When you crumple up your gift-wrapping paper this year, you'll create a shape so complex that it has defeated the most sophisticated computers

Who’d have thought paper could be so mysterious? (Image: Ballyscanlon/Getty)

WHEN you throw out your Christmas wrapping paper this year, don’t tell Narayanan Menon and Anne Dominique Cambou. You’ll be throwing away examples of their painstaking research.

That’s because they study the physics of crumpled balls of paper, which contain deeper mysteries than you might expect. Take a sheet of A4, scrunch it up and throw it at a colleague. You’ll notice that even though paper is flimsy, it becomes sturdier in the form of a ball.

How can a sheet of paper become an unaccountably tough projectile simply by the act of crushing? The answer might seem simple, but it turned out that finding a sound explanation required complex instruments and a lot of brain power. Now, though, Cambou and Menon, physicists at the University of Massachusetts in Amherst, have come up with some unexpected answers.


There is something of a niche research field in paper folding. One of its original defining experiments was testing the assertion that it is only possible to fold a sheet of paper seven times. This was shown to be false on the Discovery Channel programme Mythbusters (episode 72, first aired in 2007). The actual number turned out to be 11, though getting to that required a steamroller and a very thin grade of paper akin to parachute material. The dimensions of the folding material were also on a huge scale – the size of a football pitch. In the context of the office, however, the seven folds assertion stands, unbusted.

Another aspect of paper folding is that it is highly unpredictable. If you place a sheet of paper over a coffee cup and poke it down into a cone, it can fold in myriad ways. Researchers were eventually able to mathematically predict how a given sheet of paper would fold in this situation (Nature, vol 401, p 46).

One property of crumpled paper remained, though, resisting all forms of analysis. No matter how tightly you crumple paper into a ball, you’ll be hard-pressed to come up with a structure composed of less than about 90 per cent air. “It’s technically possible to compress them further,” says Cambou, “but that will take a lot more force because the crumpled sheet increasingly opposes the external force as it’s crushed.” Menon and Cambou wanted to know why.

Despite their insubstantial constitution, wadded paper balls are capable of feats of considerable strength. They are the ultimate packing material, for instance, able to support and cushion objects far heavier than themselves. That’s unexpected, given their lack of internal buttressing. A house, by contrast, has supporting structures such as beams built into the architecture to explain why it is so rigid. “This is not stiffness you have designed into the ball,” says Menon. “You’ve just crushed it.”

Considering that lack of uniform structure, a ball’s stiffness is also surprisingly consistent throughout, even though no two are likely to have the same configuration of folds inside. Each crumpled ball may even be unique, though researchers have not yet examined them in sufficient numbers to determine whether they can be compared on the lines of snowflakes, fingerprints and dust particles (see “A library that’s deliberately gathering dust”). Furthermore, numbers aren’t the only barrier to understanding paper balls.

Each crumpled ball may even be unique on the lines of snowflakes, fingerprints and dust particles

It’s a wrap

Despite technological advances, it is still extremely difficult to peer inside a simple scrunched-up paper ball with any detail. Computer science hasn’t been much help. It has been impossible to pinpoint the physics involved because even the most sophisticated hardware and software fail when trying to recreate the sheer complexity involved. There are simply too many variables.

Neither is it possible to make a paper ball and then reverse engineer the structure from reading the patterned wrinkles in the unfurled paper. Various groups have analysed such patterns and been driven to frustration.

“You can ask very simple questions that have surprisingly complex answers,” says Menon. He had hoped some kind of 3D imager would do the trick. For example, an X-ray tomography machine – a piece of kit normally used to hunt for tumours or to look inside delicate artefacts of archaeological digs – bounces X-rays off the internal surfaces of an object to create thousands of 2D cross-sections that can be reassembled into a 3D image.

There was just one tricky problem, which is that X-rays sail right through paper. Menon and Cambou realised that they could get what they wanted with a different material that comes in sheets: aluminium foil. Their plan worked, and they created the world’s first image of the internal geometry of a crumpled-up sheet.

The image yielded answers immediately. The first thing the researchers noticed were the ridges throughout the insides of the ball. They are the paper’s strongest points, and what fortifies them is a quality that you might not expect from paper.

It might rip easily, but it is very robust in one particular way, says Tom Witten, a solid-state physicist at the University of Chicago. To demonstrate, he picks up a flat piece of paper and tries to stretch it until it rips. It is really difficult to do.

This is tensile strength. It also manifests when you fold paper and thereby impart tension at the resulting crease. Like these creases, the ridges inside the paper ball hold the energy you imparted by folding the paper. They are also the reason that a paper ball cannot be compressed beyond around 90 per cent air without superhuman effort. Through a combination of the rigidity of the ridges and the energy they concentrate, they prop up the structure as well as any deliberate design.

This wasn’t an entirely unexpected finding, but the second thing the 3D images revealed was. You might imagine ridges would arrange themselves randomly inside the balls, but the reality is very different. Weirdly, the interior of the ball was a series of orderly layers.

The strengthening effects of these layers recalls the difficulty of folding paper more than seven times. The first fold is easy, but by the fifth time you find yourself exerting a fair degree of force. In fact, the force required is proportional to the cube of the number of sheets. “So if you have five things stacked, you’ve increased the strength 125 times,” says Menon. Crumpled paper follows a similar power law. The layers inside act like folds: not only is the paper ball harder to deform, the layers trap air, which according to Cambou, could be one more variable that adds to the structure’s strength. As a result, the multilayered walls acted as structural pillars (Proceedings of the National Academy of Sciences, vol 108, p 14741).

The wrinkly balls did not yield all of their mysteries. It remains unclear, for example, why paper balls make such a good packing material; after all, by themselves, neither air nor sheets of paper provide anything like adequate cushioning. Menon thinks the key is their ability to absorb the vibrations that pass through them.

Do the balls absorb vibrations by trapping pressure waves or by dissipating them? Nobody knows, says Menon, “but it means there are still plenty of beautiful problems to keep me interested”.