Tony’s Take December 2023

This month’s topics:

Progress in game-of-life studies.

The “Game of Life” was invented by the late mathematical genius John Conway around 1970, and brought to general attention in Martin Gardner’s “Mathematical Games” column in Scientific American, October 1970. Mathematically speaking, it is a cellular automaton, but it doesn’t hurt to think of it as a game. Working on a large grid, you mark a few squares in black. These are the live cells; the others are dead. That is Step 1. For Step 2, live cells are created or destroyed according to the following rules. (A cell’s neighbors are the eight closest cells).

  • A live cell with 0 or 1 live neighbors dies.
  • A live cell with 2 or 3 live neighbors persists.
  • A live cell with 4 or more live neighbors dies.
  • A dead cell with exactly 3 live neighbors comes to life.

Applying the rules to the new configuration leads to Step 3, etc. You can experiment at

The Game of Life proved to be a mathematically fascinating universe, with many remarkable inhabitants, starting with the glider, the Herschel and the Gosper glider gun. Some are periodic, returning to their original shape after a certain number of steps. For example, three squares in a row has period 2. The question arose, can any positive integer occur as the period of a game-of-life configuration? That is, for any integer $n$, is there a shape in the Game of Life that returns to itself after precisely $n$ steps? Over the years it was shown that every period was possible except perhaps 19 and 41 and, as Matthew Sparkes reports in New Scientist (December 25, 2023), those two last numbers have just been accounted for (“Conway’s Game of Life is Omniperiodic”, posted on ArXiV, December 5, 2023).

The new period-19 configuration. The team named it “cribbage.” Image by Tony Phillips.

Ominous shakeup for WVU mathematics.

The University of West Virginia is closing down its master’s and doctoral programs in mathematics, according to “An ‘Academic Transformation’ Takes on the Math Department,” by Oliver Whong, writing in the November 28, 2023 New Yorker. Whong’s story starts in December 2020, with WVU president Elwood Gordon Gee proclaiming that “the perceived value of higher education has diminished” and that WVU should “focus on market-driven majors, create areas of excellence, and be highly relevant to our students and their families.”

“Gee has long argued that land-grant universities, which were created in 1862 by an act of Congress, are meant to ‘prioritize their activities based on the needs of the communities they were designed to serve,’ as he puts it in the book ‘Land-Grant Universities for the Future,‘” writes Whong. The nation’s 105 public and 7 private land-grant institutions include Cornell, MIT, Wisconsin, the University of California … and WVU.

Last September, it was decided that the master’s and PhD programs in mathematics at WVU would be discontinued and a third of the department’s faculty positions eliminated. When Whong asked Gee about the driving force behind the changes, he was told that “it came down to assuring the public, which finances the school, that the university was serving the public’s needs.” A more specific diagnosis came from Maryanne Reed, the WVU provost. The overwhelming majority of math enrollments were in service courses (5000 versus 65 math majors and 23 grad students). Some of those service courses had many low grades: D’s and F’s, along with withdrawals. In an email, Reed told Whong that higher grades “drive first-year student retention and are a primary factor in students’ ability to complete their degrees in a timely fashion. The key point here is that we need to focus on what our students and their future employers want and need.”

Whong reports conversations with several WVU math faculty members, including John Goldwasser, who thinks that students’ mathematical talent can be awakened by a high-quality college program. “I’ve had students in my honors math classes who could be good students anywhere. And many of them didn’t know what their potential was before coming to my class,” Goldwasser told Whong. Goldwasser also mentions the essential belief, shared by many of us in the profession, that “a public university should help to create and shape values, not just reflect the things the majority of people already care about.”

Magnetic skyrmions and hopfions.

Two kinds of nanoscale physical structures, magnetic skyrmions and hopfions, are attracting attention right now because of their possible application to data storage. Both structures exist inside magnets. We will focus on Hopfion rings in a cubic chiral magnet, contributed to Nature (November 22, 2023) by a team at Forschungszentrum Jülich (Fenshan Zheng, Nikolai Kiselev, Filipp Rubakov and collaborators), which reports the first realization of hopfions inside a single magnetic crystal. Getting an idea of what these experiments mean, and how they relate to topology, requires some background. Since skyrmions are simpler to describe, and are necessary ingredients and companions of hopfions, we’ll start there.

