This is the way the paper crumples
While working on his doctoral thesis at Harvard University over the last few years, Omer Gottesman spent a lot of time at his desk crumpling sheets of paper, especially when he was stuck. He would crumple a sheet, uncrumple it, stare into its depths, and think, “There must be something that would make all this mess look a little less messy.”
Crumple, uncrumple, crumple. Sheet after sheet landed in the recycling bin, each one blank but for its chaotically creased geography. In time, a semblance of order emerged.
Crumpled wads of paper are no doubt as old and commonplace as paper itself — “graves for failed theories,” Gottesman, a physicist, has called them. But for him, the crumpled paper itself was the research.
The dynamics of crumpling are in play everywhere: in the initial unfolding of an insect’s wing, in the way DNA packs into a cell nucleus, in the challenge of how best to cram a giant solar sail into a small satellite so that it unfurls successfully. Scientists, in turn, devote considerable energy to deciphering, and trying to reduce, this complexity and disorder. Paper is an ideal model.
“Despite the apparent ease with which sheets of paper are crumpled and tossed away, crumpling dynamics are often considered a paradigm of complexity,” Gottesman noted in a research paper published recently in the journal Communications Physics.
“One of the key assumptions physicists make is that there are some universal properties that are shared between many disordered complicated systems,” he said recently. “Studying one complicated system could teach us a lot about other systems as well.”
An unfolding history
British conceptual artist Martin Creed once said, “I feel like you can have a microcosm of the world in a work.” He achieved as much with his 1995 creation, Work No. 88, a single piece of A4 paper crumpled into a ball. Within that crinkled sphere — like a tectonically wonky planet seen from afar — Creed wrangled complexity and chaos, deformation and disorder.
As did, that same year, two French physicists at the École Normale Supérieure in Paris, Martine Ben Amar and Yves Pomeau, with a three-page journal article, “Papier froissé.” In it they introduced the atom of paper crumpling, the d-cone: the tip of the cone that forms when you place a piece of paper over a cup and press into it with a pencil. (A crumpled ball is a collection of d-cones connected by ridges.)
“Papier froissé” was followed by a 26-page English version, “Crumpled paper.” The authors concluded by wondering if the same topological ideas about curvature might apply to general relativity.
So began an early chapter in the physics and mathematics of crumpled paper. And now the latest advancement arrives with Gottesman’s recent contribution, “A state variable for crumpled thin sheets,” which proposed that crumpling dynamics may not be hopelessly complex after all.
“The surprising thing about the result is that it’s very, very simple,” said Shmuel M. Rubinstein, a physicist and the study’s principal investigator, although he emphasized that Gottesman did most of the work. “What Omer showed is that perhaps the most important aspect of a phenomenon that’s considered to be really chaotic — a paradigm of disorder and complexity and uncertainty, like the butterfly-flapping-its-wings metaphor — is remarkably predictable, deterministic and simple.”
How to ‘kvetch’ paper
The methodology was straightforward, anyway. In the lab, Gottesman crushed hundreds of sheets of paper in a cylindrical container. This was scientific paper, elastoplastic Mylar sheets, which were less likely to inflict paper cuts, or to wilt into a tissue when subjected to repeated crumpling. Some early trial runs, posted on his website as “fun paper stuff,” involved “kvetching” a vertical tube of paper with an empty coffee can. (“Kvetch,” a Yiddish word that usually means “complain” but translates literally as “squeeze” or “press,” became a term of endearment around the lab, as stacks of paper complained about their fate.)
Like a palm reader intuiting a life line, Gottesman analyzed the creases of the crumpled paper and sought to tease out a variable, an equation, a law — something that predicted what would occur with the next crumple.
As Gottesman crumpled, he scanned each sheet into his computer, and then, with an algorithm, he measured the sum total of all the creases. He found that if he crumpled two separate sheets, each sheet would, as expected, accumulate damage in a unique way. But the total crease lengths of the two sheets stayed remarkably similar. Length seemed to be a deterministic variable, a so-called state variable, predicting how the network of creases would evolve.
“The detailed history of the crumpling dynamics is written into the intricate pattern of creases,” Gottesman and his co-authors wrote. “No two crumpled sheets are identical.”
And yet the paper is effectively devoid of memory. At each state of crumple, the intricate crease patterns, and the events that led to them, are irrelevant. All that is needed to predict the paper’s next state is the total length of creases in the current one. “You just care about the current state,” Gottesman said.
A single-state philosophy
The news that crumpled paper obeys a state variable — or a crease law, or a damage law — has been received in the academy with wonderment and delight, since, as the authors noted, it represented “a remarkable reduction in complexity.”
For Rubinstein, the damage law is inspiring. It suggests that other complex phenomena might reveal themselves in a comparable way — “systems that are more mysterious, where you can’t so easily see the scarring and the breakage,” he said. For instance, why do different proteins fold so reliably into similar shapes, and under what conditions do they fail to fold?
With paper, he said, “we’re doing something very arbitrary. We’re crumpling, flattening, crumpling, flattening. Basically, we are just cycling the system.” Many systems, including the human body, work the same way, he said.
“We’re looking at how damage and defects are accumulating, and that is a big question in materials science and in engineering. When will something break? How will it break? These are the most uncertain statistical things in nature. We are helpless against them. But at least for the crumpled paper, it seems like nature is transcending this uncertainty.”