      • A social media network
      • A brain
      • A map

—Max Levy

Mathematicians still don’t know the fastest way to multiply numbers

Scientific American, July 13, 2026

If you’ve ever tried to multiply two four- or five-digit numbers, you know how tedious it can get. For decades, researchers have been searching for ways to make the process easier on computers, which are routinely asked to multiply numbers with hundreds of digits or more. So far, the fastest multiplication algorithm takes around $n\log(n)$ steps to multiply two $n$-digit numbers. But, as Jack Murtagh writes for Scientific American, a better way could still be out there.

Classroom Activities: arithmetic, algorithms, complexity

  • (Mid-level) Multiply the following pairs of numbers using your normal techniques. Count the number of steps, that is, the number of individual computations. Do the first two examples as a class, and the rest individually.
    • $11 \times 14$
    • $29 \times 17$
    • $12 \times 14$
    • $109 \times 31$
    • $597 \times 131$
    • $1,012 \times 2,004$
    • $(10a + b) \times (10c + d)$
    • Describe the procedure you used. Did you find any shortcuts? How many steps do you think it would take to multiply two five-digit numbers? Two $n$-digit numbers?
  • (Mid-level) Read the article. Use Karatsuba’s algorithm to multiply the following pairs:
    • $11 \times 14$
    • $1,012 \times 2,004$
    • $2,331 \times 4,102$
    • $1,008 \times 4,500$
  • (Mid-level) Using Desmos’ graphing calculator, plot the curves $n^2$ and $n^{1.585}$, and $n \log(n)$. In words, describe how these quantities differ at $n = 10$, $n = 1,000,000$, and $n=10^{50}$.
  • (High level) Reflect:
    • Was Karatsuba’s algorithm easier than your usual techniques? Harder? What did you notice?
    • Describe how you would go about multiplying two eight-digit numbers.

—Leila Sloman

What Is Entropy, Really?

Wired, July 24, 2026

Unless something has gone horrendously wrong (or, perhaps, you’re scuba diving), at this moment you’re surrounded by trillions upon trillions of air molecules. Just one cubic inch of air contains so many molecules that no scientist, no matter how dedicated, could begin to untangle their individual movements. Luckily, it’s far easier to predict the collective motion of these molecules. That collective motion determines the quantities that matter at the human scale, like heat and air pressure. In this article for Wired, Rhett Allain explains why that is.

Classroom Activities: probability

  • (Mid-level) Read the article. Based on your reading, answer the following questions.
    • What happens to hot coffee when you add cold milk? What does this have to do with entropy?
    • What would happen to a sealed vacuum—containing no air and no particles of any kind—if you opened it right now? What does this have to do with entropy?
    • Explain why “macroscopic” quantities like heat and air pressure are easier to study than the “microscopic” motion of molecules.
  • (Mid-level) Imagine rolling two dice.
    • If the dice both have six sides, how many “microstates” are there? Describe them.
    • If your score is the sum of the two dice, what are the possible scores?
    • Find a formula $S(k)$ for the number of microstates that correspond to a score of $k$. Plot $S(k)$.
    • (High level) Write a computer program that repeats the activity for any number $N$ of six-sided dice. How does $S(k)$ change with $N$? (Hint: Write a recursive program, where the base case has two dice.)

—Leila Sloman

Tracking the largest cyclosporiasis outbreak in the US

CNN, July 24, 2026

This summer, a parasitic disease called cyclosporiasis spread across the Midwest. Cyclosporiasis causes severe and sometimes fatal diarrhea. As of early August, nearly 12,000 people in Michigan had been affected. When outbreaks like this occur, public health administrators race to identify the source and track hospitalizations and fatalities. This article from CNN provides an update on cyclosporiasis statistics.

Classroom Activities: public health, statistics

  • (Mid-level) Calculate the following public health statistics based on information provided.
    • 11,500 people have been infected in Michigan, a state with 10.1 million inhabitants. What is the “per capita incidence rate” (per 1000 people) of cyclosporiasis in Michigan?
    • Two people in Michigan have died. What is the mortality rate of the disease?
    • 100 people have been hospitalized around the country. What is the hospitalization rate, assuming an estimated 23,000 cases nationwide this season?
  • (High level) We can model an outbreak with equations like $y = a^{x/b}$, where $x$ is the number of days since the outbreak started, and $a$ and $b$ are growth factors of the pathogen. Use a spreadsheet to calculate the differences in outbreaks 20 days out if they have following growth factors:
    • $a=3 ; \; b=2$
    • $a=2 ; \; b=3$
    • $a=2 ; \; b=2$
    • $a=1.5 ; \; b=1$
    • Describe in words what happens to the outbreak if $a$ increases, and if $b$ increases.

—Max Levy

Scrawled signature names a Maya astronomer for the first time

Science, July 13, 2026

The Maya civilization populated the Yucatan peninsula for 3600 years until the Spanish conquered them around 1540 C.E. Archaeological records have long shown that Maya scientists predicted the motion of planets and stars. But the identities of those scientists were thought to be lost to time. “Maya astronomy nerds,” writes Laura Martín Agudelo, “can’t name their heroes.” That has now changed. Agudelo covers new research decoding an eighth-century signature scribbled beside mathematical formulas—a signature belonging to “Sak Tahn Waax.”

Classroom Activities: number system, base, calendar

  • (Mid-level) The Maya used a base-20 numbering system, rather than the base-10 system that we use today. Convert the following numbers from base 10 into base 20. You may use standard numerals, or the Maya notation of dots and lines, linked here.
    • 1
    • 10
    • 400
    • 401
    • 1000
  • (High level) The Maya had multiple calendars: a 260-day sacred calendar, a 365-day solar calendar, and one continuous “long count” that began from the Maya mythical day of creation.
    • Describe how one could use a base-20 numbering system to track each of these three timescales.
    • Compare your answers to the actual operations described here.
    • Identify two ways in which the math of this system differs from the calendars that you are accustomed to.

—Max Levy

More of this month’s math headlines