This month’s topics:
The Prime Minister’s geometry problem
French Emperor Napoleon had a theorem, U.S. President Garfield had a proof and now Russian Prime Minister Mikhail Mishustin has a geometry problem. The story was reported by Alex Bellos in his Monday puzzle column in The Guardian on September 20, 2021. “Earlier this month, Russia’s Prime Minister, Mikhail Mishustin, marked the first day of the school year by visiting a sixth form maths class at one of his country’s top science-oriented schools.” The school is the Kapitsa Physics and Technology Lyceum in Dolgoprudny, a town about 20 kilometers north of Moscow; sixth form corresponds roughly to Senior year in high school. During his visit he posed a geometry problem to the students.
Bellos shows us a photograph of Mishustin at the board with a diagram
of the problem in which he has drawn lines through the point and the
ends of the diameter and marked the inscribed angle thus formed as a right angle.
Bellos gives us a hint: it may help to remember that the three altitudes of a triangle (altitude: “the line from a corner that meets the opposing side at a right angle”) meet in a single point. Give yourself a few minutes to think about the problem and the hint before looking at Bellos’s presentation of the solution.
Amazing mathematical sandpiles
University of Wisconsin professor Jordan Ellenberg posted “The Math of the Amazing Sandpile” in the online magazine Nautilus on October 6, 2021. A mathematical sandpile is essentially a cellular automaton (like Conway’s“Game of Life”) where grains of sand populate the squares of a grid according to the following rule: No square can have more than three grains, and if a fourth grain is added, then the four disperse, one to each of the four adjacent squares. Ellenberg sketches out what happens if two adjacent squares on an otherwise empty grid both start with 4 grains:
Ellenberg first shows us the amazing complexity this simple rule can engender. Suppose we put a large number of grains on a single square in the middle of an infinite, empty grid. “You might imagine you’d end up with a big smooth pile of sand, with a big area near the center of dots maxed out with three grains of sand. You’d imagine wrong. Here’s what you get:”
This is one of several such pictures in Ellenberg’s article. He explains: “These images were generated by Wes Pegden, a math professor at Carnegie Mellon whose work with Lionel Levine and Charlie Smart […] of Cornell stands at the leading edge of sandpile studies. Pegden has interactive pictures of the billion-grain sandpiles on his website. There, you can zoom in and wander to your heart’s content.”
Ellenberg next tells us about the dynamics of sandpile behavior. A stable configuration turns out to have a density of about 2.125 grains/square. As he remarks, this critical threshold is “the dividing line between quiet and chaos.” The phenomenon of this critical state can be investigated on a finite grid, where a grain dispersed over the edge just disappears. Suppose we start with an empty grid, and add sand, grain by grain, at a square in the center. “For a while, the pattern of sand expands, looking a lot like Pegden’s pictures above […] . But once the sand gets to the rim, the story changes. The pile approaches an equilibrium, where sand drops off the edge at the same rate you add sand to the center, and the density holds steady at the critical value. Of course, there may be local fluctuations, denser and less dense patches that come and go as the system evolves; but on averge over the whole table, the number of grains per dot will hover around 2.125.”
Ellenberg links to a “hypnotic movie” of a sandpile at its critical state, which he credits to R. M. Dimeo at NIST. He remarks: “This looks like a living process to me. And that’s no coincidence. The notion of 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.”
[You may enjoy working out some sandpile moves with pencil and paper, for example
or an avalanche (both on a potentially infinite grid)
on a 5$\times$5 table.]