Math Digests October 2021

Just how fat are the fat bears?

High Country News, September 29, 2021

It’s fat bear season. Every fall, bears in Alaska’s Katmai National Park and Preserve gorge themselves before hibernating. Hundreds of thousands of people tune in to livestreams and vote for their favorite fat bear. But how fat are they really? In the spring, scientists have to tranquilize the bears and use helicopters to get an accurate measurement. In the fall, when they reach peak plump, it’s often too hard to reach them. But, as Kylie Mohr writes in High Country News, there may be a clever workaround: lasers. Light detection and ranging devices, or “lidar,” can scan a fat bear and tell us its volume, which can be used to calculate its weight if its density is known. Katmai’s Joel Cusick came up with the idea and tried it on a bear named Otis. “I got a laser return from the butt of Otis,” Cusick told Mohr. “I thought, ‘Wow, this just might work.’”

Classroom activities: density, pre-algebra, algebra

  • (All levels) Meet the bears on the Fat Bear Week website.
  • (Lower level) In the article, we learn that one past competition winner, Bear 747, weighed 1,416 pounds and had a volume of 22.6 cubic feet.
    • What is the average density of this big bear? Hint: the units for density for this problem are pounds per cubic foot.
    • How much would a bear with the same density weigh if it measured 20 cubic feet in volume?
    • What would be the volume of a bear with the same density that weighs 1000 pounds?
  • (Middle Level) Bodies are mostly water. The density of water is 62.4 pounds per cubic foot. How does this compare to the density that you calculated for Bear 747? Discuss how different parts of the body, such as fat, might affect density.
  • (Higher level) Now, let’s assume that we don’t know a bear’s overall density, but we do know its volume and the density of three components.
    • Density of water: 62.4 pounds per cubic foot
    • Density of fat: 56.2 pounds per cubic foot
    • Density of fur: 75.0 pounds per cubic foot
    • Volume of bear: 21.1 cubic feet

    Assume that a bear’s volume is 60% water, 30% fat, and 10% fur. How much does the bear weigh?

—Max Levy

Gnarly, Centuries-Old Mathematical Quandaries Get New Solutions

Scientific American, October 14, 2021

When do polynomial equations have solutions made up of integers or rational numbers? In this article for Scientific American, Rachel Crowell describes the current state of affairs on these questions. In 1970, mathematicians proved that there is no algorithm for showing whether or not certain systems of equations could be satisfied by integers. But the same question in systems with several variables is still unanswered. Although these problems have remained unsettled for a long time, Crowell writes, what’s really interesting are the methods that mathematicians develop to solve them, which often end up having unexpected connections to other areas of mathematics.

Classroom activities: polynomials, geometry, Diophantine problems

The following exercises are designed to give students a taste of Diophantine problems using a simple example: a unit circle centered at the origin. For a slightly more difficult version, try using $x^2+y^2=5$.

  • (Algebra 2) Consider the polynomial equation $x^2 + y^2 = 1$. Can you find a solution $(x,y)$ where $x$ and $y$ are integers? How confident are you that you found all the integer solutions?
  • (Algebra 2) Draw the curve $x^2 + y^2 = 1$ on a graph (either on paper or digitally). Does this make it easier to find all the integer solutions?
  • (Number theory, pre-calculus) Watch this video by Michael Penn showing how to find rational solutions to $x^2 + y^2 = 1$. Have students use the technique to find 3 more rational solutions.
  • (Introductory programming) Have students write a computer program that checks all combinations of $x,y,z$ each running from 1 to 25 to find integer solutions of $x^2 + y^2 = z^2$.

