This Is the Way the Paper Crumples

In a ball of paper, scientists discover a landscape of surprising mathematical order.

The New York Times, November 26, 2018
Jens Mortensen

While working on his doctoral thesis at Harvard 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’d 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,” Mr. 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,” Mr. Gottesman noted in a research paper published earlier this month 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

The 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 Mr. 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 Mr. 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, Mr. 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, Mr. 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.

He toyed with a number of variables: the range of individual crease lengths; the distances between creases; the largest patch without creases; the sharpness of creases, and the amount of energy needed to cause crumpling.

From one crumple to the next, he observed that a piece of paper never stopped forming new creases, although the rate of their formation slowed logarithmically. With each new crumple, the paper creased along some of its existing scars, but there always came a point when new creases were needed for crumpling to continue.

“In the lab, I crumpled as many as 70 times,” Mr. Gottesman said. “Usually after four or five times you can’t really easily see a difference between one crumple to the next.”

He did, however, notice a trend involving the cumulative total length of all creases on a sheet after each crumple. In the lab, this variable was called “the mileage.”

As Mr. 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,” Mr. 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,” Mr. 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.”

“The idea that crumpling can be characterized by as simple a quantity as the length of creases is extremely interesting,” said Dr. Ben Amar. “It will be marvelous if this experimental work is continued by a theoretical treatment” — that is, with equations derived from the laws of elasticity.

Thomas Witten, a physicist at the University of Chicago, found the result striking and puzzling. “It could prove very exciting,” he said. Witten’s research has shown that every material crumples in roughly the same way — a tectonic plate, a cell membrane, the graphene in a Buckyball, the fabric of Mona Lisa’s right sleeve as depicted by Leonardo da Vinci, which, for Dr. Witten, folds in a way reminiscent of the human ear.

For Dr. 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 it will 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.”

The demise of paper has been predicted for some time now, but paper, it seems, may not be done with us. Paper defies history, in more ways than one. Its scars and creases may even hold a potential philosophy.

“It’s a good metaphor,” Dr. Rubinstein said. “Because it’s really hard to imagine that the future depends only on the current situation. We are such a strong function of our history — personal history, global history, whatnot. It is really counterintuitive to say, Well, our future really only depends on our current state of mind, and not how it developed, how we got to this point.

“We have the same intuition for any complex system: that its evolution is something very strongly dependent on many points in time, many degrees of freedom, and you have to know so many of them. Finding out that you don’t is a pretty big deal.”