Inside a magnet, there is a magnetization vector at each point in space. A skyrmion is a localized 2-dimensional pattern of orientations of the magnetization vector; it is a kink in the magnetization vector field.

single skyrmion
A schematic image of a skyrmion. In a small disc (typically around 50 nanometers in diameter) the otherwise uniformly pointing upward (red)
magnetization vector field is replaced with one that rotates clockwise along radii until it points down (blue) at the center. Image adapted from Figure 1 in Koshibae, W. and Nagaosa, N., Scientific Reports 10 20303 (2020). Used under CC by 4.0 DEED license.

In this illustration the magnetization vector field points straight up everywhere, except within a disk. If you trace along the diameter of this disk, the magnetic field vector will gradually rotate clockwise, pointing straight down at the center and pointing up again when you get to the opposite edge. This kink cannot be removed by a continuous deformation –in physical terms, it would cost a lot of energy– and this stability is a topological phenomenon. To understand that, first imagine mapping the disc containing the skyrmion to a sphere about the origin in 3-space: each point of the disc maps to the vector representing the field at that point. Then following the vectors around concentric circles in the disc shows that each circle maps to a circle of latitude on the sphere.

The topology of a skyrmion. a. The magnetization field along one diameter in the skyrmion image above, using the same color-coding. b. The result of mapping the skyrmion to the sphere. Image by Tony Phillips.

You can imagine the map as stretching the disk over the sphere, with the center of the disk at the South Pole and the periphery at the North Pole. Topology teaches us that, holding the periphery fixed, this map cannot be continuously deformed so as not to cover the sphere. (In topological notation, it represents a non-trivial element of $\pi_2S^2 = Z$, the second homotopy group of the 2-sphere). The skyrmion is trapped in the field, just as a knot cannot be removed from a string if the ends are held fixed.

In a 3-dimensional magnetic solid, skyrmions usually appear aligned together in skyrmion strings.

skyrmion string
A schematic image of a skyrmion string, adapted (cropped) from a figure by Mari Ishida in the open access article by Koshibae and Nagaosa cited above and used under CC by 4.0 DEED license.

Now for hopfions: a skyrmion string can be bent around to join itself and form a loop. If it is twisted one or more times in the process, the resulting 3-dimensional structure is a hopfion.

hopfion and skyrmion string
A schematic image of a skyrmion string with another one twisted once and bent around to form a degree-1 hopfion. The color (coding different from that used in the Koshibae-Nagaosa images) depends on the direction of the magnetization vector field. Hopfions can only occur surrounding one or more skyrmion strings in this way. Image courtesy of Nikolai Kiselev.

For the hopfion illustrated above it is necessary to use a skyrmion string more complicated than the one described by Koshibae and Nagaosa. In particular the magnetization vector field, instead of being constant along the boundary cylinder, rotates as the base point goes around a circumference. (The colors in this image encode the direction of the magnetic vector field at those points; note that the direction-color correspondence is completely different from the one used in the Koshibae-Nagaosa article). The same color-coding is used on the sides of the three skyrmion strings enclosed by a hopfion, and on the surface of the hopfion itself, in the left-hand image below.

In this case the piece of skyrmion string forming the hopfion has been given one complete twist. As a consequence, the colored stripes on the surface corresponding to different directions of the magnetic field are topologically linked, like consecutive links on an anchor chain. That makes the configuration stable under small deformations and guarantees the permanence of the hopfion structure. (The proof is by modern algebraic topology; the magnetization-vector map on the whole solid torus represents a nontrivial element of another homotopy group of $S^2$, this one 3-dimensional.)

hopfion images
Two images from the open access article by Zheng, Kiselev, Rubakov et al. mentioned at the start of this item. a. Schematic picture of three skyrmion strings surrounded by a hopfion. Image excerpted from Fig. 1. b. Cross-section, obtained by Lorentz transmission electron microscopy, of a similar configuration. This image, magnifying an area of 450 $\times$ 450 square nanometers, is excerpted from Fig. 2. Used under CC by 4.0 DEED license.