—Leila Sloman

The Math of the Amazing Sandpile

Nautilus, October 6, 2021

The “abelian sandpile” is an abstract model of a pile of sand. Despite its simple laws, the abelian sandpile gives rise to beautiful and complicated behavior. With the help of images and videos, Jordan Ellenberg discusses this behavior and the resulting “self-organized criticality.” When the sandpile has limited room—like if it spills over the sides of a table—it will always end up at a critical density of 2.125 grains of sand per spot. At this density, writes Ellenberg, “There’s constant activity, but the activity is somehow organized and structured.” This may just seem like another curiosity of mathematics, but “self-organized criticality is one popular way to think about how the rich structures of life might have emerged from simple systems that automatically seek the critical threshold,” writes Ellenberg. Maybe abstract math isn’t so different from real life, after all.

Classroom activities: mathematical modeling, dynamics

  • (Pre-calculus) Introduce the rules of the abelian sandpile in class. Assign the question Ellenberg poses in the article: “Check that two adjacent dots in the interior sandpile can never be empty at once.”
  • (Pre-calculus) Ellenberg links to Wes Pegden’s interactive gallery of abelian sandpile simulations on an infinitely large grid. Pegden shows the abelian sandpile on different types of grids, with different numbers of grains of sand (“chips”). Have students create an abelian sandpile by hand using any small tokens as chips:
    1. Place 16 chips on the center point of a square lattice.
    2. Move 1 chip from that point to each of the neighboring lattice points.
    3. Repeat step 2 with any lattice point that has 4 or more chips. Continue until every lattice point has fewer than 4 chips.
    4. If desired, repeat with 32 chips or on a triangular or hexagonal lattice. (For a triangular lattice, the number in bold in step 3 should be 6. For a hexagonal lattice, it should be 3.)
    5. Compare the resulting patterns to the ones in Pegden’s gallery.
  • (Pre-calculus) Ellenberg writes: “The abelian sandpile model doesn’t even try to capture the behavior of actual physical materials.” Ask students to identify one way in which real sandpiles differ from the abelian sandpile, and have them modify the rules of the abelian sandpile to try to account for this difference. Is their new model still abelian?

Related Mathematical Moments poster and interview: Piling On.

—Leila Sloman

Physics Nobel goes to complexity, both general and climatic

Ars Technica, October 5, 2021

The latest Nobel Prize in Physics recognizes chaos and complexity. To physicists, chaos is when seemingly simple systems begin acting erratically. Similarly, complex systems have many moving parts that interact, yielding unpredictable consequences. This year’s Nobel recipients helped prove that these ideas explain and predict how our climate is changing. But this complexity affects more than just physics: “mathematics, biology, neuroscience, laser science, materials science, and machine learning, to name a few,” according to writers at Ars Technica. For example, geometric patterns emerge in flying flocks of birds. Climate patterns emerge when gases mix and heat under radiation. “All of these systems seem very different on the surface, but they share a common underlying mathematical framework.” Making predictions in complex systems is extremely difficult—the math can be too hard for the world’s most powerful computers to handle. But such wide-ranging applications, this math is crucial.

Classroom activities: complexity, chaos, synchrony, butterfly effect

  • (All levels) Watch this YouTube video by Veritasium about the science of synchronization, which is related to chaos. As you will see in the video, some systems that begin in total disorder will always drift into a coherent pattern—synchronized behavior like thousands of fireflies flickering simultaneously.
  • (All levels) Read more about the Nobel Prize in Physics awarded to Giorgio Parisi, Syukuro Manabe, and Klaus Hasselmann. In this article on The Conversation, an atmospheric scientist writes about how mathematical modeling made all weather and climate forecasting possible. In this Quanta Magazine article, students can learn more about how complex physics and math have helped create climate models.
  • (Middle school) Use this spreadsheet-based example to explain the butterfly effect.
  • (Middle school / high school) Use this lesson plan from Shodor to lead students through interactive web games related to chaos and probability.
  • (Middle school / high school) Play the Chaos Game. This game shows how patterns can emerge from a chaotic or complex system with seemingly random rules. (You will need some dice and markers.) The same website has additional chaos/complexity games.

Related Mathematical Moments poster and interview: Predicting Climate.

—Max Levy

Some more of this month’s math headlines: