Postdoctoral fellow Sho Yaida has helped solve a mystery about the physical nature of glass. But as interesting as the breakthrough is, perhaps equally interesting is Yaida’s choice of tools—the same ones used by Newton and Darwin. For research recently published in *Physical Review* *Letters*, Yaida spent a month at his desk calculating, using nothing more technological than printer paper and free pens from the supply cabinet.

Yaida studies physics in the chemistry lab of Patrick Charbonneau, where among the things they analyze is what physicists and chemists call “the glass problem”: how glass works. “Surprisingly, even though we know how to make them,” says Yaida, “we really don’t understand the physics of glasses.” Unlike crystals, whose ordered molecular structure makes them easy to understand, molecules in glasses (ordinary windows and tumblers and stuff) are disordered, so trying to understand and predict their behavior makes physicists go bonkers. Though at common temperatures glass is obviously solid enough to keep your cabernet from running all over the table, glass is not exactly either a liquid or a crystalline solid. It’s called an amorphous solid, which is what makes it so perplexing.

Or perplexing in three-dimensional reality. If you take glass into a place of limitless dimensions, you can write equations that work just great. In infinite-dimensional spaces, some glass makes a transition from normal to marginally stable glasses with exotic properties. The goal has been to translate those equations from unlimited spatial dimensions into our quotidian three. “Because if it only happens in infinite dimensions,” Yaida asks pragmatically, “why would you care?”

Yaida has a background in particle physics, and he got to thinking. Particle physics uses “renormalization- group flow calculations,” which he calls “a way to determine how the thing you observe can survive” as you make your way into, in this case, fewer and fewer dimensions, with each limitation making the equation more complex. The contact point between infinite dimensional space and physical reality is called a “fixed point” of the renormalization-group flow, and Yaida got Charbonneau’s permission to take a month and try to find it.

Thirty pages of handwritten calculations later, with cross-outs and corrections and diagrams and symbols and all the stuff that math has, he did. Actually, the pages he spreads onto a work table number more like a hundred. “In writing a paper, you tend to do more wrong calculations than correct ones,” Yaida says. “This process of doing wrong calculations is pretty important. Doing the wrong calculations is kind of organizing your mind and getting to the right calculations. In this case, it was long.”

Yaida smiles. “That’s life.”

One advantage of handwritten calculations is that Yaida can quickly scan columns of equations and graphs to see whether he’s made mistakes. “Also I try not to blindly trust people,” he says, starting his calculations from scratch. “Other people could have made mistakes.” The same applies to computer calculations. Building from pieces of other source code may accumulate errors. “It’s very important not to have any black boxes in your scientific investigation, whether in calculation or in computer simulation.” Even so, his approach is not an attack on computers. He once would have said computers should not be used at all, but writing code helps organize his thoughts, as it requires breaking things down into pieces simple enough for a computer to understand. And running equations through a computer is a good check as well. “That’s if you code it yourself,” he emphasizes. Unchecked equations are just as dangerous as unchecked blocks of code.

Finally, Yaida says, “Most people prefer to do it by hand—it’s faster that way.” Which led to that month with pen and paper. “The calculation was not inventing new techniques. I was just carefully cranking the knob. You don’t want to go to the computer each time.” And though handwritten calculation is charming, Yaida does not romanticize it.

“The important point is not that the calculations took thirty pages,” Yaida says. “The point is to not be scared of calculations like this.”

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