Column: Don’t dimension it

Editor’s note: This is the first in a series of columns by Physics Today‘s Search & Discovery editor, Johanna Miller. In each “Miller’s Diary” column she will reflect on something from the latest issue of the magazine, the editorial process behind the scenes, or life in general.

For the January 2019 issue of Physics Today, I wrote about the experimental construction of a surface-electron system in the shape of a Sierpinski triangle . Like other fractals, the Sierpinski triangle can be mathematically characterized as having neither one dimension nor two, but something in between.

The story was a homecoming for me. My first published paper, the product of a Research Experiences for Undergraduates program the summer after my freshman year in college, was on fractals and their non-integer dimensions. Rather than examining a fractal like the Sierpinski triangle, made up of several smaller copies of itself, my research partner and I considered the case of several interrelated fractals, composed of smaller copies of one another. You can read our paper and admire the journal editors’ patience with our attempts to be funny.

I’d long assumed I would be a mathematician. Mathematics was awash in beautiful ideas, and to the extent that I could understand them, those ideas were all at my command. I wasn’t quite sure what some of them were good for, but that didn’t bother me too much.

Eventually I ended up in physics, and dimensional effects in the Ising model, also mentioned in my Physics Today story, were an important part of how I got there. In a chemistry class I took during my sophomore year, we did a two-part laboratory. First we played with a Metropolis Monte Carlo simulation of the two-dimensional Ising model, watching the pixels flicker between spin up and spin down and eventually congregate—or not—into domains of long-range order. Then we measured the temperature-dependent heat capacity of ammonium chloride and saw that it underwent a phase transition described by the same math. (Each NH₄⁺ ion has two possible orientations in its cubic crystal site, and neighboring ions interact just like magnetic spins do.) This was a revelation: The same spin-flickering process we were watching on the computer screen was actually happening inside the little calorimeter.

We hadn’t had to program the simulation ourselves, but we’d been given the equations that described what it was doing. Because I was an overconfident 19-year-old, and because nobody had told me I couldn’t, I thought I’d see if I could solve them analytically. I chugged through the math and came up with an expression for some quantity—perhaps the expected domain size—as a function of temperature. But something was wrong: My solution showed no sign of a phase transition, and I knew that in two dimensions there should be one. It turned out I’d made an unwarranted assumption and had solved the 1D version of the model, which lacks a phase transition.

As I learned more about the physical sciences, I took many long walks around campus, marveling at how the concepts I was learning about in class were really happening in the world all around me. The light from that streetlamp was a stream of photons shooting out in all directions. This patch of grass was green because molecules were absorbing some photons and reflecting others. The atoms in that iron railing were aligned into miniature magnetic domains.

So it was with both delight and nostalgia that I learned of, and wrote about, the surface-electron Sierpinski triangle. It neatly unites beautiful mathematics with observations that can be made in the physical world, and that’s what physics is all about. String theory, you know, notwithstanding.